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1. INTEGRAL CALCULUS
Two mathematicians in a restaurant were
arguing about the mathematical knowledge of the public. The cynic
said: I will bet you the cost of this dinner that the waitress can't
answer a simple math question. The cynic excused himself to visit the
men's room. The
other called the waitress over and said: Here is $10. When I ask you a
question, say one third x cubed. She agreed.
The cynic returned, called the waitress
over, and said his friend had a question. The question asked
was: What is the integral
of x squared? After fidgeting and squirming a long time, she said:
One third x cubed. The cynic
paid the check. The waitress walked away, and muttered under her breath:
Plus a constant..
- Person 1: What's the integral of 1/cabind cabin?
- Person 2: A log cabin.
- Person 1: No, a houseboat. You forgot to add the C!
2. LOGARITHMS
God asked Noah to build an ark and later asked Noah to load the ark with a
male and a female of each species. After a flood occurred and the
water subsided, God said the ark could be unloaded. After this was
done, Noah went into the hold to see if the ark was empty and found hundreds
of snakes on a wooden table. Noah exclaimed "but I only placed two of
you here 40 days ago:. Mrs. Snake said " we are Adders and when Adders
are placed on a log table they multiply".
3. ANALYTICAL GEOMETRY
There once was a horse named DAY. Who liked numbers better than hay.
He could add and subtract, multiply and
divide. His mathematical ability filled his master
with pride.
But, when given a book, on geometry -- ANALYT.
Gave a loud whinny, and in his teeth took the
bit.
Galloped away, faster than lightning. His master wondered, what caused the
frightening.
But, we know the answer, of course.
No one should put DECARTES before DAY
horse.
4. ALGEBRA
A math professor arrived at his classroom 15
minutes early. He looked in his classroom and did not see any student so
he waited in the corridor. A few minutes later, the door to the
classroom opened and one student came out. The professor thought: "If I enter the room
it will now be empty".
A physicist, a biologist and a mathematician are sitting in a street cafe
watching people entering and leaving the house on the other side of the
street. First they see two people entering the house. Time passes. After a
while they notice three people leaving the house. The physicist says, "The
measurement wasn't accurate." The biologist says, "They must have
reproduced." The mathematician says, "If one more person enters the house
then it will be empty."
5. GEOMETRY
An Indian chief had three daughters who were given permission to marry three
braves, who had to provide tepees for their brides. The first
brave placed a deerskin on the floor of the tepee he built, the second used
a buffalo hide, and the third (who was a world traveler) used the hide of a
hippopotamus. Nine months after the three marriages had occurred, the
first wife had a baby boy, the second wife had a baby boy, and the third
wife had twin boys. This was predicted by Euclid: the sons of the
squaws on the two hides must equal the sons of the squaw on the
hippopotamus.
What is the ratio of the
Circumference of a Jack O Lantern to its Diameter? Pumpkin Pi, of course. In
the artic, the
ratio of the distance around an igloo to the distance across, is Eskimo Pi. However,
Eskimo
Pi is only 3.00, as everything shrinks in the cold.
Pi goes on and on, and e is just as cursed. I wonder which is larger,
when their digits are reversed?
6. BASE 10 & BASE 8
Mathematicians confuse
Halloween & Christmas. 25 Dec = 31Oct That is, 25 in base 10, equals 31
in base 8.
But, in the Octal System of
Base 8 12x12 =144 just like in the Decimal System of Base 10. The
number 144 in the Octal System is really the number 100 in the Decimal
System.
100 in the Decimal System
is 144 in the Octal system and in base 6 it is 244. 12 in the Decimal
system is 14 in the Octal System and 20 in Base 6. So in Base 6, 12x12 =
400.
7. INFINITY Richard Phillips Feynman (1918-1988) Did
you know there are twice as many numbers as numbers?
8. NEGATIVE & IMAGINARY
NUMBERS
Messages to leave on your telephone answering machine
are:
If you received a negative response,
hang up, rotate your phone 180 degrees, and try again.
If you think you have reached an imaginary number,
rather than a real number, hang up, rotate your phone ninety degrees, and
try again.
Here is an Argand diagram.
An Argand diagram is a plot of
complex numbers as
points (z = x + iy) in the
complex plane using
the
x-axis
as the
real axis and
y-axis
as the
imaginary axis.
While Argand (1806) is generally credited with the discovery, the Argand
diagram (also known as the Argand plane) was actually described by C. Wessel
prior to Argand. Historically, the geometric representation of a
complex number as a
point in the plane was important because it made the whole idea of a complex
number more acceptable. In particular, this visualization helped "imaginary"
and "complex" numbers become accepted in mainstream mathematics as a natural
extension to negative numbers along the
real line. The x axis
represents "real numbers" and the y axis represents "imaginary numbers", so
0 is both real and imaginary.
9. HUMOR
One day an actuarial student came into my
office and asked if I would tell him how I became Chief Actuary.
I replied: "Good judgment". The next day he asked how I got good
judgment. I replied: "Good experience." The next
day he returned and asked how I got good experience. I replied: "Bad
judgment."
A patient was told by her doctor she only had six months to live. She asked
the doctor what she should do. He said marry an actuary. She
wanted to know if that would make her live longer. The doctor said:
no, but it will seem much longer.
The difference between God and an actuary is that God does not think he is
an actuary.
An American actuary can tell you the number of people in a room will die
within one year. An Italian actuary can give you their names.
A CEO, an
actuary, an accountant, and an insurance salesperson are riding in a car.
The CEO has his hands on the steering wheel, salesperson has his foot on the
gas, the underwriter has his foot on the brake, and the actuary is looking
out the back window telling them where to go.
An actuary is a
place where they bury dead actors.
An actuary is a professional who can solve a
problem you didn't know you had in a way that you can't understand. An
actuary is someone who wanted to be an accountant but did not have enough
personality for the job.
Actuaries never die, they just get broken
down by age and sex.
Johnny, whose father was an actuary, asked
his mother where he came from. His mother said: ask your father.
Johnny replied: I don't want to know that much about it.
A lady asked the postmaster to weigh a
package. When told it was too heavy and needed one more stamp, she said: I don't
understand. Will adding a stamp make it lighter?
An infinite crowd of mathematicians enters a bar.
The first one orders a pint, the second one a half pint, the third one a
quarter pint...
"I understand", says the bartender - and pours two pints.
A mathematician is flying non-stop from Edmonton to Frankfurt. The
scheduled flying time is nine hours. Later, the pilot announces that one
engine had failed.
"Don't worry - we're safe. The only effect
is that our total flying time will be ten hours instead of nine." A
few hours later, the pilot informs the passengers that another
engine had failed. "But don't worry -
we're still safe. Our total flying time will go up to twelve hours." Some
time later, a third engine fails. The pilot
reassures the passengers: "Don't worry - even with one engine, we're
still perfectly safe. It just means that it will take sixteen hours total
for this plane to arrive in Frankfurt." The mathematician remarks to
his fellow passengers: "If the last engine breaks down, too, then we'll be
in the air for twenty-four hours altogether!"
Teacher: What is 2k + k?
Student: 3000!
Teacher: "Who can tell me what 7 times 6 is?"
Student: "It's 42!"
Teacher: "Very good! - And who can tell me what 6 times 7 is?"
Same student: "It's 24!"
Teacher: Expand (a+b)n
Student: (a + b)n
(a + b)n
(a +
b)n (a
+ b)n
Q: What did the zero say to the the eight?
A: Nice belt.
Q: What do you get if you add two apples and three apples? A: A high school math problem!
Q: What is the difference between a Ph.D. in mathematics and a large
pizza? A: A large pizza can feed a family of
four.
Q: Why did the chicken cross the Moebius strip? A: To get to the other ... er, um ...
A topologist is a person who can't tell the the
difference between a
coffee cup and a
doughnut.
An introverted actuary looks at his shoes while talking to you.
An extroverted actuary looks at your shoes.
There are three kinds of actuaries in the world: "Those who can
count and those who can't."
1 + 1 =3, for sufficiently large one's.
A circle is a round straight line with a hole in the middle.
Ernst Eduard Kummer (1810-1893), a German algebraist, was sometimes slow
at calculations.. Whenever he had occasion to do simple arithmetic in class,
he would get his students to help him. Once he had to find 7 x 9. "Seven
times nine," he began, "Seven times nine is er -- ah --- ah -- seven times
nine is. . . ." "Sixty-one," a student suggested. Kummer wrote 61 on the
board. "Sir," said another student, "it should be sixty-nine." "Come, come,
gentlemen, it can't be both," Kummer exclaimed. "It must be one or the
other."
One cat has nine tails.
Proof. No cat has eight
tails. Since one cat has one more tail than no cat, it must have nine tails.
An investment firm is hiring mathematicians. There are three applicants:
a theoretical mathematician, an applied mathematician, and an actuary.
They are asked
what starting salary they are expecting.
The theoretical mathematician: "Would $30,000 be too much?"
The applied mathematician: "I think $60,000 would be OK."
The actuary: "$100,000." The personnel officer gasped: "A pure mathematician will do the work for
only $30,000."
The actuary replied: "I will keep $35,000, pay you $35,000, and pay $30,000
to the theoretical mathematician to do the work."
A CEO is interviewing three candidates for the position of CFO.
Each candidate is asked the same question: "How much is one and one?"
The accountant said: "Two". The pure mathematician said: "It depends
upon what base you are working in". The actuary whispered in the CEO's
ear: "What do you want it to be?"
A firm is hiring mathematicians. Three recent graduates are interviewed:
one has a degree in pure mathematics, one in applied math, and the third one
obtained his degree. in statistics.
All three are asked the same question: "What is one third plus two thirds?"
The pure mathematician: "It's one."
The applied mathematician uses his pocket calculator and replies: "It's
0.999999999."
The statistician: "What do you want it to be?"
George W Bush warned math professors not to misuse their position to
give their political views to young Americans. "It is my understanding", the
president said, "that you are frequently teaching algebra classes in which
your students learn how to solve equations with the help of radicals. I
can't say that I approve of that..."
Donald Rumsfeld gave George W Bush a briefing and
concluded by saying: Yesterday, 3 Brazilian soldiers were killed. Oh no, the
President exclaimed, That's terrible! Exactly how many are in a
brazillion?
Four friends have been doing really well in their calculus class. When
it's time for the final, they decide to go to a weekend party in another
city. They finally arrive on campus, hung over and sleepy, but the
exam is already over.
They go to the professor's office and say: "We went to our friend's birthday
party, and when driving back this morning we had a flat tire. We had
no spare one, and since we were driving on back roads, it took hours until
we got help." The professor nods sympathetically and says: "I see that it
was not your fault. I will allow you to make up the exam tomorrow morning."
When they arrive next morning, the students seated so far apart from each
other they had no chance to cheat. The exam booklets are already in place
and the students begin. The first question -
five points out of one hundred - is a simple exercise in integration, and
all four finish it within ten minutes. When the first of them has
completed the problem, he turns over the page of the exam booklet and reads:
Problem 2 (95 points out of 100): Which tire went flat?
If brute force
doesn't
work, you're not using enough of it.
Mathematicians never die - they only loose some of their functions.
A father who is very much concerned about his son's bad grades in math
decides to register him at a catholic school. After his first term there,
the son brings home his report card: He's getting "A"s in math.
The father is, of course, pleased, but wants to know: "Why are your math
grades suddenly so good?"
"You know", the son explains, "when I walked into the classroom the first
day, and I saw that guy on the wall nailed to a plus sign, I knew one thing:
This place means business!"
A mother is expecting her fourth child.
One evening, the eldest daughter says to her dad: "Do you know, daddy, what
I've found out?"
"No."
"The new baby will be Chinese!"
"What?!"
"Yes. I've read in the paper that statistics shows that every fourth child
born nowadays is Chinese...
In a class, a math professor claims that he can prove everything under
the assumption that 1+1=1.
A student challenges him: "Then prove that you're the pope!"
The professor replies: "I am one, and the pope is one. Therefore, the pope
and I are one."
After two hours, the professor called for the exams, and the
students filed up and handed them in. All except the late student, who
continued writing. Twenty minutes later, the last student came up to the
professor who was sitting at his desk preparing for his next class. He
attempted to put his exam on the stack of exam booklets already there. No
you
don't, I'm not going to accept that.
It's late. The student
looked incredulous and angry. Do you know who I am? No, as a matter of fact I don't, and I
don't care! replied the professor. Good, replied the student, who quickly
lifted the stack of completed exams, stuffed his in the middle, and walked
out of the room.
