proof by contradiction

All Numbers are Interesting

There are clearly many interesting whole numbers. For instance, 2 is the only even prime number, 3 is the first odd prime, 6 is a perfect number (the sum of the proper divisors is the number itself), etc. But did you know that all whole...

Continue reading...

How many Primes?

Are there¬†infinitely many primes? We’ll give a proof, due to Euclid, to show that there must be infinitely many primes. We will show that if there were only finitely many primes, it would lead to a contradiction. First note that if two numbers differ by...

Continue reading...

e is irrational

If e were rational, then e = n/m for some integers m, n. So then 1/e = m/n. But the series expansion for 1/e is 1/e = 1 – 1/1! + 1/2! – 1/3! + … Call the sum of the first n terms of this alternating series S(n). How good is...

Continue reading...