## Numbers, An Introduction

### Introduction

Building upon the results of earlier Greek and Egyptian scholars,
Euclid (biography)
laid the foundations for number theory and geometry in his comprehensive book, The Elements.
It is difficult to know how much of this work was his creation -
most of it was a careful compilation of earlier results.
The accomplishment is impressive nonetheless.
We still talk about Euclid's gcd algorithm,
euclidean geometry, and euclidean domains.
The beauty of his work stands today, and for all time.
Euclid and his contemporaries made the same assumptions about numbers that you and I would make today.
We know what numbers are, we know how to add and subtract,
we know that the order of addition doesn't matter,
we know how to multiply,
we know that some numbers are divisible by others,
we know that some numbers cannot be divided by any others (called primes),
and somehow we know that primes are important, or at least beautiful.
Euclid knew this too, but he didn't know that cryptography,
the first practical application of prime numbers,
would not arrive for another 2,000 years.