Many of my SciBlings have been doing posts in which they define basic concepts in various scientific fields. For example, physicist Chad Orzel has done posts on Force and Fields, biologist P. Z. Myers has covered Genes, computer scientist Mark Chu-Carroll offers up wise words on Margin of Error and Standard Deviation, and philosopher John Wilkins discusses fitness.
And, in the few minutes it took me to put together that list, I notice that Wilkins has just put up this post, gathering all of the basic concepts posts together.
I figured it was high time I weighed in with some basic concept in my own field. And what could be more basic to a mathematician than the idea of a number. So here goes.
What is a number? Well, rather than try to provide an abstract definition, how about we decide what it is we want numbers to do. Then we can define a number as anything that does that.
We begin with the counting numbers, by which I mean the numbers 1, 2, 3, ... We use these numbers to distinguish between the sizes of collections of objects. A collection of three apples is different in some fundamental way from a collection of five apples, and the counting numbers allow us to give a name to that difference.
We might also think of the counting numbers as things we use to label lengths and distance. This, however, is merely a special case of the previous paragraph. In labelling lengths, say, it is understood that we have previously chosen a unit, and then the number in question simply refers to how many units are needed to cover the length.
One of the nice things about using counting numbers to label collections is the possibility of deducing things about the objects simply by manipulating the numbers. The abstraction from five apples plus three apples is eight apples, to 5+3=8 is not to be treated lightly. This latter equation is telling us something crucical and wide ranging. It tells us that any collection with the property of fiveness, when combined with a collection having the property of threeness, produces a collection with the property of eightness.
What about zero? Certainly the property of having nothing is common sensical enough, but why should the thing that describes this property be considered a number? I think our previous paragraph provides the answer. We want to draw inferences about collections of real-world objects simply by manipulating numbers. But given a collection of real-world objects, one thing we might do is give them all to someone else. If we do no thave a number that represents the property of having nothing, then we will have a system of arithmetic that is not equipped to handle certain familiar real-world situations. For this reason it is convenient to allow zero to be a bona-fide number.
But what about negative integers? We are now getting somewhat removed from the idea of labelling collections of tangible objects. After all, in the real world if you try to remove ten objects from a collection that only contains seven objects, you quickly find yourself stymied. You can take away seven objects, thereby leaving you with zero, but that is where the process ends. It's not as if you can keep going until you are left with minus three objects.
That notwithstanding, it is not too difficult to think of real-world situations in which it is useful to talk about numbers that are less than zero. Someone with nothing in his bank account who owes $100 to another person can reasonably be said to have less than no money. He would need to obtain $100 just to have nothing. No doubt you can conjure up other examples.
Negative numbers also serve a mathematical purpose. If zero is the smallest number we know, then subtraction is a tricky operation. The expression a-b is then meaningful only if a is larger than b. That means there are simple equations we can write down that have no solutions. For example, x+7=3. If we could define negative numbers in a way that is consistent with everything we know about positive numbers, then we would have a more powerful system of arithmetic.
So it seems reasonable to let negative numbers into the pantheon. What about rational numbers?
By a rational number we mean any fraction whose top and bottom are both integers (mathematicians, incidentally, routinely refer to the top and bottom of a fraction, rather than the numerator and the denominator.) We should also mention that the word “rational” comes from “ratio.” It is not because these numbers make sense, whereas other numbers, like pi, do not.
At any rate, it is a pretty common thing to divide an object into several smaller pieces. Perhaps you have a pizza that has been sliced into eight pieces. You take three of them. The number of slices you have is then described by the counting number three. But how many pizzas do you have? Our current number system, consisting of the integers; positive, negative and zero; is not up to the task. Nonetheless, we still have a perfectly common sensical notion that relates to actions taken on collections of objects. We really ought to have numbers for describing how much you have when you only have part of a greater whole, and the rational numbers are well-suited to that purpose.
So everything up to the rational numbers has earned its status by telling us something about collections of real world objects (or perhaps collections of abstract objects like units of measure). We next come to the irrational numbers. These are not hard to motivate. Draw a square whose side length is one unit. Then the length of the diagonal is the square root of two units. But, as was known to the Greeks, the hypothesis that the square root of two is a rational number leads quickly to a contradiction. Since labelling lengths is one of the tasks with which numbers are charged, it seems we have not yet produced everything worthy of the name.