The math teacher saw that little Johnny wasn't paying
attention in class. She called on him and said, Johnny! What are 2 and 4 and
28 and 44? Little Johnny quickly replied: NBC, CBS, HBO, and the
Cartoon network!
An impatient math teacher snarled, And just how far
are you from the correct answer? The boy replied,
Three seats.
At a nursery school a teacher talks to a four-year-old
applicant.
"Mike, can you count for me?"
Mike counts very fast and with a lot of enthusiasm, "Fifty-nine,
fifty-eight, fifty-seven"
"Super," says the teacher, "But how did you learn to count backwards?"
Mike replies proudly, "I can heat my own lunch in the microwave."
Teacher: What are
whole numbers?
Student: 0, 6, 8, 9.
Teacher: And what about 10?
Student: It is half-whole, 1
doesn't have a hole.
Dad, will you do my math for me tonight? No, son, it
wouldn't be right. Well, you could try.
A little girl asked an elderly woman: "Can you help me find
the lowest common denominator?"
The woman answered: "Haven't they
found that yet? They were looking for it when I was in school."
Father, to his daughter returning home at 3 a.m. said: "I
told you to be home by a quarter of 12!" The daughter answered: "But I
learned in math that a quarter of 12 is 3!"
What type of math is often discussed at the beach? ...intergull calculus!
Where can you buy a ruler that is 3 feet long?
At a yard sale.
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Math Types |
Sign Errors |
When you have a 50-50 chance of getting something right,
there's a 90% probability you'll get it wrong.
The probability of an open faced jelly sandwich falling
jelly side up varies inversely with the price of the rug.
A military instructor says, "There is a 40% chance that we
will hit our target."
One student asks, "What happens if we aim away from the target?"
The instructor replies, "We would have a 60% chance of hitting
the target."
"Excuse me Professor, how can we possibly compute a kurtosis in a
minute?" The Professor looks at the class very reassuring: "No need to
be worried, kids, it only takes a moment"
A team of statisticians has recently published a monumental finding. The
team discovered what the leading cause of divorce is marriage.
Everyone who has been divorced has been married first.
A statistician who took a Dale Carnegie course and improved his
confidence from .95 to .99.
Statisticians never have to say they're certain.
Statistics is the art of never having to say you're wrong.
"There are lies, damned lies, and statistics."
97.3% of all statistics are made up.
62% of people have a below average IQ.
This is due to the fact that there is a limit to human intelligence, but no
limit to human stupidity.
A statistician would always accelerate hard before coming to an
intersection, whizz straight through it, then slow down again. One day a
passenger asked him why he went so fast through intersections. The
statistician replied: "Statistically speaking, you are far more likely to
have an accident at an intersection, so I just make sure that I spend less
time there."
Two blondes are sitting in an airport hall waiting for their flight to
go. One has terrible flight panic. Her companion, a mathematician, said:
"Hey, don't worry, it's just every 100,000th flight that crashes." The
companion replied: " So much? Then it surely will be mine!" The
mathematician said: "Well, there is an easy way out. Simply take the next
plane. It's much more probable that you go from a crashing to a non-crashing
plane than the other way round."
Never show a bar chart at an AA meeting.
People believe what economists say about the future, but not what
statisticians say about the past.
Statisticians are "mean" people.
Numbers are like people; torture them enough and they'll tell you anything.
Statistics in the hands of politicians are like a lamppost to a drunk ---
used more for support than illumination.
When you smoke a fish, which end do you light?
Birthdays are beneficial for your health. A new statistical study unequivocally proved that the more birthdays one has the
longer one lives.
Medicine makes people ill, mathematics make them sad and
theology makes them sinful. (Martin Luther)
Salesman: "Ma'am, this vacuum cleaner will cut your work in
half."
Customer: "Terrific! Give me two of them."
If a man tries to fail and succeeds, which did he do?
A mathematician, a physicist, and an engineer are asked to test the
following hypothesis: All odd numbers greater than one are prime.
The mathematician: "Three is a prime, five is a prime, seven is a prime, but
nine is not a prime. Therefore, the hypothesis is false."
The physicist: "Three is a prime, five is a prime, seven is a prime, nine is
not a prime, eleven is a prime, and thirteen is a prime. Hence, five out of
six experiments support the hypothesis. It must be true."
Five actuaries and five accountants are traveling together by train to
attend a conference. Each accountant has a ticket, but only one of the
actuaries has one. Suddenly one of the actuaries shouts:: "Conductor
coming!" All the actuaries run into one washroom. The conductor checks
the ticket of each accountant and then knocks on the washroom door: "Your
ticket, please." An actuary sticks the one ticket under the door. The
conductor checks it and leaves. The accountants are impressed. On the return
trip, the accountants decide to buy just one ticket for their group. The
actuaries do not purchase any tickets. Again one of the actuaries
shouts: "Conductor coming!". This time all the accountants rush off to a
washroom. One of the actuaries goes
to that washroom, knocks at the door, and says: "Your ticket, please..."
A professor goes through the airplane security check, and a bomb is found
in his carry-on-baggage. The security man asked: "Why do you want to blow up
this airplane?" "Sorry", the professor interrupts him. "I had never
intended to blow up the plane. Statistics shows that the probability of one
bomb being on an airplane is 1/1000. The chance that there are two
bombs is 1/1000000. If I already bring one, the chance of another bomb being
around is actually 1/1000000, and I am much safer..."
Albert Einstein was walking across the quad at Princeton and met a
student. After chatting for a few minutes, Einstein asked: "Which
direction was I coming from?" The student pointed at one path.
Einstein said:
"Then I must have had lunch."
Albert Einstein had just about finished his work on the theory of special
relativity, when he decided to take a break and go on vacation in Acapulco.
Each day, late in the afternoon, sporting dark sunglasses, he walked in the
white Mexican sand and breathed in the fresh Pacific sea air. On the last
day, he watched the sun set. When the large orange ball was just
disappearing, the last beam of light seemed to radiate toward him. The event
brought him back to thinking about his physics work. "What symbol should I
use for the speed of light?" he asked himself. The problem was that nearly
every Greek letter had been taken for some other purpose. Just then, a
beautiful Mexican woman passed by. He asked as he lowered his dark
sunglasses, "Do you not zink zat zee speed of light is very fast?" The woman
smiled at Einstein and replied, "Si." And know you know the rest
of the story.
A student riding in a train looks up and sees Einstein sitting next to
him. Excited he asks:
"Excuse me, professor. Does Boston stop at this train?"
A six-year-old boy spotted Albert Einstein walking down the street and
decided to try out his favorite joke on him: "Mr. Einstein! Why did the
chicken cross the road?" To which the famous physicist replied:
"My young friend, zee question does not have a definite anzer. Vether zee
chicken crossed zee road or zee road crossed zee chicken depends on your
frame of reference."
A little boy refused to run anymore. When his mother asked him why, he
replied:
"I heard that the faster you go, the shorter you become."
There was an old lady called Wright
who could travel much faster than light.
She departed one day
in a relative way
and returned on the previous night.
The Heineken Uncertainty Principle says "You can never be sure how many
beers you had last night."
There is a sign in Munich that says, "Heisenberg might have slept here."
What is so special about 6.9? It is 69 ruined by a
period.
Statistics show that teenage pregnancies drop off significantly after age
25.
If nothing can go wrong, something will. There's never time to do it right, but always time to do it over. The man who can smile when things go wrong, has thought of someone he can blame it on.
An Amish boy and his father went to a large
department store for the first time. They saw an elevator, but did not
know what it was. An old lady with a cane, hobbled up to the elevator,
and pushed a button on the wall, and a door opened. She went into a
small room and the door closed. Then lights flashed above the doorway
showing numbers. The numbers stopped flashing at number five.
After a little while the numbers started flashing again, and stopped
at number one. The door opened and out walked a beautiful blonde. The
father shouted to his son: Go get your mother!
Tom Lehrer was a mathematician, songwriter and
satirist.
Lehrer
stated that: "Base 8 is just like Base 10, if you're missing 2
Fingers."
See
http://en.wikipedia.org/wiki/Tom_Lehrer
10. INTERESTING NUMBERS The interesting
number paradox is a paradox that arises from attempting to classify
natural numbers as "interesting" or "dull". The paradox states that all
natural numbers are interesting. The "proof" is by contradiction.
If there were uninteresting numbers, there would be a smallest
uninteresting number But the smallest uninteresting number is itself
interesting by being so, producing a contradiction.
1729 is also known as the Hardy-Ramanujan
number. Godfrey Hardy, an English mathematician was visiting Srinivasa
Ramanujan in a hospital when Ramanujan told him about this number.
Here are Hardy's words: I remember
once going to see him when he was ill at Putney. I had ridden in taxi cab
number 1729 and remarked that the number seemed to me rather a dull one, and
that I hoped it was not an unfavorable omen. "No," he replied, "it is a very
interesting number; it is the smallest number expressible as the sum of two
cubes in two different ways." Notice that 1729 = 1^3 + 12^3 =
9^3 + 10^3.
111,111,111 x 111,111,111 = 12345678987654321
2 raised to the 5th power times 9 raised to the 2nd power = 2592
"A man is a person who will pay two
dollars for a one-dollar item he wants. A woman will pay one dollar
for a two-dollar item she doesn't want..." -- William Binger
Three The
only three consecutive integers whose cubes sum to a cube are given by the
Diophantine equation:
3 cubed + 4 cubed + 5 cubed = 6 cubed.
"Cubes": 153, 370, 371 and 407 are all the "sum of
the cubes of their digits". In other words 153=13+53+33
4 One half of five is four,
as one half of five is iv. Write any number from 1 to
100 using four 4's: (See
http://mathforum.org/ruth/four4s.puzzle.html)
6
The problem of finding two rational numbers whose cubes sum to
six was "proved" impossible by Legendre. However, Dudeney found the simple
solutions 17/21 and 37/21. Did you know that snowflakes have six sides?
9
Dudeney found two rational numbers other than 1 and 2 whose
cubes sum to nine:
[415280564497 / 348671682660] and
[676702467503 / 348671682660]
Nine is the maximum number of cubes that are needed to sum to any
positive integer. Nineteen is the maximum number of fourth
powers needed to sum to any positive integer.
10 10! =6! x 7! These are the only
consecutive integers, 6 and 7, that solve the equation N! = A! x (A+1)!
11 British mathematician J J Sylvester said:
"Mathematics is the music of reason." In 1884 at age 70, he proved
that the highest number that cannot be created from using two numbers x
and y equals xy - x - 7. In Rugby, where drop goals are
scored as 3 and converted tries are scored as 7, there cannot ever be a
score of 11.
That is 3x7 - 3 - 7 = 11.
13 The next number in the sequence 345 is 1. The next number in
the sequence 543 is 1. The next number in the sequence 222 is 7.
The next number in the sequence 123 is 7. The next number in the
sequence 333 is 4. These are all bridge hand distributions.
19 Every single positive integer can be written as the sum of
at most 19 powers.
22, 23, and 24 are the only positive integers
(other than 1) for which n! has precisely n digits. .
23 Its digits are consecutive prime numbers. It is
the smallest odd prime that is not a twin prime. 23 is the smallest prime
for which the sum of the squares of its digits is also a prime.
There were 23 problems on
David Hilbert's
famous
list of unsolved mathematical problems, Beckham chose the number 23 on his shirt to play for Real
Madrid. Michael Jordan wore the number 23. We have 23 pairs of
chromosomes. Caesar was stabbed 23 times. 23mph is the maximum
speed of an American crow.
25 is a Friedman Number: A positive integer which can be
written in some non-trivial way using its own digits, together with the
symbols + - x / ^ ( ) and concatenation. See:
http://www2.stetson.edu/~efriedma/mathmagic/0800.html
Here, 25 = 52
Forty is the only number whose letters are in
alphabetical order.
55 is the largest two digit number used in the
NBA. This makes it easy for referees to communicate a player's number
using hand signals using the fingers on each hand. Each digit is 0 thru 5.
88 The number of keys on a piano, 52 white
keys and 36 black keys. . There are 7 white keys and 5 black keys to an
octave. 88 The number that is called "two fat
ladies" in Bingo. 88 The number of feet per second, when
driving 60 miles per hour.
93 You can chop a
big lump of cheese into a maximum of 93 bits with 8 straight cuts
128 A cord of wood is 4 feet by 4 feet by 8
feet or 128 cubic feet. A tennis tournament with 2n
players will have n rounds. A Grand slam
Singles Tournament has 128 entrants and 7 rounds. 128 is the largest number
that cannot be expressed as the sum of three distinct squares. Like 25, 128
is another Friedman number: 128 = 28-1
129 can be expressed as the sum of three distinct squares in
two different ways. 129 = 100 + 25 +4. Also 129 = 64 + 49 + 16.
