Hello friends....

This blog contains all the facts, Tricks and the funniest things about Mathematics. As you all know that maths is the most common and important subject in our Education.

I haven't invented these - I just collect them.

1. Definitions

Let's start with general definitions.

Mathematics is made of 50 percent formulas, 50 percent proofs, and 50 percent imagination.

"A mathematician

is a device for turning coffee into theorems" (P. Erdos)
Addendum: American coffee is good for lemmas.

An engineer thinks that his equations are an approximation to reality. A physicist thinks reality is an approximation to his equations. A mathematician doesn't care.

Old mathematicians never die; they just lose some of their functions.

Mathematicians are like Frenchmen: whatever you say to them, they translate it into their own language, and forthwith it means something entirely different. -- Goethe

Mathematics is the art of giving the same name to different things. -- J. H. Poincare

What is a rigorous definition of rigor?

There is no logical foundation of mathematics, and Gödel has proved it!

The Evolution of Math Teaching

1960s: A peasant sells a bag of potatoes for $10. His costs amount to 4/5 of his selling price. What is his profit?
1970s: A farmer sells a bag of potatoes for $10. His costs amount to 4/5 of his selling price, that is, $8. What is his profit?
1970s (new math): A farmer exchanges a set P of potatoes with set M of money. The cardinality of the set M is equal to 10, and each element of M is worth $1. Draw ten big dots representing the elements of M. The set C of production costs is composed of two big dots less than the set M. Represent C as a subset of M and give the answer to the question: What is the cardinality of the set of profits?
1980s: A farmer sells a bag of potatoes for $10. His production costs are $8, and his profit is $2. Underline the word "potatoes" and discuss with your classmates.
1990s: A farmer sells a bag of potatoes for $10. His or her production costs are 0.80 of his or her revenue. On your calculator, graph revenue vs. costs. Run the POTATO program to determine the profit. Discuss the result with students in your group. Write a brief essay that analyzes this example in the real world of economics.

Top ten excuses for not doing homework:

I accidentally divided by zero and my paper burst into flames.
Isaac Newton's birthday.
I could only get arbitrarily close to my textbook. I couldn't actually reach it.
I have the proof, but there isn't room to write it in this margin.
I was watching the World Series and got tied up trying to prove that it converged.
I have a solar powered calculator and it was cloudy.
I locked the paper in my trunk but a four-dimensional dog got in and ate it.
I couldn't figure out whether i am the square of negative one or i is the square root of negative one.
I took time out to snack on a doughnut and a cup of coffee.
I spent the rest of the night trying to figure which one to dunk.
I could have sworn I put the homework inside a Klein bottle, but this morning I couldn't find it.

Warning! It is against the rule to use these excuses in my classes! A. Ch.

Dictionary of Definitions of Terms Commonly Used in Math. lectures.

The following is a guide to terms which are commonly used but rarely defined. In the search for proper definitions for these terms we found no authoritative, nor even recognized, source. Thus, we followed the advice of mathematicians handed

down from time immortal: "Wing It."

