An attempt to demonstrate the beauty of maths by taking numerically knowledgeable students to one university level maths idea.

Draw a straight line. Label the left hand end as 0, the right as 1. Pick any number between 0 and 1. Lets use 0.5, which is the same as 5/10. Mark this on the line. Can we find a number between 0 and 0.5? Yes. e.g. 0.45, or 45/100. Can we find a number between 0.45 and 0.5? Yes, e.g 0.455, or 455/1000. Extrapoloating this, or repeating the process, you should be able to see that whenever we have two numbers m/n and x/y (where m,n,x,y are whole numbers) we can always find another number of the same form between the original two numbers. So we can fill the line with these types of numbers, apparently without leaving any gaps. (These types of numbers ie. produced by m/n, are called rational numbers. Whole numbers are also rational numbers since e.g. 13 is the same as 13/1).

Examples: 2*2=4, 3*3=9 . 4 is called the square of 2, 9 is called the square of 3. How do we describe 2 and its relationship with 4. We call 2 the 'square root' of 4. 3 is the square root of 9. 1.2 is the square root of 1.44.

An even number times an even number is always even.

odd*odd is always odd

odd*even is always even

An even number can always be divided by 2.

Can we find the square root of 2? That is can we find the rational number, m/n, that when multiplied by it self results in 2?

m/n can always be reduced to odd/even or even/odd (think about dividing by 2 if in any doubt)

if odd/even then (odd*odd)/(even*even)=2 which is

odd/even=2 (see part 3) or

odd=2*even or

odd=even. Hopefully it is clear this is impossible.

If even/odd then (even*even)/(odd*odd)=2 or

even*even= 2*(odd*odd) or

(even*even)/2 = odd*odd or

(even/2)*even = odd or

even=odd (use part 3 if necessary).

Hopefully, again, it is clear this is impossible.

Hence we have a contradiction ie. it is impossible to find a number of the form m/n such that when multiplied by itself equals 2. So there must be other numbers hidden between the rational numbers. Strange, what looked like completely filling the number line (rational numbers) doesn't. There are other numbers there as well. These are called (inventing names isn't always a easy) irrational numbers.

So what? The above is fairly simple, yet it required some genius's intuition to even suspect there was something hidden; it required his (and possibly others) creativity to find/create the proof. It demonstrates one method of arguing logically from a beginning to and end without any iffy steps - mathematical thinking. It shows a way of producing mathematical tools which we can use in technology (there is a direct line (albeit a long one)) from this to your PC. Hence mathematical thinking is beautiful, useful,relevant and important.

This is over 2000 years old. There are several versions of the immediate result of the discovery of irrational numbers: a goat was sacrificed, the discoverer and others were sworn to secrecy, the discoverer was expelled, the discoverer was killed.

An alternative, and better, introduction to mathematical beauty is in Chapter 1 of Life by the Numbers by Keith Devlin.