Systems of NumbersEdit
Geometric algebra is an example of a system of numbers. In particular, it's an example of what mathematicians call an algebra over a field. But since this is not intended to be a book solely for mathematicians, this page is here as an attempt to explain what exactly this means, give a general impression of what systems of numbers are and how they can be built up, and show where geometric algebra fits in. You probably already know most of what's in this section, and it is presented primarily for comparison's sake, but seeing it presented in this way may raise some interesting questions.
Perhaps the most basic system of numbers is the natural, or counting, numbers. This is the familiar set of numbers 0, 1, 2, 3, etc. One of the fundamental properties of the natural numbers is that for every natural number n, there is a natural number n + 1 (which is never equal to n) that comes after it. Since this is true no matter how big n is, there are infinitely many natural numbers. We'll call the first number that comes after n the successor of n. One reason this property is fundamental to the natural numbers is that any natural number can be represented by the successor function applied to 0 some number of times. In fact if you have another set of things in which each one has a successor and there is only one that is not the successor of any other, your set of things can be identified with the natural numbers -- just call the thing that isn't the successor of anything else 0, and go from there. Another reason the successor operation is fundamental is that the basic operations of addition and multiplication can be reduced to it. The sum of two natural numbers a and b is the bth successor of a. Likewise, the product of a and b is the sum of a and itself repeated b times. a to the power of b is defined similarly, as the product of a with itself repeated b times.
The operations of addition and multiplication that we've introduced have a few important properties in common. If you add three numbers, it doesn't matter whether you start by adding the first and second numbers, and then add the third one to your answer, or if you start by adding the second and third numbers, and then add your answer to the first number. In equation form, this is (a + b) + c = a + (b + c). The corresponding statement is true for multiplication. That is, a(bc) = (ab)c. This is called the associative property, since it is a statement of the fact that it doesn't matter in which order the terms (or factors) are associated. Both versions of the associative property are true for all natural numbers a, b, and c. A statement that is true for all values of the variables in it is called an identity, and mathematicians sometimes say that it is identically true or holds identically. This can be confusing because it conflicts with the everyday definition of identical, but the meaning is usually clear in context.
Another important identity that's true for both addition and multiplication is the commutative property, which is the statement that the order in which you add (or multiply) two numbers doesn't matter. That is, for all natural numbers a and b, a + b = b + a and ab = ba. Also, for all natural numbers a, 0 + a = a + 0 = a and 1a = a1 = a. Because of this, 0 is called the identity element for addition and 1 is called the identity element for multiplication -- yet another sense of the word identity. An important identity that relates multiplication to addition is the distributive property, the statement that, for all natural numbers a, b, and c, a(b + c) = ab + ac.
In the fairly recent past, mathematicians have begun to study sets of "numbers" that share some or all of these properties (including ones we'll discuss later on), and have come up with names for certain combinations of properties. Any set of elements with an operation that satisfies the associative property and has an identity element is called a monoid. In particular, the natural numbers with addition form what is called (for rather obvious reasons) a commutative monoid. The natural numbers with multiplication also form a commutative monoid, but of a rather different sort. To see the difference, notice that any natural number can be formed by adding 1 and 0 in various combinations, but there is no (finite) set of numbers that you can build all natural numbers out of using multiplication in a similar way. There are, however, infinite sets with this property -- take, for instance, the set containing 0, 1, and all prime numbers.
Inverses and GroupsEdit
One of the most basic questions we can ask about natural numbers is this: given two natural numbers a and b, what is a number that, when added to a, gives b? Let's call this the subtraction problem for the pair (a, b). There is never more than one natural number that answers the subtraction problem for a given pair (a, b); we call that number, if it exists, b - a. For example, the answer to the question "What number, when added to 6, gives 13?" is 13 - 6 = 7. But suppose we ask "What number, when added to 12, gives 9?" There is no answer. None of the numbers n in the set we have just constructed has the property that 12 + n = 9. But suppose there were such a number. If it obeyed all the same rules as the natural numbers, we would be able to say that it also satisfies 20 + n = 17, since 9 = 9, and therefore 8 + 9 = 8 + 9 -- but 12 + n = 9, so 8 + (12 + n) = 8 + 9. Using the associative property, we can show that this means (8 + 12) + n = 8 + 9, and so, since we know 8 + 12 is 20 and 8 + 9 is 17, 20 + n = 17. By running the same kind of reasoning in reverse, we can show that our number n must also satisfy 3 + n = 0. Symbolically, then, we can say that our number is 0 - 3, since it solves our original problem for 3 and 0 just as 13 - 6 = 7 solves it for 6 and 13. Usually, we drop the 0 and write this number as -3. By adding the numbers -n for all natural numbers n greater than 0 to our set, we end up with a solution for the subtraction problem, not only for the pairs (n, 0), but for all pairs of numbers (a, b) in our new set. (Notice that we don't have to add -0 because the subtraction problem already has a solution for the pair (0, 0) - namely 0.)