CLEARLY:: I don't want to write down all the "in- between" steps.
TRIVIAL:: If I have to show you how to do this, you're in the wrong class.
OBVIOUSLY:: I hope you weren't sleeping when we discussed this earlier, because I refuse to repeat it.
RECALL:: I shouldn't have to tell you this, but for those of you who erase your memory tapes after every test...
WLOG (Without Loss Of Gener
ality):: I'm not about to do all the possible cases, so I'll do one and let you figure out the rest.
IT CAN EASILY BE SHOWN:: Even you, in your finite wisdom, should be able to prove this without me holding your hand.
CHECK or CHECK FOR Y
OURSELF:: This is the boring part of the proof, so you can do it on your own time.
SKETCH OF A PROOF:: I couldn't verify all the details, so I'll break it down into the parts I couldn't prove.
HINT:: The hardest of several possible ways to do a proof.
BRUTE FORCE (AND IG
NORANCE):: Four special cases, three counting arguments, two long inductions, "and a partridge in a pair tree."
SOFT PROOF:: One third less filling (of the page) than your regular proof, but it requires two extra years of course work just to understand the terms.
ELEGANT PROOF:: Requires no previous knowledge of the subject matter and is less than ten lines long.
SIMILARLY:: At least one line of the proof of this case is the same as before.
CANONICAL FORM:: 4 out of 5 mathematicians surveyed recommended this as the final form for their stude; nts who choose to finish.
TFAE (The Following Are Equivalent):: If I say this it means that, and if I say that it means the other thing, and if I say the other thing...
BY A PREVIOUS THEOREM:: I don't remember how it goes (come to think of it I'm not really sure we did this at all), but if I s; tated it right (or at all), then the rest of this follows.
TWO LINE PROOF:: I'll leave out everything but the conclusion, you can't question 'em if you can't see 'em.
BRIEFLY:: I'm running out of time, so I'll just write and talk faster.
LET'S TALK THROUGH IT:: I don't want to write it; on the board lest I make a mistake.
PROCEED FORMALLY:: Manipulate symbols by the rules without any hint of their true meaning (popular in pure math courses).
QUANTIFY:: I can't find anything wrong with your proof except that it won't work if x is a moon of Jupiter (Popular in applied math courses).
PROOF OMITTED:: Trust me, It's true.

Cat Theorem:
A cat has nine tails.
Proof:
No cat has eight tails. A cat has one tail more than no cat. Therefore, a cat has nine tails.

Salary Theorem
The less you know, the more you make.
Proof:

Postulate 1: Knowledge is Power.
Postulate 2: Time is Money.

Power = Work / Time

Knowledge = Power

Time = Money

nowledge = Work / Money

Mon

Money = Work / Knowledge

Knowledge

Money

Work

Q: How do you tell that you are in the hands of the Mathematical Mafia?

A: They make you an offer that you can't understand.

Funny formulas

The limit as 3 goes to 4 of 3^2 is 16.
(For native LaTex speakers: $$\lim_{3 \rightarrow 4} 3^2 = 16$$)

1 + 1 =3, for sufficiently large one's.

The combination of the Einstein and Pythagoras discoveries:
E= m c^2= m ( a^2 + b^2)

2 and 2 is 22

The limit as n goes to infinity of sin (x) /n is 6.
Proof: cancel the n in the numerator and denominator.

As x goes to zero, the limit of 8 /x is 00 (infinity), then the limit (as x goes to zero) of Z /x is N

Examples of inverse proble

ms:

Q: To what question is the answer "9W."
A: "Dr. Wiener, do you spell your name with a V?"

Q: To what question is the answer "Dr. Livingstone, I presume."
A: "What is your full name, Dr. Presume?"

A Neanderthal child rode to school with a boy from Hamilton. When his mother found out she said, "Wha

t did I tell you? If you commute with a Hamiltonian you'll never evolve!"

Pope has settled the continuum hypothesis!
He has declared that cardinals above 80 have no powers.

In modern mathematics, algebra has become so important that numbers will soon only have symbolic meaning.

A circle is a round straight line with a hole in the middle.

In the topologic hell the beer is packed in Klein's bottles.

He thinks he's really smooth, but he's only C^1.

Moebius strip no-wear belt drive! (Please see other side for warranty details.)

Q: Why couldn't the moebius strip enroll at the school?
A: They required an orientation.

Q: What is the world's longest song?
A: "Aleph-nought Bottles of Beer on the Wall."

Q: Why do Computer Scientists get Halloween and Christmas mixed up?
A: Because Oct. 31 = Dec. 25.

Q: Why did the chicken cross the Moebius strip?
A: To get to the other ... er, um ...

Q: Why did the mathematician name his dog "Cauchy"?
A: Because he left a residue at every pole.

