### Math Post the First: Dividing and Primes

Well, here goes... math post the first. Not background, real stuff.

I thought we'd start off with a little number theory; most people do actually know what integers are, and it's not as though this early chunk of math is hard or anything. It should be easy enough to understand what I'm saying here without anything like background.

The basic object of study here is the collection of integers: zero, one, two, three, and so on. Negatives, but not fractions. For at least the next while, I'll never even hint that fractions exist (much though it pains me to not be able to share with you, right now, the simple, brilliant, unforgettable, and world-shaking for the ancient Greeks at least, proof that the square root of 2 is irrational). This is important, because it's required to sensibly parse our first term, "divides," which is the same concept you called "divides evenly" back in elementary school. Formally, we say that

There are a few first easy results to be deduced. First, if a given number

Some numbers don't divide any number not themselves: zero, for instance, can never be multiplied by anything to make something other than zero. Zero is actually the only number with this property, as should be pretty clear. Other numbers divide every number: since for any

The reason I bring all this up at the moment is the concept of a prime number, whose elementary school definition is a number only divisible by itself and one. We'll have to go slightly further: a non-unit

I thought we'd start off with a little number theory; most people do actually know what integers are, and it's not as though this early chunk of math is hard or anything. It should be easy enough to understand what I'm saying here without anything like background.

The basic object of study here is the collection of integers: zero, one, two, three, and so on. Negatives, but not fractions. For at least the next while, I'll never even hint that fractions exist (much though it pains me to not be able to share with you, right now, the simple, brilliant, unforgettable, and world-shaking for the ancient Greeks at least, proof that the square root of 2 is irrational). This is important, because it's required to sensibly parse our first term, "divides," which is the same concept you called "divides evenly" back in elementary school. Formally, we say that

*a*divides*b*if there is a number*k*such that*ak=b*. Two divides six because 2*3=6, for instance. Five doesn't divide six, as I'm sure is clear.There are a few first easy results to be deduced. First, if a given number

*a*divides a number*b*,*a*also divides any multiple of*b*. This follows more or less immediately from the associative law, a property of multiplication and addition that you probably learned without thinking about it in middle school. Multiplication and addition are binary operations: two numbers go in, one comes out. But you can also multiply or add a big line of things: 1+2+3+4+5, for instance. If I were being completely technical, this is utterly incorrect; should we add the two and three first, or start from the left, or what? Handily, it doesn't matter. As you almost certainly already know, the order in which I do the adding doesn't matter, just which numbers are involved, and this is the associative law. If you toss in the distributive law (that is, a(b+c)=ab+ac), you can also see that if*a*divides two numbers, it divides their sum. These two facts together are the "linear comination lemma."Some numbers don't divide any number not themselves: zero, for instance, can never be multiplied by anything to make something other than zero. Zero is actually the only number with this property, as should be pretty clear. Other numbers divide every number: since for any

*x*, 1**x*=*x*, one is one of these. Similarly, negative one divides all the numbers because every number has a negative; such numbers are called "units." I mention this seemingly trivial situation because I'll need it soon, but also because some large number of posts down the road the concept of unit will become deep and interesting and important, but only after it gets generalized to the many more complex objects on the docket.The reason I bring all this up at the moment is the concept of a prime number, whose elementary school definition is a number only divisible by itself and one. We'll have to go slightly further: a non-unit

*p*is prime if whenever*ab=p*, either*a*or*b*is a unit. We need this extension because I also want to talk about negative numbers; seven is divisible by minus seven, so "itself and one" isn't quite going to cut it. I want to exclude units because the nice property of integers we all remember is that they uniquely factor into products of primes. I'll say a bit more about it on Wednesday, but for now I'll just give the basic property of primes: if a prime divides a product, it divides one of the factors. It's not that hard to understand why, but I'll leave it as an exercise until next time.