We need to be careful here to make sure that what we're talking about -- the set of integers -- actually exists, and that we can define addition and multiplication in a meaningful way on this new set of numbers. (Of course, the integers are a well-known system, and you almost certainly already know how to add and multiply them, but this is intended as a demonstration and a clarification.) One way to do this is to represent a number by the set of pairs for which it satisfies the subtraction problem. We do this by putting together a set of rules that tell us whether two pairs represent the same number -- what mathematicians call an equivalence relation. In this case, we'll say that the pair (a, b) is equivalent to the pair (c, d) if and only if (a + d = b + c). Then, writing b - a for the set of pairs related to (a, b), we define addition and multiplication by the rules we want them to follow: (b - a) + (d - c) = (b + d) - (a + c) and (b - a)(d - c) = (ac + bd) - (ad + bc). Now all we have to do is name the sets n and -n for each natural number n, and this is easy - just let n be the set of pairs we've been calling n - 0 and -n be the set of pairs 0 - n. (Can you see how this all fits together? Hint: Try drawing a picture.)
Our new set of numbers, with addition, has a new, important property -- every integer has an inverse element, or a number that it can be added to to get the identity (that is, 0). A monoid with this property is called a group -- a name which can be confusing, especially since the everyday meaning of group, unlike the mathematical definition, has no notion of structure associated with it. A commutative group, like the integers, is also called an Abelian group, after the mathematician Niels Henrik Abel.
One question we may want to ask about integers is the equivalent of the subtraction problem for multiplication: the division problem. Given a pair of integers (a, b), it asks what number, when multiplied by a, gives b. We call the answer to the division problem for a pair (a, b), if it exists, b/a. If we treat the division problem just as we treated the subtraction problem for natural numbers, we end up with a new set of numbers, the rational numbers, in which every number other than 0 has a multiplicative inverse element 1/a. A set, like the rational numbers, with an addition operation with which it is an Abelian group and a multiplication operation with which its set of nonzero elements is a group is called a field.
Magnitudes and the ContinuumEdit
The process by which we extended the natural numbers to the integers and then the rational numbers was one of abstraction and generalization: If a solution to an equation existed and obeyed the rules we had already learned about numbers, it had to have certain properties. Once we knew all the properties it would have to have, we could build a new set of numbers in which it did exist. We can carry this process even further. Some equations cannot be solved by rational numbers, for example . In introducing irrational numbers into our system, let's go a bit further than we strictly have to by allowing numbers with any decimal expansion as part of our set, which we'll call the real numbers. The real numbers have some interesting properties: every positive real number has a real root (of every order) and a real logarithm, and they are complete in the sense that for every set of real numbers, you can find exactly one number that's greater than every other number in the set and smaller than every other number with this property. You can't do this for all sets of rational numbers -- for example, it fails for the set 3, 3.1, 3.14, 3.141, 3.1415, ..., which contains infinitely many numbers that approach . One of their most interesting properties is that you can assign exactly one real number to every point on a line and cover the entire line. Somehow along the way in this process of abstraction and generalization, numbers have turned from something we use only for counting to something we can use to describe shapes, and this is the concept geometric algebra is based on.
As you may know, the real numbers aren't the end of the story. There are still equations that can't be solved by any real numbers, for example . In order to solve equations like this, we will introduce a number . The new set of numbers we build using , which includes all numbers of the form , where and are real numbers, is called the complex numbers. is quite a remarkable invention. For one thing, it's the last hoop we have to jump through -- every algebraic equation has a solution in the complex numbers. For this reason, we say that the complex numbers are algebraically closed.
Since complex numbers aren't as familiar as the other sets of numbers we've discussed, I'll talk briefly about how to work with them. To find the sum of two complex numbers, we use the distributive property and the associative and commutative properties of addition:
To find their product, we use the distributive property and the fact that :
Finding the difference of two complex numbers is simple as long as you remember to distribute the negative sign to both parts of the number, but division is a little more complicated. You have to use a little trick to turn the bottom of the fraction into a real number:
We call the complex number , which you had to multiply by on the top and the bottom as part of that trick, the complex conjugate of .
Like the real numbers, the complex numbers have a natural geometric interpretation -- they can be thought of as points in a plane. The standard representation of the complex plane is with the real line running horizontally and the line of real multiples of running vertically, above the real line and below it. Geometrically, the sum of two complex numbers can be constructed in the same way as the sum of two vectors -- draw a parallelogram with two of its sides running from the origin (that is, 0) to the numbers you are adding. The point of the parallelogram opposite the origin is the sum of the two complex numbers. The rule for multiplication is a little more complicated: first, construct a right triangle for each of the numbers being multiplied, with one of its vertices at the origin, one at the endpoint of the complex number, and one along the real axis (at the point representing the real part of the complex number). Rotate one of these triangles until the leg that was on the real axis is parallel to the hypotenuse of the other triangle, and then scale it by a factor equal to the length of the hypotenuse of the other triangle. The tip of this triangle represents the product of the two complex numbers.
When we interpret complex numbers as points (or vectors) in a plane, it makes sense to introduce the concept of absolute value, which we will define exactly as we did for real numbers: the distance between a number and 0. The absolute value of a complex number is also called its modulus. Notice that there's a complication: more often than not, a complex number is situated along a diagonal, and so we have to use the Pythagorean theorem to find this distance. For example,
Complex conjugates come in handy here. As we saw in the division rule, the product of a complex number and its conjugate is always real. What's more, it's equal to the square of the complex number's absolute value. So we can simply multiply a complex number by its conjugate and then take the square root to find its absolute value. Here's a little demonstration of why this works algebraically:
See if you can work out the geometric proof for yourself. Note that taking the complex conjugate of a number amounts to reflecting it across the real axis.