Q: What do you get when you cro

ss an elephant and a banana?
A: | elephant | * | banana | * sin(theta)

Q: What do you get if you cross a mosquito with a mountain climber.
A: You can't cross a vector with a scalar.

Q: What is a compact city?
A: It's a city that can be guarded by finitely many near-sighted policemen.

Two mathematicians are studying a convergent series. The first one says: "Do you realize that the ser

ies converges even when all the terms are made positive?" The second one asks: "Are you sure?" "Absolutely!"

Q: What does the zero say to the the eight?
A: Nice belt!

Life is complex: it has both real and imaginary components.

Math problems? Call 1-800-[(10x)(13i)2]-[sin(xy)/2.362x].

"Divide fourteen sugar cubes int

o three cups of coffee so that each cup has an odd number of sugar cubes in it." "That's easy: one, one, and twelve." "But twelve isn't odd!" "Twelve is an odd number of cubes to put in a cup of coffee..."

A statistician can have his hea

d in an oven and his feet in ice, and he will say that on the average he feels fine.

Q: Did you hear the one about

the statistician?
A: Probably....

The only dirty joke in maths

Q. What's the square root of 69?

A. Eight something.

I failed every subject except for algebra.
How did you keep from failing that ?
I didn't take algebra !

Teacher: Are you good at math ?
Pupil: Yes and no
Teacher: What do you mean ?
Pupil: Yes, I'm no good at math !

Dad, can you help me find the lowest common denominator in this problem please ?
Don't tell me that they haven't found it yet, I remember looking for it when I was a boy !

Teacher: Did you parents help you with these homework problems ?
Pupil: No I got them all wrong by myself !

Teacher, I can't solve this problem.
Any five year old should be able to solve this one.
No wonder I can't do it then, I'm nearly ten !

Teacher: What's 2 and 2
Pupil: 4
Teacher: That's good
Pupil: Good ?, that's perfect !

Teacher: If 1+1=2 and 2+2=4, what is 4+4 ?
Pupil: That's not fair you answer the easy ones and leave us with the hard one !

Teacher: How much is half of 8
Pupil: Up and down or across ?
Teacher: What do you mean ?
Pupil: Well,up and down makes a 3 or across the middle leaves a 0

Teacher: Now class, whatever I ask, I want you to all answer at once. How much is six plus 4 ?
Class: At once !

If there are ten cats in a boat and one jumps out, how many are left ?
None, they were all copycats !

Great

$1 = 10cents

Dollars equal ten cents

Theorem: 1$ = 10 cent

Proof: We know that $1 = 100 cents

Divide both sides by 100

$ 1/100 = 100/100 cents

=> $ 1/100 = 1 cent

Take square root both side

=> squr($1/100) = squr (1 cent)

=> $ 1/10 = 1 cent

Multiply both side by 10

=> $1 = 10 cent

Ha Ha Ha

Painful Section (read at own risk!--you have been warned!)

Q: Why do truncated McLaurin Series fit the original function so well?
A: Because they are "tailor" made.

Q: Why is it valid for people dining at a Chinese restaurant to ask for the leftovers to be taken home?
A: This is valid by the "Chinese Remainder Theorem."

Q: What is a monad?
A: A person who moves around a lot in the desert.

Professor: This term I offer topology.
Student: No problem. Topology accepted.

Rather than Laplace transform, would it not have been better to transform Laplace?

When the American Indian Chief Cochise was ill, most members of the tribe were amazed at how many other chiefs came to visit him, some from over a thousand miles away. But one Indian brave stated that he expected such attention. When asked why, he replied that he knew everyone would converge there because of Cochise condition.

Q: If a quotation by Reagan is a Reaganism, and a quotation by Clinton is a Clintonism, what would you call a quotation by Al Gore?
A: An Algorism.

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!"

Mathematical Tricks

Effect
You write down the following 8 digit number on a piece of paper:

1 2 3 4 5 6 7 9

Then ask a friend to circle one of the digits. Say that they circle number 7.

