Dear Nrich-ers

I have recently been teaching Rational and Irrational numbers to a Year 10 top set at an 11-18 comprehensive (where I am Head of Maths).

I have used the following method to demonstrate whether a root is rational or irrational (and practice Prime factorisation and Negative powers work)

All integers can be written uniquely in prime factors, with counting number powers, eg 12 = 2² .3¹

All rational numbers are the ratio of two integers, which themselves can be
written in prime factors, eg 12/35 = (2² .3¹ ) / (5¹ .7¹ )

This can now be written as the product of prime factors, with integer powers
12/35= 2² .3¹ .5^-1 .7^-1

Square to get
(12/35)² = 2² .3¹ .5^-1 .7^-1 .2² .3¹ .5^-1 .7^-1
= 2⁴ .3² .5^-2 .7^-2 , and note all the powers were bound to be even.

Conclusions:
If a number has a rational square root, then it's powers when written in prime factors must all be even.
Consequently, sqrt(2) is irrational because 2 = 2¹ when written in prime factors
I expect pupils then to be able to deduce a similar rule for rational cube roots.

I don't claim this is easy, but it does tie together several higher level number ideas. Hopefully it may prove useful to some of you.


My problem (well, one of them) is that I don't know a proof that Pi is irrational (which is probably a terrible thing to admit to with a degree in Maths!) Can anyone supply one please?

Chris Davis

There is a proof on page 69 of Ian Stewart's book Galois theory (published by Chapman and Hall - used to be an Open University set text). It involves integration and would be difficult to copy by e-mail - I could scan it or send a photocopy if you have difficulty getting hold of the book. It's several levels more sophisticated than the irrationality of square roots, but uses the same principle of proof by contradiction. He goes on to give proofs that pi is not only irrational but transcendental, ie not the root of any polynomial equation with rational coefficients.

Jo Tomalin
Sheffield College

I have been reading your letters. They are quite interesting. I feel good that you pay so much attention towards teaching the kids, which is generally very dificult to find.

Regarding your difficulty about Pi, my opinion is that it will be extremely hard to teach the students in School level. Pi is not just irrational, it is not even an Algebraic number. It is a Trancendental number.

First of all to prove any thing about Pi, you need to give them its definition (one which is given in terms of 'e', whose definition is to be given in terms of series, which is generally taught in Universities).

Yours
Ravi Shankar Gautam.

Yours,

Andrew Rogers

NRICH Student Team

I don't know of any proofs of the irrationality of pi that don't involve Galois Theory. Teaching Galois Theory to explain that pi is irrational would be somewhat perverse even if it were possible to teach it to a class of schoolchildren!

When I was at school I was told that pi and e were irrational but that it was too hard to prove. In first year at university I was told that it was too hard to prove that pi was irrational. I didn't like it at the time, but now that I have some idea of how to prove it I have to agree.

Jonathan Kirby

for e^p, p^e and ep it is not known if any of these numbers are rational, irrational or transcendental. So don't start trying to use them :)

I'm just starting my second year at uni and I can't prove p is irrational. If they really want to see a proof, direct them to any book on Galois theory, there will be one if there, running to about 15 pages...

It is known that e^p is transcendental.

Gelfond's theorem states that if a is rational and not 0 or 1 and b is algebraic and irrational then a^b is transcendental. Now use the fact that

(-1)ⁱ=e^p

to see that e^p is transcendental.

Also a proof that pi is irrational can be fitted into a page: see here for example.

AlexB

Well, I've found another proof that pi is irrational, but again it's a tad hard for Y10!!
In fact it's a tad hard for me!

It uses differentiation, induction and the mean value theorem! (but at least no Galois thoery!)

Not very helpful, but never mind!
I suspect there isnt anything that Y10 will understand. Nice idea though!

Dave Lynch

The irrationality of pi was proved by J. H. Lambert in 1761 as follows. Lambert first gives the continued fraction expansion of tan x and then shows, by an argument of infinite descent, that if x is rational and nonzero then tan x is irrational.

Since tan (pi/4) = 1, this implies that pi is irrational.

Actually the idea behind the proof of the irrationality of pi isn't too difficult and it might be possible that Y10 would understand it. Granted they will almost certainly not understand all the algebra. Again I refer to the page that I gave earlier Pi Proof . Now I'll explain the ideas behind it:

It is a type of proof called proof by contradiction . This means that we assume it to be false and from this logically derive some clearly absurd fact. This technique is also used to prove that root 2 is irrational and this was on A-level when I did it.

So we assume that pi is rational. And we say that it is equal to the rational a/b.

How are we going to get an absurd fact from this? Well the idea is to construct from the fact that pi is a/b a function (which will be sin(x) x p(x) for some polynomial p --- but this isn't too important) which I'll call f(x). f(x) will have the following properties:

(1) that the integral from 0 to pi will be a positive integer.

(2) the function will never be less than 0.

(3) the function isn't always 0.

(4) the integral will be less than 1.

Hopefully you can see that this is a problem... The integral is less than 1 and is positive so it must be equal to 0. But the only way to integrate a function that is always at least 0 to get 0 is if the function is constantly 0. But (3) says this isn't true.

Now the actual polynomial that you pick is sort of irrelevant - what really matters is the idea (this is true for most proofs in mathematics, in my opinion). And it turns out that by being even more smart you can prove that pi is transcendental in EXACTLY the same way.

If you want all the details of what the polynomial is then look at the page I reference - it is a complete proof.

Of course the message above about the original proof is as good for putting over the general ideas. The only reason my way is better is that it is considerably shorter to prove everything.

AlexB.