One of the greatest mathematical proof in history is that there are infinitely many Prime Numbers.  It was a proof that was simple yet in a way brilliant.  Brace yourself and lets journey into the mind of Euclid...

What is a prime number?

C’mon, a high school student knows this.  In high school we are taught that any number that can only be divided by 1 and itself is a prime number.  Almost right, but of course, don’t forget that 1 is not a prime number.  My definition of a prime number is an atom of a composite.  When I say atom, I mean that it is the smallest possible object that can no longer be divided (Of course we all know that atoms can be divided releasing an enormous amount of energy.  But the atoms were once believed to have been the smallest form of matter and cannot be divided).  So the fact that atoms were once been believed to be the smallest form of matter, then everything in this world can be divided into atoms.  So are composite numbers.  All composite numbers can be divided into prime numbers. 

Consider the composite number 24.

24 = 2x2x2x3

In the example above 24 can be divided into a group of 2s and a 3, which are prime numbers.

But there is a property of prime numbers that intrigues most mathematicians. This property is that as long as a group of positive whole numbers moves from 2 to positive infinity, the number of prime numbers in a particular range decreases. This is quite vague is it?  Okay let’s try to illustrate it.  How many prime numbers are there in the range of 1 to 20?  You might be surprised to know that almost half are prime numbers.  However you cannot say that same is true for the range 21 to 40.  In fact, as long as the number increases even if you maintain the range as constant, you will notice that the number of prime numbers decreases.  In other words, it is more difficult to determine larger prime numbers.  So now, the question is, are there a finite set of prime numbers?

Euclid Says No!

Sorry to spoil the surprise too soon.  But the answer is “No”, there is no such thing as a finite set of prime numbers.  Why?  Ask the master…

This problem has puzzled most great thinkers before Euclid proved that prime numbers where indeed infinite.  But my goal is not to present the conclusion.  What amazed me was the solution Euclid presented - an extremely creative and brilliant solution.  So, enough with this crap and let’s start our journey…

Who says all assumptions are wrong?

I might argue that all of us are fit to be a mathematician.  In fact, gossipers are fitted to be mathematician or a scientist.  Okay, maybe I’m exaggerating it, but I would only like to present my point.  Gossipers like to assume based from unsubstantial yet kind’a logical issues.  They tend to rename the term “it may be” to “it is”.  We hate these people (Don't worry, God will judge these people). But never deny the fact that in one point in our lives we have been gossipers as well. Scientific or mathematical discovery always starts with an assumption, a hypothesis.  It may even exist without any substantial evidence.  Michael Faraday thought that an invisible string attaches a bar of magnet from one side to another.  Who would have thought that magnetic fields are really invisible forces acting the same way Michael Faraday described it based on his unsubstantial assumption.  But nevertheless it was one of the greatest discoveries.  But what happens if a mathematician makes a false assumption?  Then you mean he’ll be proving something all his life without even reaching a conclusion?  Nice question, but I would say “No”.  I would argue that false assumption is not at all a weakness for a mathematician.  In fact, in certain instances this could even produce positive results.  In the case of Euclid’s proof for the infinite prime numbers, believe it or not, was based on false assumptions.  I’m not saying that he discovered an infinite prime number proof based on a false assumption Euclid thought was true, but rather by reconstructing the problem with false assumptions even if deep within he knew it was false.  Just like those people who are gossipers.  They probably know they’re saying things that are not true, but still continue to insist that there is a truth in what they are saying. 

The Assumptions

 I’m sure you are already excited to know that proof.  Alright then, here goes…

 The first instinct of any person or even a mathematician is to start plotting a list of all the prime numbers in their head or in a piece of paper.  In this case he might start listing all the possible prime numbers as illustrated below:

2, 3, 5, 7, 11, 13, 17, 19 …

And of course he will notice that “This seems to go on forever, and it will take me a lifetime to prove this one!”

Anyway don’t feel bad if you knew you where going to start at proving using this strategy.  I have a strong feeling even Euclid tried this one.  It is just a first step in order to acquire a proof.  Of course you need to provide your assumptions, and one way to do this is by analyzing a concrete illustration.

In this case Euclid might have wrote this illustration and tried to acquire some patterns from it.  Maybe, he thought:

“What will happen if I add 2 with 1? Hey, it produced 3 and 3 is a prime number!”

“What if I multiply 2 with 3?  It produced 6?  It doesn’t make any sense. 6 Isn’t a prime number.  Wait, what if I add 1 with 6, then it will produce 7, which is a prime number… hmmm… wait a second…”

“If I multiply 2, 3, 5 then I get 30.  If I add 30 with 1 then I get 31 which is a prime number!”