144 is the largest Fibonacci square.
37 (666)/(6 + 6 + 6) = 111/3
3 x 37 = 111
6 x 37 = 222
9 x 37 = 333
12 x 37 = 444
15 x 37 = 555
18 x 37 = 666
21 x 37 = 777
24 x 37 = 888
27 x 37 = 999 |
1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111=12345678987654321 |
1 x 9 + 2 = 11
12 x 9 + 3 = 111
123 x 9 + 4 = 1111
1234 x 9 + 5 = 11111
12345 x 9 + 6 = 111111
123456 x 9 + 7 = 1111111
1234567 x 9 + 8 = 11111111
12345678 x 9 + 9 = 111111111
123456789 x 9 +10= 1111111111 |
2519 Mod 2 = 1
2519 Mod 3 = 2
2519 Mod 4 = 3
2519 Mod 5 = 4
2519 Mod 6 = 5
2519 Mod 7 = 6
2519 Mod 8 = 7
2519 Mod 9 = 8
2519 Mod 10 = 9 |
0 x 9 + 8 = 8
9 x 9 + 7 = 88
98 x 9 + 6 = 888
987 x 9 + 5 = 8888
9876 x 9 + 4 = 88888
98765 x 9 + 3 = 888888
987654 x 9 + 2 = 8888888
9876543 x 9 + 1 = 88888888
98765432 x 9 + 0 = 888888888
987654321 x 9 - 1 = 8888888888
9876543210 x 9 - 2 = 88888888888 |
1 x 8 + 1 = 9
12 x 8 + 2 = 98
123 x 8 + 3 = 987
1234 x 8 + 4 = 9876
12345 x 8 + 5 = 98765
123456 x 8 + 6 = 987654
1234567 x 8 + 7 = 9876543
12345678 x 8 + 8 = 98765432
123456789 x 8 + 9 = 987654321 |
|
|
|
142857 x 2 = 285714
142857 x 3 = 428571
142857 x 4 = 571428
142857 x 5 = 714285
142857 x 6 = 857142 |
|
91
|
times |
1 |
= |
0 |
9 |
1 |
7x7=49 |
|
91 |
times |
2 |
= |
1 |
8 |
2 |
67x67=4489 |
|
91 |
times |
3 |
= |
2 |
7 |
3 |
667x667=44889 |
|
91 |
times |
4 |
= |
3 |
6 |
4 |
6667x6667=44448889 |
|
91 |
times |
5 |
= |
4 |
5 |
5 |
66667x66667=4444488889 |
|
91 |
times |
6 |
= |
5 |
4 |
6 |
666667x666667=444444888889 |
|
91 |
times |
7 |
= |
6 |
3 |
7 |
6666667x6666667=44444448888889 |
|
91 |
times |
8 |
= |
7 |
2 |
8 |
etc. |
|
91 |
times |
9 |
= |
8 |
1 |
9 |
|
| 1x9+2 = 11 |
9 x 9 + 7 = 88 |
9 x 9 = 81 |
6 x 7 = 42 |
| 12x9+3 = 111 |
98 x 9 + 6 = 888 |
99 x 99 = 9801 |
66 x 67 = 4422 |
| 123x9+4 = 1111 |
987 x 9 + 5 = 8888 |
999 x 999 =
998001 |
666 x 667 = 444222 |
| 1234x9+5 =
11111 |
9876 x 9 + 4 = 88888 |
9999 x 9999 =
99980001 |
6666 x 6667 = 44442222 |
| 12345x9+6 =
111111 |
98765 x 9 + 3 = 888888 |
etc |
etc |
| 123456x9+7 =
1111111 |
987654 x 9 +2 =
8888888 |
|
|
| 1234567x9+ =
11111111 |
9876543x9+1= 88888888 |
|
|
| 12345678x9+9=111111111 |
98765432x9 = 888888888 |
|
|
| 1x1=1 |
4x4=16 |
|
11x11=121 |
34x34=1156 |
|
111x111=12321 |
334x334=111556 |
|
1111x1111=1234321 |
3334x3334=11115556 |
|
11111x11111=123454321 |
33334x33334=1111155556 |
|
111111x111111=12345654321 |
etc. |
|
1111111x1111111=1234567654321 |
|
|
11111111x11111111=123456787654321 |
|
|
111111111x111111111=12345678987654321 |
|
Santa's Options: Assuming Rudolph was in front, there are 40320
ways to arrange the other eight reindeer.
Roman Numerals The original Roman year had 10 named months
Martius "March", Aprilis "April", Maius "May", Junius
"June", Quintilis "July", Sextilis "August", September
"September", October "October", November "November",
December "December", and probably two unnamed months in the dead of
winter when not much happened in agriculture. The year began with Martius
"March". Numa Pompilius, the second king of Rome circa 700 BC, added the two
months Januarius "January" and Februarius "February". He also
moved the beginning of the year from Marius to Januarius and
changed the number of days in several months to be odd, a lucky number.
After Februarius there was occasionally an additional month of
Intercalaris "intercalendar". This is the origin of the leap-year day
being in February. In 46 BC, Julius Caesar reformed the Roman calendar
(hence the Julian calendar) changing the number of days in many months and
removing Intercalaris.
The numerals are: 1=I (unus); 5=V (quinque); 10=X (decem); C=100 (centum),
and M=1000 (mille). They also used 50=L (quinquaginta); and
500=D (quingenti).
When a small number comes before a larger
number, the smaller number is subtracted. 4 = IV or 5-1. When a smaller
number follows a larger one, the two are added together: 7 = VII, or 5 + 2
and 19 = XIX, or 10 + 9. Although the Romans used a decimal
system for whole numbers, they used a duodecimal system for fractions
because the divisibility by 12 made it easier to handle the common fractions
of 1/3 and 1/4.
Chinese Multiplication
See
http://www.youtube.com/watch?v=8iIU9EDC2GQ and
http://www.youtube.com/watch?v=maRN2fUOF0o
11. NUMBER THEORY
Sums of Natural Numbers
The sum of the first n natural numbers is: 1 + 2 + 3 + 4 ....... + n
= n(n+1)/2 Gauss developed this formula when in primary school:
the average of the first number and the last number times the number of
numbers!
If n = 6: 1 + 2 + 3 + 4 + 5 + 6 = 21 = (6x7)/2
Sums of Even Numbers The sum of the first k even natural
numbers is: 2 + 4 + 6 ..... + 2k = k(k+1).
If k = 3: 2 + 4 + 6 = 12 = (3x4)
Sums of Odd Numbers The sum of the first k odd natural
numbers is: 1 + 3 + 5 ..... + (2k - 1) = k2
If k = 3: 1 + 3 + 5 = 9
= (3x3)
-
Sums of Squares The sum of the squares of
the first n natural numbers is:
30 = (12 + 22 + 32 + 42) =
1 + 4 + 9 + 16
Also see:
http://www.takayaiwamoto.com/Sums_and_Series/sumsqr_1.html and
http://www.math.utah.edu/~palais/sums.html
365 = ( 102 + 112 + 122) = (132
+ 142)
Sums of Cubes The sum of the cubes of the first n natural numbers is:
-
As you can see, the sum of the cubes of the first n natural numbers
is equal to the square of the sum of the first n natural numbers.
Sum of 1 - 2 + 3 - 4 +
5 - 6 ... =
= 1/4
http://en.wikipedia.org/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%C2%B7_%C2%B7_%C2%B7
Triangular
Numbers
The numbers which can be arranged in a compact triangular pattern are
termed as triangular numbers. The triangular numbers are formed by partial
sum of the series 1+2+3+4+5+6+7......+n. So
T1 = 1
T2 = 1 + 2 = 3
T3 = 1 + 2 + 3 = 6
T4 = 1 + 2 + 3 + 4 = 10
So the nth triangular number can be obtained as Tn
= n(n+1)/2, where n is any natural number. In other words triangular numbers
form the series 1,3,6,10,15,21,28.....
n2 = the sum of two consecutive triangular numbers,
because Tn + Tn-1 = n(n+1)/2 +
(n-1)(n)/2 = n2
Cubics, Quartics, and Quintics
Niccolo Tartaglia, who solved the cubic, failed miserably for the
rest of his life (mainly because he spent it trying to discredit Cardano).
Giralamo Cardano, who
published the solution, lived a long unhappy life. His only son was executed
for murder; he was later put on trial by the Inquisition for attempting to
cast the horoscope of Christ.
Lodovico Ferrara, who solved the general quartic, was poisoned,
probably by his sister, over an inheritance dispute.
Evariste Galois, who showed the general quintic was unsolvable,
died in a duel at the age of 29.
Niels Henrik Abel, who duplicated and extended Galois' proof
independently, finally managed to receive his first faculty position.
The notification letter arrived a few days after Abel had died of pneumonia.
He was 29.
Prime Number Spiral See what Stanislaw Ulum discovered in 1963.
http://www.numberspiral.com/ and
http://mathworld.wolfram.com/PrimeSpiral.html and
http://www.ulamspiral.com/ and
http://en.wikipedia.org/wiki/Ulam_spiral and
http://www.hermetic.ch/pns/pns.htm
Some Prime Numbers:
31, 331, 3331, 33331, 333331,
3333331, and 33333331 are primes. But 333333331 =
17x19607843
73,939,133
is also a prime number If you
keep removing a digit from the right hand end of the number, each of the
remaining numbers is also prime. It's the largest number known with this
property.
"Check Digit" in a book's
ISBN: An ISBN (International Standard Book Number) is a
controlled, 10- or 13-digit identification number allowing publishers,
libraries, and book dealers to locate books. An International Standard Book Number consists of 4 or 5 parts. For a
13 digit ISBN:
- the
GS1 prefix: 978 or 979 (indicating
the industry (978 denotes book publishing)
- the
group identifier
(language-sharing country group)
- the publisher code
- the item number (title of
the book)
- a
check digit.
The calculation of an ISBN-13 check digit
begins with the first 12 digits of the thirteen-digit ISBN (thus excluding
the check digit itself). Each digit, from left to right, is alternately
multiplied by 1 or 3, then those products are summed
modulus 10 to give a value ranging from 0 to 9. Subtracted from 10, that
leaves a result from 1 to 10. A zero (0) replaces a ten (10), so, in all
cases, a single check digit results.
For example, the ISBN-13 check digit of 978-0-306-40615-? is
calculated as follows:
s = 9x1 + 7x3 + 8x1 + 0x3 + 3x1 + 0x3 + 6x1 + 4x3 + 0x1 + 6x3 + 1x1 + 5x3
= 9 + 21 + 8 + 0 + 3 + 0 + 6 + 12 + 0 + 18 + 1 + 15
= 93 93 / 10 = 9 remainder 3 Then 10 - 3 = 7
Thus, the check digit is 7, and the complete sequence is ISBN 978-0-306-40615-7.
The ISBN-13 check digit calculation is:
-
This check system does not catch all errors of adjacent digit
transposition. If the difference between two adjacent digits is 5, the check
digit will not catch their transposition. For instance, the above example
allows this situation with the 6 followed by a 1. The correct order
contributes 3x6 +1x1 = 19 to the sum; while, if the digits are transposed (1
followed by a 6), the contribution of those two digits will be 3x1+1x6 = 9.
However, 19 and 9 are congruent modulo 10, and so produce the same check
digit. The ISBN-10 formula uses the
prime modulus 11 avoids this blind
spot, but requires more than the digits 0-9 to express the check digit. A
10 digit ISBN develops a check digit computed so that multiplying
each digit by its position in the number (counting from the right) and
taking the sum of these products
modulus 11 is 0. The furthest digit to the right (which is multiplied by
1) is the check digit, chosen to make the sum
correct. It may need to have the value 10, which is represented as the
letter X. For example, take the
ISBN 0-201-53082-1. The sum of products is 0x10 + 2x9 + 0x8 + 1x7 + 5x6
+ 3x5 + 0x4 + 8x3 + 2x2 + 1x1 = 99 ≡ 0 modulo 11. So the ISBN is valid.
While this may seem more complicated than the first scheme, it can be
validated very simply by adding all the products together then dividing by
11. More info on frequency of errors is at
http://www.augustana.ab.ca/~jmohr/algorithms/checkdigit.html
Brocard's Problem: N factorial + 1 = X squared. This is
true for X = 5, 11, and 71, but that may be all.
