## very first thing you need to learn before going into advanced mathematics

Posted: November 21, 2012 in Mathematics
Tags: ,

Like the title has stated, there is one thing you need to learn before you go into the second year of mathematics in university.  That one thing is certainly neither the first year Calculus class nor the Linear Algebra class.  But the definition of the terms you will see a lot in those advanced math classes such as Analysis, Abstract Algebra, or Number Theory, etc.  So here they are.

Caution:  The followings are stolen from an undisclosed source in the American Mathematical Society aka AMS. 😀

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 Generality):
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 YOURSELF:
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 IGNORANCE):
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 students 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 stated 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 in case 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.

## prove that all odd integers are prime

Posted: November 21, 2012 in Mathematics
Mathematician:
3 is a prime, 5 is a prime, 7 is a prime,
9 is not a prime - counter-example - claim is false.

Physicist:
3 is a prime, 5 is a prime, 7 is a prime,
9 is an experimental error, 11 is a prime, ...

Engineer:
3 is a prime, 5 is a prime, 7 is a prime,
9 is a prime, 11 is a prime, ...

Computer scientist:
3's a prime, 5's a prime, 7's a prime, 7's a prime,
7's a prime, ...

Computer scientist using Unix:
3's a prime, 5's a prime, 7's a prime,
segmentation fault

## which kind are you?

Posted: November 21, 2012 in Mathematics

There are three kinds of people in the world;
those who can count,
and those who can’t.

## a cat has 9 lives

Posted: November 18, 2012 in Mathematics
Tags: , , , ,

Have you ever wondered why there is a saying that cats have nine lives?  Well today I am gonna prove this statement and make it a mathematical theorem.  This is a huge breakthrough in the history of cats and mathematics.  So here is the proof.

Theorem.  A cat has nine lives.

Proof.  No cat has eight lives.  Since a cat has one life more than no cat, a cat must have nine lives.  🙂  $\square$

## the sum of all natural numbers is negative one-twelfth?

Posted: November 16, 2012 in Mathematics
Tags: , , ,

No, I haven’t gone crazy yet.  And yes, I am telling you that the sum of all natural numbers (aka positive integers) is -1/12.  This maybe a little bit hard to swallow because what I am saying is that if you add up all the counting numbers (ie. 1+2+3+…), you will get a negative number and it’s finite and a “fraction” too.  This sounds crazy I know but I can show you how it could be done, so read on.  First of all let me give you some concepts on summations.

Cesaro Summability

Cesaro summation could be used when one wants to assign a meaning to the sum of a divergent series.  In simple English, if an average of the nth partial sum of a series converges to a limit, then that limit is the Cesaro sum of that series.

If $\displaystyle\lim_{n\to\infty} \frac{s_1+s_2+\ldots+s_n}{n}=L$ exists, then $L =$ Cesaro sum.

Now let’s examine the following alternating sequence,

$\{t_n\}=\{1,-1,1,-1,1,\ldots\}$.

The sequence of the partial sum is,

$\{s_n\}=\{1,0,1,0,1,0,\ldots\}$.

Then a sequence of $\displaystyle\left\{\frac{s_1+s_2+\ldots+s_n}{n}\right\}$ is,

$\displaystyle\left\{\frac{1}{1},\frac{1}{2},\frac{2}{3},\frac{2}{4},\frac{3}{5},\frac{3}{6},\ldots\right\}$.

Hence the limit,

$\displaystyle \lim_{n\to\infty}\frac{s_1+s_2+\ldots+s_n}{n} = \frac{1}{2}$.

If you think this is too much math and too difficult to understand, I can give you a naive way of obtaining the same result of that alternating series.  Let’s ignore the rigorous part of summability here at the moment, and be open minded, assuming a divergent series can be rearranged.

$S = 1-1+1-1+\ldots = 1-(1-1+1-\ldots) = 1-S$,

then,

$\displaystyle S=1-1+1-1+\ldots=\frac{1}{2}$.

If you believe this is true, then let small s be the sum of all natural numbers,

$s=1+2+3+4+5+6+\ldots$

$s-4s = -3s = 1+2+3+4+5+6+\ldots -4-8-12-\ldots$

$-3s = 1-2+3-4+5-6+\ldots$

$-3s= 1- (2-3+4-5+6-\ldots)$

$-3s= 1- (1-2+3-4+5-\ldots + 1-1+1-1+1-\ldots)$

$-3s= 1- (1-2+3-4+5-\ldots) - (1-1+1-1+1-\ldots)$

Since the sum of that alternating series is $\displaystyle\frac{1}{2}$,

$\displaystyle -3s=1+3s-\frac{1}{2}$

$\displaystyle \therefore s = -\frac{1}{12}$ 😀

Remarks:

This result is somewhat related to Riemann Zeta Function,

$\displaystyle\zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}$.

If we try to compute $\zeta(-1)$, we get the following result.

$\displaystyle \zeta(-1) = \sum_{n=1}^\infty \frac{1}{n^{-1}} = \sum_{n=1}^\infty n=1+2+3+\ldots$,

which is the sum of all natural numbers.  Now if you want to compute $\zeta(-1)$, you can click the link here.

Wolframalpha tells us $\displaystyle \zeta(-1) = -\frac{1}{12}$.

## more math jokes

Posted: November 4, 2012 in Mathematics
Tags: , ,

First of all let me make it clear that I have nothing against contravariant functors.  Some of my best friends are cohomology theories!  But now you aren’t supposed to call them contravariant anymore.  It’s Algebraically Correct to call them “differently arrowed” !!

In the same way that transcendental numbers are polynomially challenged?

Manifolds are personifolds (humanifolds).

Neighborhoods are neighbor victims of society.

It’s the Asian Remainder Theorem.

It isn’t Politically Correct to use “singularity” – the function is “convergently challenged” there.

## some calculus jokes

Posted: November 4, 2012 in Mathematics
Tags:

The guy gets on a bus and starts threatening everybody: “I’ll integrate you! I’ll differentiate you!!!”

So everybody gets scared and runs away.  Only one person stays.

The guy comes up to him and says: “Aren’t you scared, I’ll integrate you, I’ll differentiate you!!!”

And the other guy says: “No, I am not scared, I am $e^x$.”

~~~~~~~~~~~~~~~~~~~~~~~~

So the next day $x$ and $e^x$ are walking down a road when they see a derivative coming towards them.

“Ahh!” exclaims $x$.  “Help!  It’s a derivative!  He’s gonna got me!”, $x$ screams as he runs away in fear.

“Ha, I’m $e^x$, that derivative can’t hurt me,” said $e^x$.  Confidently, $e^x$ walks up to the derivative.  “Hello, I’m $e^x$!”

“Hello, ” says the derivative.  “I’m $\displaystyle\frac{d}{dy}$. “