This is also from The Irrationals - though I had to ask for assistance over at Stack Exchange to get the right answer (as is so often the case the solution is reasonably obvious when you are presented with it).
Anyway, the question is: given a regular pentagon (of sides with length of 10 units) which is inscribed by a circle, what is the diameter of the circle?
This figure helps illustrate the problem:
We are trying to find and we know that .
If we knew then we could answer as . We do not know that but we do know that and hence, from the identity: .
From our knowledge of the pentagon with sides of unit length (you’ll have to trust me on this or look it up – it’s too much extra to fit in here) we also know that , where is the golden ratio.
Hence … well, the rest is left as an exercise for the reader :)
In a fortnight my Code Club will end with the last Friday of the school year – and I want to leave my small (4) group with a useful present – a book about maths or computing that will engage and stretch them (they are 10/11 and bright but, as far as I can tell not budding Nobel winners – though I could always be wrong!).
If is rational then and where and are the smallest nominator and denominator possible, i.e, in the lowest terms.
Hence and so must be even (as two odd numbers multiplied always give an odd number – let and be even numbers then ). Hence .
But we also have and thus and and so is even.
So we have and sharing a common factor of 2, so they cannot be the nominator and denominator of the fraction in the lowest terms.
But if and are both even then they share a common factor of 2 and : implying that and , an obvious contradiction: hence cannot be a rational.
Update: I have made the final step of this shorter and clearer.
Further update: I have been told (see comments below) I would have been better sticking with a clearer version of the original ending ie., we state that are the lowest terms. Then, plainly as they have a factor in common (2) they cannot be the lowest terms and so we have a contradiction. Would be great if someone could explain why we cannot use the contradiction.
And another update: Should have stuck with the original explanation – which I have now restored in a hopefully clearer way. The comments below are really interesting and from serious mathematicians, so please have a look!
As the most prolific mathematician of all time I am rather disappointed he hasn’t been deemed worth a “Google doodle“.
Erdős was victimised in the McCarthy era in the US and certainly seems to have had a warm relationship with the Hungarian Communist authorities (though there is nothing to suggest he was a spy), though eventually boycotted the country over its treatment of Israeli citizens.
I am puzzled, though, by its treatment of Euclid’s famous proof of the infinite order of the set of primes.
Not because it gets the proof wrong – but because I do not understand the answer it gives to one part of one exercise.
Now the proof (this is my summary not the book’s) runs like this:
Suppose you decide that the number of primes is finite, the set (where is 2 and so on). You then form a number from the product of these primes plus 1 – . This number cannot be a prime and yet is not divisible by any of the primes. Hence we have a contradiction and the set of primes must be infinite.
Now the book asks the following:
The proof of [the theorem] gives a method for finding a prime number from any in a given list of prime numbers:
(a) Use this method to find a prime different from 2, 3, 5 and 7
(b) Use this method to find a prime different from 2, 5 and 11
Now, the first seems easy enough:
and the book agrees with my calculation.
But what of the second?
Well, , but that’s not a prime at all. The book gives the answer as “3 and 37″ (which obviously are primes) but I cannot see what formal method is used from the list of “2, 5 and 11″ to generate these numbers.
The obvious suggestion is that I have missed the “method” in the theorem as set out in the book, but I have reread it many times and I cannot see where I have missed anything (it is really just a longer version of my summary).
Can someone set out the formal method that would work with a partial list such as 2, 5 and 11?