“I think I see a pattern.  If I multiply 2, 3, 5, 7 then I get 210, adding 1 will yield 211, which is also a prime number!”

“Could it be true that if N = 2x3x5x7x…P where P is infinite, then N+1 will always be a prime number?”

So from the way I presented it, Euclid had arrived to an assumption that N = 2x3x5x7x…P where P is infinite will yield a prime number for N+1.

We have not proven anything yet, this was just an assumption.  So from the given, how did Euclid proved that prime numbers are infinite?

First he formulated an assumption that he might have known to be false.  In this case he thought, “What if I assume that P is not infinite but rather the largest prime number, then what will happen?”

So in this case we have the following given:

N = 2x3x5x7…P
P = largest prime number
N+1 = A prime number

But you will notice, that the given itself is incomprehensible.  If P is the largest prime number, and if N+1 is a prime number, then in this case N+1 should rather be the largest. So this problem goes on and on proving that P should be infinite.

Okay I’m sure you’re not totally convinced.  How about the other factor such that N+1 is a prime number?  Euclid only assumed that N+1 is a prime number.  How can he prove that N+1 is indeed something that he claims it to be?  Good point. So now let’s move on to the next step of proving that N+1 is a prime number.  Again, he proved this using another false assumption.  He assumed that N+1 is not a prime number but rather a composite number.  I have mentioned that composite numbers can always be divided into several combinations of prime numbers such as shown in the example 24 = 2x2x2x3.  If N+1 is a composite number then we know that N+1 can be divided by a prime number M which is smaller than P.  But of course, we also know that we can divide N by M since N is the combination of all prime numbers.  From these given we can almost conclude that the assumption is indeed false.

Before I further proceed, I would first like to introduce the concept of a remainder.  I’m sure most of us know what a remainder is.  It may have been even thought to you since elementary.  Anyway let’s review.  If you divide 5 by 2, then you get a remaining 1.  1 is called the remainder of 5 divided by 2.  Likewise the remainder of 23 divided by 6 is 5.  Let us use a notation symbol “%” known as “modulo” or “the remainder of”.  So we know that 5 % 2 = 1 and 23 % 6 = 5.

Let’s go back to the given that was recently derived:

Since we know that N is the set of all prime numbers, then we can conclude that by dividing N by M will yield a remainder 0:

N % M = 0

Since we assumed that N+1 is a composite number then N+1 divided by M will also yield a remainder 0:

N+1 % M = 0

So we have N % M = N+1 % M?

 Outrageous!

If you will further examine the equation above, the only possibility for this equation to be true for whole numbers in the range 1 to positive infinity is if M = 1.  However, we defined M as any prime number smaller than P, and that I have mentioned that 1 is not at all a prime number.  Again, obviously we have a conflict.  So this proves that N+1 can never be a composite number.  So if N+1 is not composite then what is it?  We can therefore conclude that it is a prime number.  So proving N+1 must be prime number will in turn prove the first statement to be true.  We therefore conclude that prime numbers are infinitely many.

 Often times we are faced with problems and we tend to look for a solution using the normal approach that is by proving the truthfulness of a situation.  But don’t forget that there is also a term called disproving.  Proving and disproving are both important in facing problems.  Take this situation for example when taking a multiple choice exam:

 Given a problem you will have to solve it with only 4 possible choices.  Say, you are given a vocabulary problem.  The question is to find the word that best describes the given sentence.  Our first instinct is to look for the word from the list that you could directly associate for the given sentence.  But since you did not find any word that can be associated with the sentence, then the next step is to scan the word choices and disprove its association with the given sentence. Once you’ve done this, you might be surprised to find out that there are exactly 3 words cancelled out.  Obviously, since you are left with 1 word, then that must be the answer.  This is just a simple solution that was done quite creatively.  And if you would try to analyze Euclid’s proof, it was indeed creative yet simple. 

- Taki Cuevas Oct 27, 2002

I'm sure you have heard about Mathemagics, which is always advertised in home TV shopping.  They claim that you can learn to add, multiply, or divide big numbers in your head.  I never really bought their product, but I kinda know how they do it.  I'm not sure if this is really how they do it, but nevertheless it works for me.  I only know a couple of their tricks, and I'm still trying to figure out the others.