Pierre Rene Jean Baptiste Henri Brocard: (1845 - 1922)
Primes, other than 2 or 3 are either of the form 6n + 1 or 6n - 1.
Lychrel Numbers. Most numbers become a palindrome by reversing their
digits and adding repeatedly. (349 + 943 = 1292, 1292 + 2921 = 4213,
4213 + 3124 = 7337 a palindrome. Those that do not convert, are Lychrel Numbers. The name "Lychrel" was coined by Wade Van Landingham: a
rough anagram of his girlfriend's name Cheryl.
Catalan's conjecture (occasionally now referred to as
Mihăilescu's theorem) was conjectured by the mathematician Eugene
Charles Catalan in 1844 and proven in 2002 by Preda
Mihăilescu.
To understand the conjecture, notice that 23 and 32
(i.e. 8 and 9) are two
powers of
natural numbers,
whose values 8 and 9 respectively are consecutive. The conjecture states
that this is the only case of two consecutive powers.
That is to say, that the only
solution in the natural numbers
of
- xa − yb = 1
for x, a, y, b > 1 is x = 3, a
= 2, y = 2, b = 3.
12. SQUARE ROOT The square root of the number 81 is 9. 81 is the only
number whose square root is the sum of its digits.
What did Pythagoras say when he was first confronted with the square root of
2? "There has to be a rational explanation for this."
13. LIMITS The limit, as n goes to infinity, of {sin x}/n
is 6. Divide numerator & denominator by n and you get six.
Not true.
14. SYMMETRY and ITERATION See
rmmsmsp.ucdenver.edu/instructormaterial/geometry-daisies.pps
15.
BIG NUMBERS See
http://www.guardian.co.uk/world/2009/mar/25/trillion-dollar-rescue-plan
16. MATH TERMS
See
http://www.cut-the-knot.org/glossary/stop.shtml
17. MUSICAL DOW In 1987, there was an article in
the Hartford Courant about Myron Schwager, a cellist, who
teaches at the Hartt School of Music, part of the University of Hartford.
He has a stock market charting system based on scales, and I wrote him a
letter. See correspondence.
A month or so later, as I was a
senior officer, Mary and I were invited to a black tie ITT/Hartford Board
dinner at the Hartford Club. A small group
played music during the dinner. I recognized the cellist from his
picture in the Hartford Courant article. After dinner I went up to
Myron Schwager and asked him what instrument he was playing. He said
"It's a cello." I asked if I could look at it and he
handed it to me. I held it up to my ear and exclaimed: "Its playing
the Dow!" He said: "You must be Don Sondergeld."
18. MATHEMATICIANS and Others (see
http://aleph0.clarku.edu/~djoyce/mathhist/chronology.html )
Pythagoras of Samos (c. 570-c. 495 BC):
(Greek)
He was an
Ionian
Greek
philosopher,
mathematician, and
founder of the religious movement called
Pythagoreanism. Most
of the information about Pythagoras was written down centuries after he
lived, so that very little reliable information is known about him.
Around 530 BC, he moved to
Croton, a
Greek colony in
southern Italy, and
there set up a religious sect. His followers pursued the religious rites and
practices developed by Pythagoras, and studied his philosophical theories.
The society took an active role in the politics of Croton, but this
eventually led to their downfall. The Pythagorean meeting-places were
burned, and Pythagoras was forced to flee the city. He is said to have ended
his days in
Metapontum. He is
often revered as a great
mathematician,
mystic and
scientist, and he is
best known for the
Pythagorean theorem
which bears his name.
Euclid of Alexandria (300 BC- )
(Greek) He was often referred to as the
"Father of Geometry." His "Elements" is one of the most
influential works in mathematics, serving as the main textbook for teaching
mathematics, especially geometry, from the time of its publication until the
late 19th or early 20th century.
Archimedes of Syracuse (c.287-c.212 B.C): (from
Sicily) A mathematician and inventor. He determined the exact value of
pi, is also known for his strategic role in ancient war and the
development of military techniques.
"Give me a
place to stand and I will move the earth" was his boast when he
discovered the laws of levers and pulleys. His mechanical inventions
defeated the Roman fleet of Marcellus.
The word "eureka" comes from the story that when Archimedes figured out a
way to determine whether the king (Hiero II of Syracuse), a possible
relative, had been duped by measuring the buoyancy of the king's supposedly
solid gold crown in water, he became very excited and exclaimed the Greek
(Archimedes' native language) for "I have found it": Eureka.
Archimedes
requested that his tombstone be decorated with a sphere contained in
the smallest possible cylinder and inscribed with the ratio of the
cylinder's volume to that of the sphere. Archimedes considered the discovery
of this ratio the greatest of all his accomplishments.
Diophantus of Alexandria
(200 and 214 --
284 and 298): (Greek) Sometimes called
"the father of
algebra", was an
Alexandrian
Greek mathematician
and the author of a series of books called
Arithmetica.
These texts deal with solving
algebraic equations,
many of which are now lost. In studying Arithmetica.
Pierre de Fermat
concluded that a certain equation considered by Diophantus had no solutions,
and noted without elaboration that he had found "a truly marvelous proof of
this proposition," now referred to as
Fermat's Last Theorem.
This led to tremendous advances in
number theory, and
the study of
Diophantine equations
("Diophantine geometry") and of
Diophantine approximations
remain important areas of mathematical research. Diophantus was the first
Greek mathematician
who recognized fractions as numbers; thus he allowed
positive
rational numbers for
the coefficients and solutions. In modern use, Diophantine equations are
usually algebraic equations with
integer coefficients,
for which integer solutions are sought. Diophantus also made advances in
mathematical notation.
Leonardo Pisano Fibonacci (1170?-1250):
(Italian) Fibonacci is considered to be one of the most talented
mathematicians for the Middle Ages. Few people realize that it was Fibonacci
that gave us our decimal number system (Hindu-Arabic numbering system) which
replaced the Roman Numeral system. When he was studying mathematics, he used
the Hindu-Arabic (0-9) symbols instead of Roman symbols which didn't have
0's and lacked place value. In fact, when using the Roman Numeral system, an
abacus was usually required. There is no doubt that Fibonacci saw the
superiority of using Hindu-Arabic system over the Roman Numerals. He shows
how to use our current numbering system in his book Liber abaci. And
he gave us the Fibonacci Series. Fibonacci was known as Leonardo of
Pisa. He was born in Pisa, home of the famous leaning tower and his statue
is located there.
In his famous "Rabbit Problem" he produces the Fibonacci Series as the answer:
1 1 2 3 5 8 13 21 34 55 etc., where each term is equal to the sum of the two
previous terms. The Fibonacci sequence obeys the recursion relation F(n) = F(n-1) + F(n-2). The ratio of the current term to the previous term
approaches the golden ratio or (1 + sq rt of 5)/2, about 1.618... This
ratio is called the "golden ratio". The German Adolph Zeising claimed
the front of the Parthenon is in proportion to the golden ratio. There is no
documentary evidence that Phidias, used the golden ratio in any of his work
related to the Parthenon.
However around 1909, the American mathematician Mark Barr, named the golden
ratio the Greek letter "phi" for Phidias.. When phi is
expressed as a continued fraction it looks like this:
Continued fractions provide mathematicians with a way of rating how
irrational a number might be. Since the expression for phi contains only 1s,
it is the purest continued fraction that there is, and hence is considered
the most irrational number.
Leonardo Da Vinci
called the golden ratio the "divine proportion" and featured it in
many of his
paintings.
Nicolaus Copernicus (1473-1543):
(Prussia) He was a
Renaissance
astronomer and the
first person to formulate a comprehensive
heliocentric
cosmology, which
displaced the
Earth from the center
of the
universe.
Copernicus' epochal book,
De revolutionibus orbium coelestium
(On the Revolutions of the Celestial Spheres), published just before
his death in 1543, is often regarded as the starting point of modern
astronomy and the
defining
epiphany that began
the
scientific revolution.
His
heliocentric model,
with the Sun at the center of the universe, demonstrated that the observed
motions of celestial objects can be explained without putting Earth at rest
in the center of the universe. His work stimulated further scientific
investigations, becoming a
landmark in the
history of science
that is often referred to as the
Copernican Revolution.
Gerolamo
Cardano (1501-1576): (French) He
was an
Italian
Renaissance
mathematician,
physician,
astrologer and
gambler. Today, he is
best known for his achievements in
algebra. He published
the solutions to the
cubic and
quartic equations in
his 1545 book
Ars Magna. The
solution to one particular case of the cubic, x3 + ax =
b (in modern notation), was communicated to him by
Niccolo Fontana Tartaglia
(who later claimed that Cardano had sworn not to reveal it, and engaged
Cardano in a decade-long fight), and the quartic was solved by Cardano's
student
Lodovico Ferrari.
Both were acknowledged in the foreword of the book, as well as in several
places within its body. In his exposition, he acknowledged the existence of
what are now called
imaginary numbers,
although he did not understand their properties (Mathematical field theory
was developed centuries later). In Opus novum de proportionibus he
introduced the
binomial coefficients
and the
binomial theorem.
Cardano was notoriously short of money and kept himself solvent by being an
accomplished gambler and
chess player. His
book about games of chance, Liber de ludo aleae ("Book on Games of
Chance") , written in 1526, but not published until 1663, contains the first
systematic treatment of
probability, as well
as a section on effective cheating methods. Cardano invented several
mechanical devices including the
combination lock, the
gimbal consisting of
three concentric rings allowing a supported
compass or
gyroscope to rotate
freely, and the
Cardan shaft with
universal joints,
which allows the transmission of rotary motion at various angles and is used
in vehicles to this day. He studied
hypocycloids,
published in de proportionibus 1570. The generating circles of these
hypocycloids were later named Cardano circles or cardanic circles and were
used for the construction of the first high-speed
printing presses.
Franciscus Vieta (1540-1603):
(French)
His work on
new algebra was an
important step towards modern algebra, due to its innovative use of letters
as parameters in equations. He was a lawyer by trade, and served as a
privy councillor to
both
Henry III and
Henry IV.
Galileo Galilei(1564-1642):
(Italian) A
physicist,
mathematician,
astronomer and
philosopher who
played a major role in the
Scientific Revolution.
His achievements include improvements to the
telescope and
consequent astronomical observations, and support for
Copernicanism.
Galileo has been called the "father of modern observational
astronomy",
the "father of modern
physics", the "father
of
science",
and "the Father of Modern Science".
Stephen Hawking says,
"Galileo, perhaps more than any other single person, was responsible for the
birth of modern science." Read about his "square cube"
law: http://dinosaurtheory.com/scaling.html
Johannes Kepler (1571-1630):
(German)
A
mathematician,
astronomer and
astrologer, and key
figure in the 17th century
scientific revolution.
He is best known for his
eponymous
laws of planetary motion,
codified by later astronomers, based on his works
Astronomia nova,
Harmonices Mundi,
and
Epitome of Copernican Astronomy.
These works also provided one of the foundations for
Isaac Newton's theory
of
universal gravitation.
Rene Descartes (1596-1650): (French)
The inventor of Analytical Geometry. He was a philosopher,
mathematician, physicist and writer. He has been dubbed the "Father of
Modern Philosophy".
Pierre de Fermat (1601-1665):
(French) A lawyer and amateur mathematician who contributed to Number
Theory and known for "Fermat's Last Theorem". Fermat was the
first person known to have evaluated the integral of general power
functions. Using an ingenious trick, he was able to reduce this evaluation
to the sum of geometric series. The resulting formula was helpful to both
Newton and Leibnitz in developing calculus.
John Wallis (1616-1703): (English)
A mathematician
who is given partial credit for the development of
infinitesimal calculus.
Between 1643 and 1689 he served as chief
cryptographer for
Parliament and,
later, the royal court. He is also credited with introducing the
symbol ∞ for
infinity.
Wallis made significant contributions to
trigonometry,
calculus,
geometry, and the
analysis of
infinite series. In
his Opera Mathematica I (1695) Wallis introduced the term "continued
fraction". He is generally credited
as the originator of the idea of the
number line where
numbers are represented geometrically in a line with the positive numbers
increasing to the right and negative numbers to the left. In 1655,
Wallis published a treatise on
conic sections in
which they were defined analytically. This was the earliest book in which
these curves are considered and defined as curves of the second degree. It
helped to remove some of the perceived difficulty and obscurity of
Rene Descartes' work
on
analytic geometry.
