Interesting maths? Surely an oxymoron (part 2)

When I last left off we were talking about extending the Natural Numbers to contain values less than zero. And some clever chap had suggested just having the same thing as the natural numbers, just going the opposite direction down the number line. This set of numbers has a name too, it's called the Integers and is defined by I. Using our set notation, we can say the integers are the set of whole numbers {-∞...∞}. So, like the Natural Numbers, the Integers extend off to infinity, however, they also extend off to negative infinity. This means that unlike the Natural Numbers, we can subtract any integer from any other integer. It also means we have to define what happens when you multiply, add and subtract negatives.

It's worth taking an aside here to define a couple of properties our number systems thus far have, which are vitally important to their use. These properties are Distributivity, Associativity and Commutatvity.
Commutativity is what tells us that 3 X 4 = 4 X 3. This might seem trivial, but it's an important property, and it doesn't hold true for all operations in I (or N for that matter); Subtraction is not commutative. 3-4 is not the same as 4-3. More formally;
If a and b are Integers, a + b = b + a, a x b = b x a for all a, b.

Associativity is similar to commutativity. It states that when we perform a series of operations like (3 x 5) x 2, it is equivalent to 3 x (5 x 2) which is equivalent to 3 x 5 x 2. Again multiplication and addition are associative.
More formally
If a, b, c exist in I, a(bc) = (ab)c = abc for all a, b, c.

Distributivity is the final property we have to explain. It's easiest shown by example, as above; 3 x (2 + 5) = 3 x 2 + 3 x 5. i.e we can distribute the multiplication operation to the numbers in the brackets then add the products. So multiplication distributes over addition of integers.

Formally

If a, b, c exist in I, a(b + c) = ab + ac for all a, b, c.

With those properties (which are pretty important) defined, we can talk about integers properly.
If you recall all the way back to the previous entry, we needed to expand our number system to stop subtraction breaking it (the natural numbers were "open" under subtraction), whereas with the Integers, we can take any number from any other, meaning that the Integers are "closed" under subtraction. This gives negative numbers, which, lets face it, are pretty weird and unituitive. Values on nothing? What? But like most unitiuitive things in maths, we can slap some rules on them and use them. We already defined the properites our number system should have, so the negative numbers should obey these. Stepping back a section, we can define each of our natural numbers as 0 + that number (or even more specifically, 0 + 1 a number of times equal to our number). Slightly confusing I know, let's use an example to illustrate;

100 = 0 + 100 = 0 + 1 x 100.

We can do this for the infinity of positive numbers. What does this let us do for our new negative numbers?

Lets let each negative number be equal to 0 minus the value of that number were it positive. Confused? Probably. Illustrative example time!

-3 = 3 subtracted from 0 = 0 - 3 = 0 - 1 x 3.

Hey look, this is basically the same as for our positive numbers with a sign switch. This means that any rules we have for positive numbers should probably apply to our negative numbers!

So lets define our operations using our new negative numbers.

0 + a = a
a + a = 2a
2a - a = a
a - a = 0
=>
0 - a = -a
-a + a = 0
-a + -a = -2a
2a + -a = a

This seems pretty trivial, but we can now use these basic rules to add or subtract any negative number we like. We also know that multiplication is just repeated addition, so we can also multiply negative numbers. Truly we are the masters of our world.

Are we? Really? There's one basic operation we haven't discussed. Division (or if you like, taking the ratio between 2 numbers). Aha, I hear you say, we can divide 4 by 2 and get 2 in both I and N. Yes, we can. In fact we can divide any number by any other number which divides it exactly with no remainder in the integers and natural numbers. Somewhat simpler; we can divide any number that gives us a result that is an integer value, not a fraction. I isn't rich enough to include fractions. It's "open" under division. We need to expand our number systems again for a system that's closed under division.
This number system isn't really interesting enough to get its own post. It's called the Rational Numbers (from ratio between 2 numbers) and it's denoted by Q. It obeys the same rules as, and has the same properties as the integers, except it is closed under division. You can take any integer, and divide it by any other and have a result in Q. It's also worth noting that there is an infinity of fractions between every integer. Yup. That's a lot of fractions.

Formally;
If a, b are integers, there exists a rational number c where c = a/b

So, we can add, subtract, divide and multiply as much as we like. Surely we're done expanding our number systems now. We could be, but before we do, have a look at the length of the hypotenuse of a triangle with 1 unit sides. Quick refresher: . You can assume a, b, c are all integers. I'll be back later when you're done ;)