Adding Large numbers

Whenever I add large numbers in my head, often time I do it from left to right rather than the conventional right to left, with the carry stuff.

for example:

569 + 478 = ?

to answer this, I do it this way in my head, in this exact order:

500 + 400 = 900

60 + 70 = 130

900 + 130 = 1030

9 + 8 = 17

1030 + 17 = 1047

so the answer is 1047.  Try to do it with other numbers. Try doing this 578334 + 42344 = ?

Adding 9's

Here is a simple trick when you add a 9 on a number.  When you add 9 the first digit goes up, and the second digit goes down.

example:

47 + 9 = 57

notice that 4 goes up to 5, and 7 goes down to 6.

But in some cases such as 69 + 5, you can see that you are adding 9 + 5.  Don't touch the 9 so 6 goes up to 7 and instead of 9, 5 should instead go down to 4.  You'll get an answer of 74.

Subtracting 9's

Well this is just similar to adding 9's.  Except you are doing it in reverse order.  In this case, the first digits goes down and the second digit goes up.

example:

47 - 9 = 38

notice that 4 goes down to 3 and 7 goes up to 8.

Some basic terms

There are some basic terms that can be useful in understanding better some calculations.

take half means divide by 2

double means multiply by 2

add a zero to the end means multiply by 10

move 1 decimal place to the left means dividing by 10

move 1 decimal place to the left means multiply by 10

get the first right digit of a given number means divide that number by 10 and get the remainder

Multiplying numbers

 When multiplying by 2 just double.

example: 5 x 2 = 10.

When multiplying 4 just double and double.

example:  4 x 4 = 16 double 4 equals 8 and double 8 equals 16

When multiplying 5 just add 0 to the end and then take half.

example: 43 x 5 = 215

add zero to 43 equals 430 then take half equals 215.

When multiplying 8 just write a zero at the end and subtract double the original number.

example: 43 x 8 = 344

add a zero to 43 equals 430, multiply 43 by 2 equals 86, subtract 430 with 86 equals 344

When multiplying by 9 just add zero at the end and subtract the original number.

example: 9 x 57 = 513

add a zero to 57 equals 570, subtract 570 with 57 equals 513

When multiplying 11 just add and insert.

example: 36 x 11 = 396

add 3 and 6 equals 9, insert it between 3 and 6 getting 396

note: if for example 78 x 11 was encountered, notice that 7 + 8 = 15.  All you need to do is insert 5 between 7 and 8 equals 758 and adding a carry to 7 making the answer 858.

When multiplying 12 just add a zero at the end and double the original number and add.

example: 36 x 12 = 432

add a zero to 36 equals 360, double 36 equals 72, add 360 with 72 equals 432.

When multiplying 15 just add a zero at the end, take half and add.

example: 15 x 56 = 840

add a zero to 56 equals 560, take half of 560 equals 280 then add 560 with 280 = 840

When multiplying 25 just add two zeroes at the end and take half then take half again.

example: 25 x 56 = 1400

add two zeroes to 56 equals 5600, take half of 5600 equals 2800, take half of 2800 equals 1400.

When multiplying 50 just add two zeroes at the end and take half.

example: 50 x 56 = 2800

add two zeroes to 50 equals 5600, take half of 5600 equals 2800

When multiplying 125 just take half and half and half and move 3 decimal places to the right.

example 125 x 56 = 7000

take half of 56 equals 28, take half of 28 equals 14, take half of 14 equals 7, move 3 decimal from the left equals 7000

Dividing numbers

Well this still quite hard.  I only know a couple of them.

When dividing a number with 5 just double the number and move one decimal place to the left.

example: 56 / 5 = 91.2

double 56 equals 112, move one decimal place to the left equals 11.2

There is also a technique in dividing.  This is quite common sense.  you just need to know how to factor.  In fact this technique is not only useful for division, but for other operations as well.

When you encounter a number such as 966 / 3, all you need to do is to factor 966 in such a way that it will be easier for you to divide the whole number by 3.  In this case dissect 966 into 900 and 66. Now we know that it is easier to divide 900 or 66 with 3. We all know that 900 / 3 equals 300 and 66 / 3 equals 22.  Now all we need to do is to add 300 and 22 and we'll get the answer 322.  See the picture, you can even use this in multiplying large numbers.  If a number is too large, just dissect it into a simpler form and work it out to get the answer.

Well now, calculating big numbers in your head is not really a magic trick.  Its just retransforming mathematical equations into a much simpler form.  These are just some of the techniques I use, and I'm sure you can create your own techniques.  That is why if you see people in home TV shopping claiming they have invented formulas for calculating big numbers, and that they are planning to put a patent on these formulas, then that's crazy.  Its just plain and simple common sense.