Arithmetica Infinitorum, the most important of Wallis's works, was
published in 1656. In this treatise the methods of analysis of Descartes and
Cavalieri were
systematised and extended. in 1659, Wallis published a tract
containing the solution of the problems on the
cycloid which had
been proposed by
Blaise Pascal. In
this he incidentally explained how the principles laid down in his
Arithmetica Infinitorum could be used for the rectification of algebraic
curves; and gave a solution of the problem to rectify (i.e. find the length
of) the semi-cubical parabola x3 = ay2,
which had been discovered in 1657 by his pupil
William Neile. Since
all attempts to rectify the ellipse and hyperbola had been (necessarily)
ineffectual, it had been supposed that no curves could be rectified, as
indeed Descartes had definitely asserted to be the case. The
logarithmic spiral
had been rectified by
Evangelista Torricelli,
and was the first curved line (other than the circle) whose length was
determined, but the extension by Neil and Wallis to an algebraic curve was
novel. The cycloid was the next curve rectified; this was done by
Wren in 1658.
Blaise Pascal (1623-1662): (French)
He helped create two major new areas. He wrote a significant treatise on
projective geometry at the age of sixteen. Pascal's
development of probability theory was his most influential contribution to
mathematics, a subject on which he corresponded with Fermat. Pascal
continued to influence mathematics throughout his life. In 1653 he described
a convenient tabular presentation for binomial coefficients, now called
Pascal's triangle.
Sir Isaac Newton (1643-1727):
(British) His theory
of gravity unified the force that keeps our feet on the ground, with
the force that holds planets in their orbits. His 1687 publication of the
Philosophiae Naturalis Principia Mathematica is considered to be
among the most influential books in the history of science. In this
work, Newton described universal gravitation and the three laws of
motion. Newton shares the credit with Leibnitz for the development of
differential and integral calculus. He also demonstrated the generalized
binomial theorem and contributed to the study of power series.
Gottfried Wilhelm Leibnitz (1646-1716):
(German) He invented infinitesimal calculus independently of Newton,
and his notation has been in general use since then. He also invented the
binary system, the foundation of virtually all modern computer
architectures.
Bernoulli Family of Swiss Mathematicians: Three were : Jacob
Bernoulli (1654-1705),
his brother
Johann Bernoulli (1667-1748) and Johann's son
Daniel Bernoulli (1700-1787).
Jacob wrote the Art of
Conjecture. In this work, he described the known results in
probability theory and in enumeration, often providing alternative proofs of
known results. This work also includes the application of probability theory
to games of chance and his introduction of the theorem known as the law of
large numbers. The terms
Bernoulli trial and
Bernoulli numbers result from this work. He.chose a figure of a
logarithmic spiral and the motto Eadem mutata resurgo ("Changed and
yet the same, I rise again") for his gravestone. He called it the
spiral mirabilis, the wonderful spiral. The spiral executed by the
stonemasons was, however, an Archimedean spiral. Just
like a fractal, a logarithmic spiral is self similar: That is, any
smaller piece of a larger spiral is identical in shape to the larger piece.
Johann studied the
function y = xx and he also investigated
series using the method of integration by parts. Integration to Bernoulli
was simply viewed as the inverse operation to differentiation and with this
approach he had great success in integrating differential equations. He
summed series, and discovered addition theorems for trigonometric and
hyperbolic functions using the differential equations they satisfy. Johann
was known as the "Archimedes of his age" and this is indeed inscribed on his
tombstone.
Daniel was a Dutch Swiss
mathematician. He is particularly remembered for his applications of
mathematics to mechanics, especially fluid mechanics and for his pioneering
work in probability and statistics. Bernoulli's work is still studied at
length by many schools of science throughout the world. The Bernoulli
Principle that was used to explain lift applicable to airplane wings was developed by
Daniel Bernoulli.
Leonhard Euler (1707-1783): One of
his many contributions was called "Euler's Formula". The formula states that, for any real number x,
where e is the base of the natural logarithm, i is the
imaginary unit, and cos and sin are the trigonometric functions, with the
argument x given in radians. The formula is still valid if x
is a complex number. Richard Feynman called Euler's formula "our
jewel" and "one of the most remarkable, almost astounding, formulas in all
of mathematics".
Joseph-Louis Lagrange (1736-1813): (Italian)
Lagrange was one of the creators of the
calculus of variations,
deriving the
Euler Lagrange equations.
Lagrange invented the method of solving
differential equations
known as
variation of parameters,
applied
differential calculus
to the
theory of probabilities
and attained notable work on the solution of
equations. He proved
that
every natural number is a sum of four squares.
His treatise Theorie des fonctions analytiques laid some of the
foundations of
group theory,
anticipating
Galois. In
calculus, Lagrange
developed a novel approach to
interpolation and
Taylor series. He
studied the
three-body problem
for the Earth, Sun, and Moon and the movement of Jupiter's
satellites. In 1772 found the special-case solutions to this problem that
are now known as
Lagrangian points. He
transformed
Newtonian mechanics
into a branch of analysis,
Lagrangian mechanics
as it is now called. One of Lagrange's more famous books is the
Analytical Mechanics, which, he boasted proudly, contains no pictures.
Caspar Wessel
(1745-1818): (Danish-Norwegian) Wessel was a mathematician who
was born in Norway. In 1763, having completed secondary school, he went to
Denmark for further studies (Norway having no university at the time). In
1778 he acquired the degree of
candidatus juris.
From 1794, however, he was employed as a
surveyor (from 1798 as Royal inspector of
Surveying).
It was the mathematical aspect of surveying that
led him to exploring the geometrical significance of
complex numbers. His
fundamental paper, Om directionens analytiske betegning, was
published in 1799 by the
Royal Danish Academy of Sciences and Letters.
Since it was in Danish, it passed almost unnoticed, and the same results
were later independently found by
Argand and
Gauss.
One of the more prominent ideas presented in "On
the Analytical Representation of Direction"
was that of
vectors. Even though
this wasn't Wessel's main intention with the publication, he felt that a
geometrical concept of numbers, with length and direction, was needed.
Wessel's approach on addition was: "Two straight lines are added if we unite
them in such a way that the second line begins where the first one ends and
then pass a straight line from the first to the last point of the united
lines. This line is the sum of the united lines". This is the same idea as
used today when summing vectors.
Wessel's priority to the idea of a complex number
as a point in the
complex plane is
today universally recognized. His paper was re-issued in French translation
in 1899, and in English in 1999 as "On the analytic representation of
direction".
Pierre-Simon, marquis de Laplace (1749-1827):
(French) He is remembered as one of the greatest
scientists of all time, sometimes referred to as a
French
Newton or
Newton of France, with a phenomenal natural mathematical faculty
superior to any of his contemporaries. Laplace's writing
of Celestial Mechanics, an enormous, five volume tome of celestial
mechanics, established him as the Prince of Celestial Mechanicians. When
presented with a copy of some of the initial volumes, Napoleon is said to
have remarked, "I see no mention of God in this work". Laplace is said to
have replied, "Sir, I have no need of that hypothesis." (In an addition to
the story, the tale was related to Lagrange, who added "Ah, but it is such a
beautiful hypothesis; it explains a great many things!"
Jean Baptiste Joseph Fourier (1768-1830):
(French) A mathematician and physicist best known for initiating
the investigation of
Fourier series and their applications to
problems of
heat transfer and
vibrations. The
Fourier transform and
Fourier's Law are
also named in his honour. Fourier is also generally credited with the
discovery of the
greenhouse effect.
Carl Friedrich Gauss (1777-1855): (German)
Called the Prince of Mathematicians and the greatest mathematician
since antiquity. He is ranked as one of history's most influential
mathematicians. He referred to mathematics as the Queen of Sciences.
Gauss proved the Fundamental Theorem of Algebra. Gauss claimed
to have discovered the possibility of non Euclidean Geometries but never
published it.
Simeon Denis Poisson
(1781-1840): (French) A
mathematician,
geometer, and
physicist. In
probability theory
and
statistics, the
Poisson distribution (or Poisson law of small numbers) is a
discrete probability distribution
that expresses the probability of a number of events occurring in a fixed
period of time if these events occur with a known average rate and
independently of the
time since the last event. (The Poisson distribution can also be used for
the number of events in other specified intervals such as distance, area or
volume.)
Baron
Augustin-Louis Cauchy (1789-1857):
(
French) He was an
early pioneer of
analysis. He started
the project of formulating and proving the theorems of
infinitesimal calculus
in a rigorous manner. He also gave several important theorems in
complex analysis and
initiated the study of
permutation groups in
abstract algebra. A
profound mathematician, Cauchy exercised a great influence over his
contemporaries and successors. His writings cover the entire range of
mathematics and
mathematical physics.
Michael Faraday (1791-1867) and James Clerk Maxwell
(1831-1879):
They proved that electric and magnetic forces are the same force in
different guises.
Nikolai Ivanovich
Lobachevsky (Никола́й Ива́нович
Лобаче́вский) (1792-1856): (Russian)
A
mathematician and
geometer, renowned
primarily for his pioneering works on
hyperbolic geometry.
Niels Henrik Abel (1802-1829):
(Norwegian) At the age of 16, Abel gave a proof
of the
binomial theorem
valid for all numbers, extending Euler's result which had only held for
rational numbers. At age 19, he showed there is no general algebraic
solution for the roots of a quintic equation, or any general polynomial
equation of degree greater than four, in terms of explicit algebraic
operations. To do this, he invented (independently of Galois) an extremely
important branch of mathematics known as
group theory, which
is invaluable not only in many areas of mathematics, but for much of physics
as well. Among his other accomplishments, Abel wrote a monumental work on
elliptic functions which, however, was not discovered until after his death.
When asked how he developed his mathematical abilities so rapidly, he
replied "by studying the masters, not their pupils."
Janos Bolyai (1802-1860):
(Hungarian)
He was known for his work in
non-Euclidean geometry.
Between 1820 and 1823 he prepared a treatise on a complete system
of
non-Euclidean geometry.
Bolyai's work was published in 1832 as an appendix to a mathematics
textbook by his father.
Gauss, on reading the Appendix, wrote to a friend saying "I regard this
young
geometer Bolyai as a genius of the first order". In 1848 Bolyai
discovered not only that
Lobachevsky had published a similar piece of work in 1829, but also a
generalization of this theory. As far as is known, Lobachevsky published
his work a few years earlier than Bolyai, but it contained only hyperbolic
geometry. Bolyai and Lobachevsky did not know each other or each other's
works. In addition to his work in the geometry, Bolyai developed a rigorous
geometric concept of
complex numbers as
ordered pairs of
real numbers.
Although he never published more than the 24 pages of the Appendix,
he left more than 20,000 pages of mathematical manuscripts when he died.
Carl Gustav Jacob Jacobi (1804-1851)
(German) A
mathematician, widely considered to be the
most inspiring teacher of his time and one of the greatest mathematicians of
all time. One of Jacobi's greatest accomplishments was his theory of
elliptic functions.
He also made fundamental contributions in the study of differential
equations. It was in algebraic development that Jacobi's peculiar
power mainly lay, and he made important contributions of this kind to many
areas of mathematics, as shown by his long list of papers in Crelle's
Journal and elsewhere from 1826 onwards. One of his maxims was: 'Invert,
always invert' ('man muss immer umkehren'), expressing his belief that the
solution of many hard problems can be clarified by re-expressing them in
inverse form. He was also one of the early founders of the theory of
determinants.
Johann Peter Gustav Lejeune Dirichlet
(1805-1859): (German) He was credited with the modern formal
definition of a
function.
Dirichlet's brain is preserved in the anatomical collection of the
University of Gottingen, along with the brain of Gauss.
Sir William Rowan Hamilton
(1805-1865): (Irish)
A
physicist,
astronomer, and
mathematician, who
made important contributions to
classical mechanics,
optics, and
algebra. His studies
of mechanical and optical systems led him to discover new mathematical
concepts and techniques. His greatest contribution is perhaps the
reformulation of
Newtonian mechanics,
now called
Hamiltonian mechanics.
This work has proven central to the modern study of classical field theories
such as
electromagnetism, and
to the development of
quantum mechanics. In
mathematics, he is perhaps best known as the inventor of
quaternions. A
striking feature of quaternions is that the product of two
quaternions is
noncommutative,
meaning that the product of two quaternions depends on which factor is to
the left of the multiplication sign and which factor is to the right.
