The story of Euclid and the infinitude of primes

Once upon a time, there lived a Mathematician named Euclid.

Euclid came up with an ingenious method of proving that there are infinitely many prime numbers. Prime numbers are whole numbers that only have two factors, one and itself. For instance, 7 is a prime number while 6=2×3 is not.

The proof is by contradiction. Suppose that there are only a finite number of prime numbers, p_1, p_2, \cdots, p_n.

Now, we consider the number P=p_1 \cdot p_2 \cdots p_n+1.

P is either a prime number, or it is a composite number.

If P is a prime number, then we have just contradicted the assumption that there are only a finite number of prime numbers p_1, p_2, \cdots, p_n!

If P is a composite number, then, it has a factor (smaller than P). However, none of p_1, p_2, \cdots, p_n can be a factor since P divided by those primes will leave a remainder of 1! Hence, P has another factor that is not in the original list, contradicting the initial assumption once again.

(Strictly speaking, Euclid’s proof is not by contradiction, instead he used a similar argument to show that given any finite list of primes, there is at least one other prime.)


As of now, the largest known prime to mankind is 257,885,161 − 1, a number with 17,425,170 digits! (http://en.wikipedia.org/wiki/Largest_known_prime_number)



Euclid’s Elements


Math Quotes

Famous Math Quotes

  1. If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.  ~John Louis von Neumann
  2. Pure mathematics is, in its way, the poetry of logical ideas.  ~Albert Einstein
  3. A mathematician is a device for turning coffee into theorems. ~Paul Erdos
  4. Give me a place to stand, and I will move the earth. ~ Archimedes
  5. Mathematics is the door and key to the sciences. ~ Roger Bacon
  6. The essence of mathematics is its freedom. ~ Cantor
  7. A youth who had begun to read geometry with Euclid, when he had learnt the first proposition, inquired, “What do I get by learning these things?” So Euclid called a slave and said “Give him threepence, since he must make a gain out of what he learns.” ~ Euclid
  8. Mathematics is the queen of the sciences and number theory is the queen of mathematics. ~ Gauss
  9. Mathematics knows no races or geographic boundaries; for mathematics, the cultural world is one country. ~ Hilbert
  10. When you can measure what you are talking about and express it in numbers, you know something about it. ~ Kelvin

 

Geometry and Abraham Lincoln; O Level Maths Tuition Group

Source: http://www.mathopenref.com/euclid.html

At age forty, Abraham Lincoln studied Euclid for training in reasoning, and as a traveling lawyer on horseback, kept a copy of Euclid’s Elements in his saddlebag.  In his biography of Lincoln, his law partner Billy Herndon tells how late at night Lincoln would lie on the floor studying Euclid’s geometry by lamplight. Lincoln’s logical speeches and some of his phrases such as “dedicated to the proposition” in the  Gettysburg address are attributed to his reading of Euclid.

Lincoln explains why he was motivated to read Euclid:

“In the course of my law reading I constantly came upon the word “demonstrate”.  I thought at first that I understood its meaning, but soon became satisfied that I did not.  I said to myself, What do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof?
I consulted Webster’s Dictionary. They told of ‘certain proof,’ ‘proof beyond the possibility of doubt’;  but I could form no idea of what sort of proof that was. I thought a great many things were proved beyond the possibility of doubt, without recourse to any such extraordinary process of reasoning as I understood demonstration to be.  I consulted all the dictionaries and books of reference I could find, but with no better results.  You might as well have defined blue to a blind man.
At last I said,- Lincoln, you never can make a lawyer if you do not understand what demonstrate means;  and I left my situation in Springfield, went home to my father’s house,  and stayed there till I could give any proposition in the six books of Euclid at sight.  I then found out what demonstrate means, and went back to my law studies.”
Iconic black and white photograph of Lincoln showing his head and shoulders.

Tips on attempting Geometrical Proof questions (E Maths Tuition)

Tips on attempting Geometrical Proof questions (O Levels E Maths/A Maths)

1) Draw extended lines and additional lines. (using pencil)

Drawing extended lines, especially parallel lines, will enable you to see alternate angles much easier (look for the “Z” shape). Also, some of the more challenging questions can only be solved if you draw an extra line.

2) Use pencil to draw lines, not pen

Many students draw lines with pen on the diagram. If there is any error, it will be hard to remove it.

3) Rotate the page.

Sometimes, rotating the page around will give you a fresh impression of the question. This may help you “see” the way to answer the question.

4) Do not assume angles are right angles, or lines are straight, or lines are parallel unless the question says so, or you have proved it.

For a rigorous proof, we are not allowed to assume anything unless the question explicitly says so. Often, exam setters may set a trap regarding this, making the angle look like a right angle when it is not.

5) Look at the marks of the question

If it is a 1 mark question, look for a short way to solve the problem. If the method is too long, you may be on the wrong track.

6) Be familiar with the basic theorems

The basic theorems are your tools to solve the question! Being familiar with them will help you a lot in solving the problems.

Hope it helps! And all the best for your journey in learning Geometry! Hope you have fun.

“There is no royal road to Geometry.” – Euclid

Animation of a geometrical proof of Phytagoras...
Animation of a geometrical proof of Pythagoras theorem (Photo credit: Wikipedia)