You then ask your friend to multiply the 8 digit number by 63, and magically the result ends up being:

   1 2 3 4 5 6 7 9
x                     6 3
7 7 7 7 7 7 7 7 7

with the answer as a row of the chosen number 7.

Multiply Up to 20X20 In Your Head

In just FIVE minutes you should learn to quickly multiply up to 20x20 in your head. With this trick, you will be able to multiply any two numbers from 11 to 19 in your head quickly, without the use of a calculator.

I will assume that you know your multiplication table reasonably well up to 10x10.

Try this:

Take 15 x 13 for an example.
Always place the larger number of the two on top in your mind.
Then draw the shape of Africa mentally so it covers the 15 and the 3 from the 13 below. Those covered numbers are all you need.
First add 15 + 3 = 18
Add a zero behind it (multiply by 10) to get 180.
Multiply the covered lower 3 x the single digit above it the "5" (3x5= 15)
Add 180 + 15 = 195.

That is It! Wasn't that easy? Practice it on paper first!

The 11 Rule

You likely all know the 10 rule (to multiply by 10, just add a 0 behind the number) but do you know the 11 rule? It is as easy! You should be able to do this one in you head for any two digit number. Practice it on paper first!

To multiply any two digit number by 11:

For this example we will use 54.
Separate the two digits in you mind (5__4).
Notice the hole between them!
Add the 5 and the 4 together (5+4=9)
Put the resulting 9 in the hole 594. That's it! 11 x 54=594

The only thing tricky to remember is that if the result of the addition is greater than 9, you only put the "ones" digit in the hole and carry the "tens" digit from the addition. For example 11 x 57 ... 5__7 ... 5+7=12 ... put the 2 in the hole and add the 1 from the 12 to the 5 in to get 6 for a result of 627 ... 11 x 57 = 627
Practice it on paper first!

Finger Math: 9X Rule

To multiply by 9,try this:
(1) Spread your two hands out and place them on a desk or table in front of you.
(2) To multiply by 3, fold down the 3rd finger from the left. To multiply by 4, it would be the 4th finger and so on.
(3) the answer is 27 ... READ it from the two fingers on the left of the folded down finger and the 7 fingers on the right of it.

This works for anything up to 9x10!

Square a 2 Digit Number Ending in 5

For this example we will use 25

Take the "tens" part of the number (the 2 and add 1)=3
Multiply the original "tens" part of the number by the new number (2x3)
Take the result (2x3=6) and put 25 behind it. Result the answer 625.

Try a few more 75 squared ... = 7x8=56 ... put 25 behind it is 5625.
55 squared = 5x6=30 ... put 25 behind it ... is 3025. Another easy one! Practice it on paper first!

Square 2 Digit Number: UP-DOWN Method

Square a 2 Digit Number, for this example 37:

Look for the nearest 10 boundary
In this case up 3 from 37 to 40.
Since you went UP 3 to 40 go DOWN 3 from 37 to 34.
Now mentally multiply 34x40
The way I do it is 34x10=340;
Double it mentally to 680
Double it again mentally to 1360
This 1360 is the FIRST interim answer.
37 is "3" away from the 10 boundary 40.
Square this "3" distance from 10 boundary.
3x3=9 which is the SECOND interim answer.
Add the two interim answers to get the final answer.
Answer: 1360 + 9 = 1369

With practice this can easily be done in your head.

Multiply By 4

To quickly multiply by four, double the number and then double it again.
Often this can be done in your head.

Multiply By 5

To quickly multiply by 5, divide the number in two and then multiply it by 10. Often this can be done quickly in your head.