Hamilton defined a quaternion as the
quotient of two
directed lines in a three-dimensional space or
equivalently as the quotient of two
vectors. It can also be represented as the sum of a scalar and a vector.
http://en.wikipedia.org/wiki/Quaternion
In four-dimensioal space the tesseract, or
hype, is the
four-dimensional
analog of the
cube. The tesseract
is to the cube as the cube is to the
square. Just as the
surface of the cube consists of 6 square
faces, the
hypersurface of the tesseract consists of 8 cubical
cells. The tesseract
is one of the six
convex regular 4-polytopes.
A generalization of the cube to dimensions greater than three is called a
"hypercube",
"n-cube" or "measure
polytope". The
tesseract is the four-dimensional hypercube' See
http://en.wikipedia.org/wiki/Fourth_dimension
Joseph Liouville (1809-1882): (French) Liouville worked in a number
of different fields in mathematics, including
number theory,
complex analysis,
differential geometry and topology,
but also
mathematical physics
and even
astronomy. He is
remembered particularly for
Liouville's theorem,
a nowadays rather basic result in complex analysis. In number theory, he was
the first to prove the existence of
transcendental numbers
by a construction using
continued fractions (Liouville
numbers). In mathematical physics,
Liouville made two fundamental contributions: the
Sturm Liouville theory,
which was joint work with
Charles Francois Sturm,
and is now a standard procedure to solve certain types of
integral equations by
developing into eigenfunctions, and the fact (also known as
Liouville's theorem)
that time evolution is measure preserving for a
Hamiltonian system.
In Hamiltonian dynamics, Liouville also introduced the notion of
action-angle variables
as a description of completely
integrable systems.
The modern formulation of this is sometimes called the Liouville-Arnold
theorem, and the underlying concept of integrability is referred to as
Liouville integrability.
The
following number
is
known as Liouville's constant.
(The exponent is negative j factorial.)
Liouville's constant is a decimal fraction with 1"s
and 0"s in each decimal place. In1844 he constructed an infinite class of
transcendental numbers
using
continued fractions,
but the above number was the first decimal constant
to be proven
transcendental by
Liouville in 1850. Cantor subsequently proved that "almost all" real
numbers are in fact transcendental.
Evariste Galois (1811-1832):
A symmetry of an object is what you can do to an object to leave it
essentially looking like it did before you touched it. Galois was
interested in the collection of all symmetries and seeing what happens if
you do one symmetry after another. He discovered that it is the
interactions between the symmetries in a group that encapsulate the
essential qualities of the symmetry of an object.
Karl Theodor Wilhelm Weierstrass (1815-1897): (German) He is often cited as the "father of modern
analysis".
George Boole
(1815-1864): (English) A
mathematician and
philosopher. As the
inventor of
Boolean logic, the
basis of modern digital
computer logic, Boole
is regarded in hindsight as a founder of the field of
computer science.
Boole said: " ... no general method for the solution of questions in
the theory of probabilities can be established which does not explicitly
recognise ... those universal laws of thought which are the basis of all
reasoning".
Arthur Cayley (1821-1895): (British)
He helped found the modern British school of
pure mathematics. He
proved the
Cayley-Hamilton theorem:
that every square
matrix is a root of
its own
characteristic polynomial.
He was the first to define the concept of a
group in the modern
way: as a set with a
binary operation
satisfying certain laws. Formerly, when mathematicians spoke of "groups",
they had meant
permutation groups.
Charles Hermite (1822-1901):
(French)
He did research on
number theory,
quadratic forms,
invariant theory,
orthogonal polynomials,
elliptic functions,
and
algebra.
Hermite polynomials,
Hermite interpolation,
Hermite normal form,
Hermitian operators,
and
cubic Hermite splines
are named in his honor. One of his students was
Henri Poincare.
Hermite was the first to prove that
e, the base of
natural logarithms,
is a
transcendental number.
His methods were later used by
Ferdinand von Lindemann
to prove that
π is transcendental.
In a letter to
Thomas Stieltjes in
1893, Hermite famously remarked: "I turn with terror and horror from this
lamentable scourge of
continuous functions with no derivatives."
Bernhard Riemann (1826-1866): If the "Riemann Hypothesis"
is true, the exact number of primes less than a given number N, or Pi(N), can
be calculated exactly. Although thought to be correct,
this hypothesis is unproven. Karl Friedrich Gauss (1777-1855) had an approximation to Pi(N), equal to N/ln(N),
where ln is the natural logarithm. Adrien-Marie Legendre
(1752-1833)
improved on Gauss's estimate using Pi(N) = N/{ln(N) - 1.08366}
Gauss then improved upon that estimate using Li(N) , which he
called the logarithmic integral. (not shown here) Leonard Euler
(1707-1783)
showed that the Riemann Zeta Function: Z(s) = The
sum of 1/n raised to the s power for n = 1 to infinity, is also
equal to a product series involving primes. Z(s) = The product
of (1 + 1/p to the s + 1/p to the 2s + 1/p to the 3s + 1/p to the 4s +
1/p to the 5s +...) over all primes. It is important to note:
"s" is a "complex number". Riemann then hypothesized that Z(s) = 0 for only
complex numbers where the real part = 1/2. The Riemann Hypothesis has not been proven, but computers have shown the first 6.3 billion zeros all lie on the line s = 1/2 +ki. If the
Riemann Hypothesis is correct, then Riemann has a formula for calculating
Pi(N) exactly! Pi(N) = R(N) minus an
Adjustment. R(N) is a formula involving the logarithmic integral and
the Adjustment is expressed in terms of the zeros of the Zeta Function. The
function R(N) was named in honor of Riemann.
Julius Wilhelm Richard Dedekind
(1831-1916): (
German) He did
important work in
abstract algebra
(particularly
ring theory),
algebraic number theory
and the foundations of the
real numbers.
James Clerk Maxwell (1831-1879): (Scottish)
A physicist and mathematician.
His most prominent achievement was formulating classical
electromagnetic theory.
This united all previously unrelated observations, experiments and equations
of electricity, magnetism and even optics into a consistent theory.
Maxwell's equations
demonstrated that electricity, magnetism and even light are all
manifestations of the same phenomenon, namely the
electromagnetic field.
Subsequently, all other classic laws or equations of these disciplines
became simplified cases of Maxwell's equations. Maxwell's achievements
concerning electromagnetism have been called the "second great unification
in physics", after the first one realised by
Isaac Newton.
Maxwell demonstrated that
electric and
magnetic fields
travel through space in the form of
waves, and at the
constant
speed of light. In
1864 Maxwell wrote
A Dynamical Theory of the Electromagnetic Field.
It was with this that he first proposed that
light was in fact
undulations in the same medium that is the cause of electric and magnetic
phenomena. His work in producing a unified
model of
electromagnetism is
one of the greatest advances in physics.
Georg Ferdinand Ludwig Philipp Cantor
(1845-1918): (German)
He is best known as the inventor of
set theory, which has
become a
fundamental theory in
mathematics. Cantor established the importance of
one-to-one correspondence
between sets, defined
infinite and
well-ordered sets,
and proved that the
real numbers are
"more numerous" than the
natural numbers. In
fact,
Cantor's theorem
implies the existence of an "infinity
of infinities". He defined the
cardinal and
ordinal numbers and
their arithmetic. Cantor's work is of great philosophical interest, a fact
of which he was well aware. Cantor's theory of
transfinite numbers
was originally regarded as so counter-intuitive, even shocking, that it
encountered
resistance from
mathematical contemporaries such as
Leopold Kronecker and
Henri Poincare
and later from
Hermann Weyl and
L. E. J. Brouwer.
Thomas Alva Edison (1847-1931):
(American) An inventor, scientist, and
businessman who developed many devices that greatly influenced life around
the world, including the
phonograph, the
motion picture camera,
and a long-lasting, practical electric
light bulb. Dubbed
"The Wizard of Menlo Park" . He was born in
Milan, Ohio. His quotations include:
"There's a way to do it better - find it!"
"Genius is one percent inspiration and ninety-nine percent perspiration."
"I have not failed. I've just found 10,000 ways that won't work." "I
never did a day's work in my life. It was all fun."
Prime Number Theorem states that if you select a large number N, the
probability of it being prime is about 1/Ln(N) was solved
independently in 1896 by Jacques-Solomon Hadamard (1865-1963) and
Charles de la
Vallee Poisson (1866-1962) by showing that the Riemann Zeta Function has no zeros
of the form (1 + ki).
Carl Louis Ferdinand von Lindemann (1852-1939):
(German) He was a noted for his proof, published in 1882,
that π (pi) is a
transcendental number,
i.e., it is not a zero of any
polynomial with
rational
coefficients.
Jules Henri Poincare (1854-1912):
(French) A mathematician, theoretical physicist, and a
philosopher of science. Poincare is often described in mathematics
as The Last Universalist, since he excelled in all fields of the
discipline as it existed during his lifetime.
George Eastman (1854-1932):
(American) An inventor and
philanthropist. He
founded the
Eastman Kodak Company
and invented
roll film
In his final two years, Eastman was in intense pain, caused by a
degenerative disorder affecting his spine. He had trouble standing and his
walking became a slow shuffle. Today it might be diagnosed as
lumbar spinal stenosis,
a narrowing of the spinal canal caused by
calcification in the
vertebrae. Eastman grew depressed, as he had seen his mother spend the last
two years of her life in a wheelchair from the same condition. On
March 14, 1932, Eastman died by suicide with a single gunshot to the heart,
leaving a note which read, "My work is done. Why wait?"
 |
The logo from 1987 to 2006. The letter "K" had
been a favorite of Eastman's, he is quoted as saying, "it seems a
strong, incisive sort of letter." He and his mother devised the name
Kodak with an anagram set. He said that there were three principal
concepts he used in creating the name: it should be short, one cannot
mispronounce it, and it could not resemble anything or be associated
with anything but Kodak. |
Andrey (Andrei) Andreyevich Markov (Андрей
Андреевич Марков) (1856-1922): (Russian) He is best
known for his work on theory of
stochastic processes.
His research later became known as
Markov chains.
Henry Ernest Dudeney (1857-1930): (English) An author and mathematician who
specialized in logic puzzles and mathematical games. He is known as one of
the foremost creators of puzzles.
Max Karl Ernst Ludwig Planck (1858-1947):
(German) A
physicist who is
regarded as the founder of the
quantum theory, for
which he received the
Nobel Prize in Physics
in 1918.
David Hilbert (1862-1943): (German)
He was recognized as one of the most influential and universal
mathematicians of the 19th and early 20th centuries. He discovered and
developed a broad range of fundamental ideas in many areas,
including
invariant theory and
the
axiomatization of geometry.
He also formulated the theory of
Hilbert spaces, one
of the foundations of
functional analysis.
Bertrand Russell (1872-1970): "Physics is mathematical not
because we know so much about the physical world, but because we know so
little; it is only its mathematical properties we can discover."
Albert Einstein (1879-1955): The
speed of light is the same, irrespective of how the source of light or the
observer is moving. Furthermore, space and time cannot be treated as
separate entities, rather they are inseparably tethered together by
symmetry. One of the known results of special relativity is that
the length of moving bodies, as measured by observers at rest, contracts
along their direction of motion. The contraction is larger, the
higher the speed. Gravity warps and bends spacetime. One of the key
predictions of general relativity was the bending of light rays under the
influence of gravity. Guided by
principles of symmetry Einstein showed that acceleration and gravity
are two sides of the same coin.(If a train is moving very fast to the north
and a man in a boxcar drops his keys, they fall to the south.)(If a man in a
stationary box car drops his keys, the keys would fall to the south, if
gravity was tilted to the south.)
Max Born (1882-1970): (German)
A
physicist and
mathematician who was
instrumental in the development of
quantum mechanics. He
also made contributions to
solid-state physics
and
optics and supervised
the work of a number of notable physicists in the 1920s and 30s. Born won
the 1954
Nobel Prize in Physics,
shared with
Walther Bothe.
Frank Albert Benford, Jr. (1883-1948):
(American) Benford's law, also called the
first-digit law, states that in lists of numbers from many (but not
all) real-life sources of
data, the leading
digit is distributed in a specific, non-uniform way. According to this law,
the first digit is 1 almost one
third of the time,
and larger digits occur as the leading digit with lower and lower frequency,
to the point where 9 as a first digit occurs less than one time in twenty.
This distribution of first digits arises logically whenever a set of values
is distributed
logarithmically.
Measurements of real world values are often distributed logarithmically (or
equivalently, the logarithm of the measurements is distributed uniformly).
This counter-intuitive result has been found to apply to a wide
variety of data sets, including electricity bills,
street addresses, stock prices, population numbers, death rates, lengths of
rivers,
physical and
mathematical constants,
and processes described by
power laws (which are
very common in nature). The result holds regardless of the
base in which the
numbers are expressed, although the exact proportions change. It is named
after physicist
Frank Benford, who
stated it in 1938, although it had been previously stated by
Simon Newcomb in
1881.
John Maynard Keynes (1883-1946):
British Economist and Mathematician. On the Law of Large Numbers
or "long run calculations", he said: "This long run is a misleading guide to
current affairs. In the long run we are all dead.
Economists set themselves too easy, too useless a task if in tempestuous
seasons they can only tell us that when the storm is long past the ocean is
flat again." Other quotes: "I do not know which makes a man more
conservative - to know nothing but the present, or nothing but the past."
" It would be foolish, in forming our expectations, to attach great weight
to matters which are very uncertain." " It is generally agreed that
casinos should, in the public interest, be inaccessible and expensive. And
perhaps the same is true of Stock Exchanges." "The outstanding faults
of the economic society in which we live are its failure to provide for full
employment and its arbitrary and inequitable distribution of wealth and
incomes."
Niels Henrik David Bohr (1885-1962):
(Danish) A
physicist who made
fundamental contributions to understanding
atomic structure and
quantum mechanics,
for which he received the
Nobel Prize in
Physics in 1922. Bohr
mentored and collaborated with many of the top physicists of the century at
his institute in
Copenhagen. He was
part of a team of physicists working on the
Manhattan Project.
Bohr married Margrethe Norlund in 1912, and one of their sons,
Aage Bohr, grew up to
be an important physicist who in 1975 also received the Nobel prize. Bohr
has been described as one of the most influential scientists of the 20th
century.
Srīnivāsa Aiyangār Rāmānujan (1887-1920):
(Indian) He was a
self taught
genius, with almost
no formal training in
pure mathematics,
made substantial contributions to
mathematical analysis,
number theory,
infinite series and
continued fractions.
Ramanujan's talent was said, by the prominent English mathematician
G.H. Hardy, to be in
the same league as legendary mathematicians such as
Euler,
Gauss,
Newton and
Archimedes.
Enrico Fermi (1901-1954):
(Italian-American)
A
physicist
particularly known for his work on the development of the first
nuclear reactor,
Chicago Pile-1, and
for his contributions to the development of
quantum theory,
nuclear and
particle physics, and
statistical mechanics.
He was awarded the 1938
Nobel Prize in Physics
for his work on
induced radioactivity.
Fermi is widely regarded as one of the leading
scientists of the
20th century, highly
accomplished in both theory and experiment.
Along with
J. Robert Oppenheimer,
he is frequently referred to as "the father of the
atomic bomb".
He also held
several patents
related to the use of nuclear power.
Andre John von Neumann
(1903-1957): (Hungarian-born)
An
American
mathematician who
made major contributions to a vast range of fields, including
set theory,
functional analysis,
quantum mechanics,
ergodic theory,
continuous geometry,
economics and
game theory,
computer science,
numerical analysis,
hydrodynamics (of
explosions), and
statistics, as well
as many other mathematical fields. He is generally regarded as one of the
greatest mathematicians in modern history. The mathematician
Jean Dieudonne called
von Neumann "the last of the great mathematicians",while
Peter Lax described
him as possessing the most "fearsome technical prowess" and "scintillating
intellect" of the century.Weil (1906-1998): (from France)
"God
exists since mathematics is consistent, and the Devil exists since we cannot
prove it."
Julius Robert Oppenheimer (1904-1967):
(American) A
theoretical physicist
and professor of physics at the
University of California, Berkeley.
He is often called the "father of the atomic bomb" for his role as
the scientific director of the
Manhattan Project,
the
World War II project
that developed the first
nuclear weapons. The
first atomic bomb was detonated in July 1945 in the
Trinity test in
New Mexico;
Oppenheimer remarked later that it brought to mind words from the
Bhagavad Gita: "Now,
I am become Death, the destroyer of worlds."
After the war he became a chief adviser to the
newly created
United States Atomic Energy Commission
and used that position to lobby for international control of
nuclear power to
avert
nuclear proliferation
and an
arms race with the
Soviet Union. After provoking the ire of many politicians with his outspoken
opinions during the
Second Red Scare, he
had his
security clearance
revoked in a much-publicized hearing in 1954. Though stripped of his direct
political influence he continued to lecture, write and work in physics. A
decade later President
John F. Kennedy
awarded (and
Lyndon B. Johnson
presented) him with the
Enrico Fermi Award as
a gesture of
political rehabilitation.
Oppenheimer's notable achievements in physics
include the
Born Oppenheimer approximation
for molecular
wavefunctions, work
on the theory of
electrons and
positrons, the
Oppenheimer Phillips process
in
nuclear fusion, and
the first prediction of
quantum tunneling.
With his students he also made important contributions to the modern theory
of
neutron stars and
black holes, as well
as to
quantum mechanics,
quantum field theory,
and the interactions of
cosmic rays. As a
teacher and promoter of science, he is remembered as a founding father of
the American school of theoretical physics that gained world prominence in
the 1930s. After World War II, he became director of the
Institute for Advanced Study
in Princeton.
Kurt Goedel (1906-1978): (from Austria)
Goedel's Incompleteness Theorem: Any consistent axiom system is necessarily
incomplete in that there will be true statements that can't be deduced from
the axioms.
Andre Weil (1906-1998): (French)
He was an influential
mathematician of the
20th century, renowned for the breadth and quality of his research output,
its influence on future work, and the elegance of his exposition. He is
especially known for his foundational work in
number theory and
algebraic geometry.
James R Newman (1907-1966):
(American) Von Neumann's first significant contribution to
economics was the minimax theorem in1928. He eventually improved and
extended the minimax theorem to include games involving imperfect
information and games with more than two players. This work culminated in
the 1944 classic
Theory of Games and Economic Behavior.
Von Neumann was one of the pioneers of computer science making significant
contributions to the development of logical design. "The Theory of Groups is a branch of mathematics in which one
does something to something and then compares the result with
the result obtained from doing the same thing to something else, or something else to the same thing."
Edward Teller (1908-2003):
(Hungarian-born
American) A
theoretical
physicist, known
colloquially as "the father of the
hydrogen bomb,"
even though he did not care for the title. In 1942, Teller was
invited to be part of
Robert Oppenheimer's
summer planning seminar at the
University of California, Berkeley
for the origins of the
Manhattan Project,
the
Allied effort to
develop the first
nuclear weapons. A
few weeks earlier, Teller had been meeting with his friend and colleague
Enrico Fermi about
the prospects of
atomic warfare, and
Fermi had nonchalantly suggested that perhaps a weapon based on
nuclear fission could
be used to set off an even larger
nuclear fusion
reaction. Even though he initially explained to Fermi why he thought the
idea would not work, Teller was fascinated by the possibility and was
quickly bored with the idea of "just" an atomic bomb (even though this was
not yet anywhere near completion). At the Berkeley session, Teller diverted
discussion from the fission weapon to the possibility of a fusion
weapon, what he called the "Super" (an early version of what was later known
as a hydrogen bomb).
Stanislaw Marcin Ulam
(1909-1984): (Polish-Jewish) He
participated in America's
Manhattan Project,
originated the
Teller–Ulam design of
thermonuclear weapons,
invented the
Monte Carlo method of computation,
and suggested
nuclear pulse propulsion.
In pure and applied mathematics, he produced many results, proved many
theorems, and proposed several conjectures.
Paul Erdős (1913-1996): (from Budapest,
Hungary) Erdős published more papers than any other
mathematician in history, working with hundreds of collaborators. His
colleague
Alfred Renyi said, "a mathematician is a
machine for turning coffee into theorems", and Erdős drank copious
quantities.
Because of his prolific output, friends created the Erdős number
as a humorous tribute; Erdős alone was assigned the Erdős number of 0 (for
being himself), while his immediate collaborators could claim an Erdős
number of 1, their collaborators have Erdős number at most 2, and so on.
Approximately 200,000 mathematicians have an assigned Erdős number, and some
have estimated that 90 percent of the world's active mathematicians have an
Erdős number smaller than 8.
It is said that Hank Aaron has an Erdős number of 1 because they
both autographed the same baseball when
Emory University awarded them honorary
degrees on the same day.
Erdős numbers have also been assigned to an infant, a horse, and
several actors.
Martin Gardner (1914-2010): (American) A mathematics and science writer specializing
in recreational mathematics, but with many interests (especially the
writings of Lewis Carroll. He wrote the Mathematical Games column in
Scientific American from 1956 to 1981, the Notes of a Fringe-Watcher
column in Skeptical Inquirer from 1983 to 2002, and published over 70 books.
See:
http://en.wikipedia.org/wiki/Martin_Gardner
Richard Feynman
(1918-1988):
Feynman said: "Mathematics is looking for patterns".
"Mathematics is only patterns". "Nature uses only the longest threads to
weave her patterns, so that each small piece of her fabric reveals the
organization of the entire tapestry." Also: "Physics is like sex.
Sure, it may give some practical results, but that's not why we do it".
Murray Gell-Mann commented to the New York Times that the Feynman Algorithm to solve a problem is:
1. Write down the problem
2. Think very hard
3. Write down the answer.
Benoit Mandelbrot (1924- 2010): (from France) The Father of Fractal Geometry.
There are many beautiful pictures to view on the web. For example:
http://sprott.physics.wisc.edu/fractals.htm Also there are
terrific videos to be found at:
http://www.fractal-animation.net/ufvp.html and
http://www.ericbigas.com/fractalanimation/index.html and
http://www.fractal-animation.net/ufvp.html and
http://fractalanimations.com/
and
http://www.google.com/images?hl=&q=fractal+animation&rlz=1B3GGLL_enUS405US405&um=1&ie=UTF-8&source=univ&ei=2-dFTfmfPI-p8AaSw42EAg&sa=X&oi=image_result_group&ct=title&resnum=6&ved=0CEoQsAQwBQ&biw=1045&bih=404
and
http://video.google.com/videoplay?docid=1619313842463920970#docid=8570098277666323857 and
http://video.google.com/videoplay?docid=1619313842463920970#docid=6460130356432628677
and
http://www.youtube.com/watch?v=34zPvmNXTYQ and
http://www.youtube.com/watch?v=G_GBwuYuOOs . Learn from Robert
Devaney at:
http://video.google.com/videoplay?docid=1619313842463920970#docid=-6460544449138143366
Fractal art is shown at:
http://www.lifesmith.com/art2008.html and at
http://www.lifesmith.com/art2006.html and
http://www.lifesmith.com/art2007.html I presented some
pictures of fractals and some background at the Annual Meeting of the
Society of Actuaries when I was president in 1992. I may have used
Robert Devaney to develop the script. See
Chaos video, but start about 40% of the way through.
(OOPs is copywrite protected so will only play on my computer)
Alexander Grothendieck (1928--
): (German) He is one of the most influential
mathematicians of the
20th century. He is known principally for his revolutionary advances in
algebraic geometry,
and also for major contributions to
number theory,
category theory and
homological algebra,
and his early achievements in
functional analysis.
He was awarded the
Fields Medal in 1966.
Murray Gell-Mann (1929-
): (American) A
physicist and
polymath who received
the 1969
Nobel Prize in physics
for his work on the theory of
elementary particles.
He is a Distinguished Fellow and co-founder of the
Santa Fe Institute
and the Presidential Professor of Physics and Medicine at the
University of Southern California.
He formulated the
quark model of
hadronic resonances, and identified the SU(3)
flavor symmetry of
the light quarks, extending
isospin to include
strangeness, which he
also discovered. He developed the V-A theory of the
weak interaction in
collaboration with
Richard Feynman. He
created
current algebra in
the 1960s as a way of extracting predictions from quark models when the
fundamental theory was still murky, which led to model-independent
sum rules confirmed
by experiment.
Stephen Hawkings (1942-
): A physicist
from Cambridge wrote "A Brief History of Time". In it he tells the story
of a lady commenting on a statement made in a lecture on astronomy.
She said: "Rubbish, The world is really a flat plate supported on the back
of a giant tortoise" When asked what the tortoise was sitting on, her
answer would have made Goedel smile: "You're very clever, young man, very
clever. But its turtles all the way down."
Persi Warren Diaconis (1945-
): He is the
statistician who demonstrated that it takes the average card player no fewer
than seven shuffles to create a random order in a deck of cards.
Marilyn vos Savant (1946-
): An
American
magazine
columnist,
author,
lecturer, and
playwright. She has
written "Ask Marilyn", a Sunday column in
Parade magazine
in which she solves puzzles and answers questions from readers on a variety
of subjects.