Multiplying by Eight

Multiplying by 8 can be achieved by doubling three times:

Example:
Q. What is 742 x 8?
A. 742 x 2 = 1484
.....1484 x 2 = 2968
.....2968 x 2 = 5936
The answer is 5936

Calculate a Tip
If you need to leave a 15% tip, here is the easy way to do it. Work out 10% (divide the number by 10) - then add that number to half its value and you have your answer:
15% of $25 = (10% of 25) + ((10% of 25) / 2)
$2.50 + $1.25 = $3.75
Tough Multiplication
If you have a large number to multiply and one of the numbers is even, you can easily subdivide to get to the answer:
32 x 125, is the same as:
16 x 250 is the same as:
8 x 500 is the same as:
4 x 1000 = 4,000

Maths Trick 1

STEP 1
Ask a friend to write down a number (any number with more than 3 digits will do, but to save time and effort you might suggest a maximum of 8 digits).
Example: 83 972 105

STEP 2
Ask them to add the digits.
Example: 8+3+9+7+2+1+0+5 = 35

STEP 3
Ask them to subtract this number from the original one.
Example: 83 972 105 – 35 = 83 972 070

STEP 4
Ask them to select any digit from this new number and strike it out, without showing you.
Example: 83 972 070

STEP 5
Ask them to add the remaining digits and write down the answer they get.
Example: 8+3+9+7+0+7+0 = 34

STEP 6
Ask them to tell you the number they get (34) and you will tell them which number they struck out.

SOLUTION
The way you do this is to subtract the number they give you from the next multiple of 9. The answer you get is the number they struck out.
Example: The next multiple of 9 here is 36 (9 x 4 =36)
36 – 34 = 2 and there you have your answer, easy isn’t it!

Note: If the number they give you after step 5 is a multiple of 9, there are two possible answers then you simply tell them that this time they crossed out either a 9 or a zero.

Maths Trick 2

Amazing 1089...

Step 1
Take two pieces of paper and hand one to a friend.
On yours, without letting them see, write the number 1089, then fold the paper to keep it hidden.

Step 2
Ask them to think of a 3-digit number but, before they write it down, ask them to put the numbers in order from greatest to smallest. Don't let them show what they've written.
Example: 543

Step 3
Below their number, ask them to write the same digits, but in reverse order, from smallest to greatest.
Example: 345

Step 4
Now, ask them to subtract the new lower number from the original one they wrote.
Example: 198

Step 5
Next, ask them to reverse the order of that number.
Example: 891

Step 6
Then, get them to add this latest number and the previous number together and show you the result.
Example: 891 + 198 = 1089

Step 7
Finally, you can reveal your own number, which (if they have calculated correctly) will be exactly what they have written...
1089

Amazing!

Maths Trick 3

Try this one in your head, using mental maths...

Step 1
Ask a friend to think of a number between 1 and 10.
Example: 8

Step 2
Get them to double it.
Example: 16

Step 3
Ask them to add 10 to the answer.
Example: 26

Step 4
Then get them to divide by 2.
Example: 13

Step 5
Ask them to tell you what number they now have.
Example: 13

Step 6
You subtract 5 from this and tell them what their original number was.
Example: 13 - 5 = 8

Note: If you wish to take turns to practice your mental maths, you can also use 2 and 3 digit numbers to make it harder!

Magic Squares

A magic square is a set of integers arranged in a square in such away that each row, each column (and often the two diagonals as well) sum to the same number.
For example: This 3 x 3 magic square's rows, columns and diagonals each add up to the number 15.

4	9	2
3	5	7
8	1	6

This 4 x 4 magic square's rows, columns and diagonals each add up to the number 34.

3	6	10	15
13	12	8	1
16	9	5	4
2	7	11	14

Squares

The result of squaring a number can also be arrived at by progressively adding consecutive odd numbers as shown below.

1²	= 1	= 1
2²	= 4	= 1+3
3²	= 9	= 1+3+5
4²	= 16	= 1+3+5+7
5²	= 25	= 1+3+5+7+9
6²	= 36	= 1+3+5+7+9+11
7²	= 49	= 1+3+5+7+9+11+13
8²	= 64	= 1+3+5+7+9+11+13+15
9²	= 81	= 1+3+5+7+9+11+13+15+17
10²	= 100	= 1+3+5+7+9+11+13+15+17+19