Her September 9, 1990 column began with a question now
called The Monty Hall problem (Suppose you are on a game show and you are
given the choice of three doors. Behind one door is a car, the
others, goats. You pick a door, say #1, and the host, who knows what's
behind the doors, opens another door, say #3, which has a goat. He says to
you: 'Do you want to pick door #2?' Is it to your advantage to switch doors?
Marilyn vos Savant answered arguing that the selection should be switched to
door #2 because it has a 2/3 chance of success, while door #1 has just 1/3.
This response provoked letters of thousands of readers, nearly all arguing
doors #1 and #2 each have an equal chance of success.
A follow-up column reaffirming her position served only to intensify the
debate and soon became a feature article on the front page of
The New York Times.
Among the ranks of dissenting arguments were hundreds of academics and
mathematicians.
In a subsequent column, vos Savant offered numerous
explanations as to why her solution is correct. She also called upon
elementary teachers to simulate the problem in their class. Numerous
elementary school math classes devoted themselves to this experiment,
playing the game hundreds of times and reporting their results. Nearly 100%
of those classes found that your odds of winning were doubled if you switch
doors.
Finally, thanks to the diligence of elementary school children, the
controversy subsided.
Dr. Keith Devlin (1947-
): This professor from Stamford defines Mathematics as the
Science of Patterns.
Robert L. Devaney (circa 1948-
): A native of Methuen, Massachusetts, is currently Professor
of Mathematics at Boston University. He received his undergraduate degree
from the College of the Holy Cross in 1969 and his PhD from the University
of California at Berkeley in 1973 under the direction of Stephen Smale. He
taught at Northwestern University and Tufts University before coming to
Boston University in 1980. His main area of research is dynamical
systems, primarily complex analytic dynamics, but also including more
general ideas about chaotic dynamical systems. Lately, he has become
intrigued with the incredibly rich topological aspects of dynamics,
including such things as indecomposable continua, Sierpinski curves, and
Cantor bouquets.
Devaney developed the 8 minute script contained in the middle of the
presentation on Chaos
that Don had presented for the 1992 Annual Meeting of the Society of
Actuaries. This link only works on Don's computer.
Edward Witten (1951-
): One of the
researchers at Princeton working on "string theory" which may help with the
clash between the central ideas of general relativity and quantum mechanics
when it comes to extremely small scales. He is regarded by
many of his peers as one of the greatest living physicists, perhaps even a
successor to
Albert Einstein.
In 1990 he was awarded a
Fields Medal
by the International Union of Mathematics, which is the highest honor in
mathematics and often regarded as the Nobel Prize equivalent for
mathematics. He is the only physicist to have received this honor.
Sir Andrew John
Wiles (1953- ):
(British) A professor at Princeton University in Number Theory. He
published a flawed proof of Fermat's Last Theorem in 1993. He
corrected the error in 1994.
Simon Kirwan Donaldson
(1957- ): (British)
An
English mathematician
famous for his work on the
topology of smooth
(differentiable) four-dimensional
manifolds. He is now
Royal Society research professor in Pure Mathematics and President of the
Institute for Mathematical Science at
Imperial College London.
He used the solutions to the Yang-Mills equations to discover a finger-print
which allowed him to distinguish whether two shapes were actually the same.
These finger-prints are called invariants.
.
Marcus Peter Francis du Sautoy
(1965 - ):
(born in London) A Professor of Mathematics at the University of Oxford. His
academic work concerns mainly group theory and number theory. He is known
for his books popularizing mathematics. In 2001 he won the
Berwick Prize of the
London Mathematical Society,
which is awarded every two years to reward the best mathematical
research by a mathematician under forty. In March, 2006, his article
Prime Numbers Get Hitched was published on
Seed Magazine's
website.
http://seedmagazine.com/content/article/prime_numbers_get_hitched/
In it he explained how the number
42, mentioned in
The Hitchhiker's Guide to the Galaxy
as the
answer to everything,
is related to the
Riemann zeta function.
See
http://www.culturenorthernireland.org/article/2836/belfast-festival-marcus-du-sautoy
Also:
http://people.maths.ox.ac.uk/dusautoy/newleft.htm and
http://people.maths.ox.ac.uk/dusautoy/newright.htm
Grigori Yakovlevich Perelman (1966-
): (from Russia) The Millennium Prize Problems
are seven problems in mathematics that were stated by the Clay Mathematics
Institute in 2000. Currently, six of the problems remain unsolved. A
correct solution to any of the problems results in a US$1,000,000 prize
(sometimes called a Millennium Prize) being awarded by the institute.
One of the problems, the Poincare' conjecture, was solved by Perelman in
2002 He was also awarded the Fields Medal in
2006. He has not accepted either prize.
MATHEMATICS and MUSIC
Symmetry and Music The six
symmetries of music refers to a set of transformations that can be
applied to music while leaving a fundamental essence of the music unchanged.
The six symmetries are:
pitch translation invariance,
time scaling invariance,
octave translation invariance,
time translation invariance,
amplitude scaling invariance,
and
pitch reflection invariance.
Also see:
http://orion.math.iastate.edu/mathnight/activities/modules/music/
Patterns by Natasha Glydon:
Musical pieces are read much like you would
read math symbols. The symbols represent some bit of information about
the piece. Musical pieces are divided into sections called measures or
bars. Each measure embodies an equal amount of time. Furthermore, each
measure is divided into equal portions called beats. These are all
mathematical divisions of time.
Fractions are used in music to indicate
lengths of notes. In a musical piece, the time signature tells
the musician information about the rhythm of the piece. A time signature
is generally written as two integers, one above the other. The number on
the bottom tells the musician which note in the piece gets a single beat
(count). The top number tells the musician how many of this note is in
each measure. Numbers can tell us a lot about musical pieces.
Each note has a different shape to indicate its beat length or time.
Notes are classified in terms of numbers as well. There are whole notes
(one note per measure), half notes (two notes per measure), quarter notes
(four notes per measure), eighth notes (eight notes per measure), and
sixteenth notes (sixteen notes per measure). These numbers signify how
long the notes last. That is, a whole note would last through the entire
measure whereas a quarter note would only last one quarter of the measure and thus
there is enough time for four quarter notes in one measure. This can be
expressed mathematically since 4 x 1/4 = 1. A note with a dot after it
lengthens the note by half. For example, a quarter note with a dot after
it would be held for 3/8
of a measure, since: 1/4 + 1/2(1/4) = 3/8.
Three eigths of a measure is midway between a quarter note and a half
note. It is important for musicians to understand the relationships and
values of fractions in order to correctly hold a note.
Fibonacci: The
Fibonacci sequence is a famous and well-known sequence that follows as: 1,
1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on, adding each term to the
one before it to create the next term. That is, 5 + 8 = 13, 8 + 13 = 21,
13 + 21 = 34, and continuing infinitely. In music, the Fibonacci sequence
can be seen in piano scales. For example, the C scale on the piano
consists of 13 keys from C to C; eight white keys and five black keys,
with black keys arranged in groups of three and two. In the
Fibonacci sequence, the ratio between each term is very close to 0.618,
which is known as the golden ratio.
Pythagoras and Frequency: It was Pythagoras who realized
that different sounds can be made with different weights and vibrations.
This led to his discovery that the pitch of a vibrating string is
proportional to and can be controlled by its length. Strings that are
halved in length are one octave higher than the original. In essence, the
shorter the string, the higher the pitch. He also realized that notes of
certain frequencies sound best with multiple frequencies of that note.
For example, a note of 220Hz sounds best with notes of 440Hz, 660Hz, and
so on.
The closest tie between music and math is patterns. Musical pieces
often have repeating choruses or bars, similar to patterns. In
mathematics, we look for patterns to explain and predict the unknown.
Music uses similar strategies. When looking at a musical piece, musicians
look for notes they recognize to find notes that are rare (high or low)
and less familiar. In this way, notes relate to each other.
Relationships are fundamental to mathematics and create an interesting
link between music and math.
Heinrich Rudolf Hertz (1857-1894): A
Hertz, or Hz, is named after this German physicist. It is a measure of
the frequency, the number of vibrations of a string per second. People
in different musical traditions have different ideas about which notes they
think sound good together. If you double the frequency, the human ear
tends to hear both notes as the same. This is called "octave
equivalency". The doubled frequency is called a higher octave. This
"octave" is two times higher, not eight times higher. In the "diatonic
scale", there are 8 notes counting both ends of the octave hence the term
"octave". In the "chromatic scale" there are 13 notes counting both
ends, and the "Arab classical scale" has 17, 19, or even 24 notes in its
"octave.
Scales can be classified as "Just" (See
http://en.wikipedia.org/wiki/Just_intonation (if the ratios between the frequencies are ratios of integers), "tempered"
(if the just scale is tempered), and "practice-based" (if it reflects
musical practice) In the Even Tempered Scale going
from one semitone to the next is the 12th root of 2, or 1.05946...
(Pythagoras discovered that frequencies whose ratio is equal to the ratio
of two simple whole numbers yield "harmonious" and pleasing sounds.
The ratio of 3:2 is 1.5. A "perfect fifth" corresponds to a separation of
seven semitones, as the seventh power of 1.05946 is close to 1.5. A
"perfect fourth" corresponds to a frequency ratio of 4/3 and five
semitones.)
|
The Just Scale
|
|
|
|
|
The Even Tempered Scale |
|
|
Ratio to Starting Frequency |
Steps |
|
Note |
|
Ratio to Starting Frequency |
Steps |
| 1
= 1.000 |
0 |
|
C |
|
= 1 |
0 |
| 16/15 = 1.067 |
.5 |
|
C Sharp |
|
= 1.05946 |
1 |
| 9/8 = 1.125 |
1 |
|
D |
|
= 1.12246 |
2 |
| 6/5 = 1.200 |
1.5 |
|
E Flat |
|
= 1.18920 |
3 |
| 5/4 = 1.250 |
2 |
|
E |
|
= 1.25991 |
4 |
| 4/3 = 1.333 |
2.5 |
|
F |
|
= 1.33482 |
5 |
| 45/32 = 1.406 or 25/18 = 1.389 |
3 or |
|
F Sharp |
|
= 1.41419 |
6 |
| 64/45 = 1.422 or 36/25 = 1.440 |
3 |
|
F Sharp |
|
= 1.41419 |
6 |
| 3/2 = 1.500 |
3.5 |
|
G |
|
= 1.49828 |
7 |
| 8/5 = 1.600 |
4 |
|
G Sharp |
|
= 1.58736 |
8 |
| 5/3 = 1.667 |
4.5 |
|
A |
|
= 1.68175 |
9 |
| 16/9 = 1.778 |
5 |
|
B Flat |
|
= 1.78174 |
10 |
| 15/8 = 1.875 |
5.5 |
|
B |
|
= 1.88769 |
11 |
| 2/1 = 2.000 |
6 |
|
C |
|
= 2.00000 |
12 |
The frequency of middle C on a piano is often set at 261.6 Hz. There
are twelve semitones in an octave. octave. A piano keyboard has 7 white
keys and 5 black keys to play notes within any octave. octave. A trumpet has 3 valves that can be all open, two closed, etc. with 7
combinations of fingering, fingering, so a trumpet can only play 7 tones within an octave.
Don Francis writes: "The
ratio of the interval ending on F sharp is sometimes called the devil's
interval, probably because there is no good ratio to define it. It was
rarely used in pre 20th century music because of perceived
dissonance. It came to typify some types of jazz (the famous flatted fifth
of bebop). A good reference for remembering the interval is the first two
notes of Maria.
Also the trumpet reference is mathematically correct
about the valve openings, but trumpet players use different techniques and
can play a full range of notes within an octave, and also over several
octaves."
THE BACKSIDE OF THE MOON
The Earth rotates once per day on its axis towards the east and orbits
around the Sun once in about 365 days.
The
Moon rotates
once in about 28 days on its axis towards the west. The Moon orbits around the Earth
once in about 28 days.
If the Moon
didn't spin
at all, then eventually it would show its far side to the Earth while moving
around our planet in orbit. However, since the rotational period is exactly
the same as the orbital period, the same portion of the Moon's sphere is
always facing the Earth.
Assume there is only one house on the Moon and you can see it on day one
from your house on Earth. After 14 days, the Moon has orbited to the opposite side of the Earth. The Moon has turned one half revolution on its
axis in those 14 days, so you and anyone else on the Earth are still looking
at that same house. This means you always are looking at the same side of
the moon, whether you are at day 3, 9, 22, etc. in the 28 day orbit of the
Moon around the planet Earth.
|