The irrational numbers fill in the gaps of the rational numbers. Defining them is not as easy as you might think. It is customary, as a first approximation, to say an irrational number is one whose decimal expansion is infinite and nonrepeating. But how dow you do arithmetic with such beasts? If the decimal expansion of pi is infinite and nonrepeating, what on Earth is pi squared supposed to be? Every algorithm for basic multiplication you have ever learned breaks down when you try to do it for numbers with infinitely many digits.
Suffice it to say that these hurdles can be overcome, and the irrational numbers can be added to our collection without messing up anything that came before. If you combine the rational numbers and the irrational numbers into one package you have the real numbers.
Is that it? Certainly we can now label any length or describe any collection that comes our way. Given a unit, I promise you that every length you can draw is describable with a real number multiple of that unit. It might seem that we can rest on our laurels. Indeed, mathematicians were cotnent to do precisely that for quite some time.
Yet the real numbers do still have an annoying defect. It is possible to write down polynomial equations with real number coefficients (indeed, integer coefficients) that do not have real number solutions. For example, x2+1=0. This is annoying, much like the limitations on our ability to perform subtraction were annoying back when we only had the positive integers. Finding solutions of polynomial equations is something that is often useful in practical situations. Why should we limit ourselves to an environment in which certain such equations lack solutions. Perhaps we can construct a larger environment in which we lose nothing of our previous hard work in constructing the real numbers, but can also factor currently intransigent polynomial functions.
This can be done of course, and the result is the field of complex numbers. By this we mean all numbers of the form a+bi, where i is defined to be the square root of minus one. It can be shown that every polynomial with complex coefficients has a root in the complex numbers. This is the famous fundamental theorem of algebra. The complex numbers contain subfields isomorphic to the real numbers and the rational numbers, and a subring isomorphic to the integers. This is usually abbreviated to the statement that the real numbers, rational numbers and integers are subsets of the complex numbers.
The complex numbers can not be viewed as accounting devices for real-world collections of objects. This does not mean their interest is purely theoretical, however. There are problems that are intractable over the reals that become solvable over the complex numbers. Since the real numbers are themselves attached to real-world objects, the complex numbers do manage to find plenty of practical applications.
And that's where it ends. These are the things we think of when we think of numbers. Ultimately we get a lot of mileage out of the basic idea that “number” is a property possessed by collections of objects, and numbers are labels that we use to evoke thoughts of this property in people contemplating such collections.
Aw, who am I kidding. That's not really the end. How about the p-adic numbers? Do they count (no pun intended)? The quaternions and octonions, are they numbers? The integers modulo n?
Describing these objects is a subject for another post. This is a basic concepts post, after all, and I think we have pondered this for long enough.
So what is a number? I'm not sure. But I know one when I see one!
Sorry, I have to correct you with the same correction that every high-school maths teacher has to make. There is no such thing as "the square root of -1". Rather, i is a number such that i^2 = -1.
This becomes very important when you look at generalisations and representations of complex numbers where, in general, there are many numbers which square to -1. For example, the matrix [ [ 0 -1 ] [ 1 0 ] ], when squared, is -1. So is any unit quaternion that doesn't have a "real" part. As is a unit "pure timelike" vector in Minkowski space (using the Clifford algebra formulation, rather than the tensor one).
The rules of complex numbers work for any "number" i such that i^2 = -1. (Technical nit: i also needs to commute with "scalar", usually real, numbers.)
Picky picky. You're technically correct, of course, but it makes for awkward writing to constnatly have to write “a number whose square is minus one” as opposed to the square root of minus one. I don't think any confusion is caused by the latter phrasing, which is what I care about.
Yeah, picky. My minor was physics, and I was always annoyed how physicists played fast and loose with maths. (Chad of Uncertain Principles seems to wear it as a badge.)
"The square root of minus one" is a phrase worthy of a physicist. Hell, quantum field theory routinely takes the square root of a derivative...
There's nothing wrong with what he said, except that "the square root of negative one" is a little misleading. Rather, it should be "a square root of negative one". In the context of the reals, it makes no difference: by the isomorphism extension theorem for fields, there exists an automorphism of C sending a + bi -> a - bi (i.e., complex conjugation is a field automorphism).
Nothing would change if i were everywhere replaced by -i, at least as far as the field operations go. The square root of negative one is unique up to field isomorphism, over any field (that doesn't already contain such a root). In practice you don't have ordering axioms or other things you want to preserve over C that get in the way of this, so in fact every significant result should be the same for i or -i.
Of course, in arbitrary rings things are a lot more messy, as you point out. But be fair: that wasn't the context. He was talking about extending the reals. Anyway, although I haven't worked with quaternions, it's not true that "the rules of complex numbers" work for any square root of -1 over any ring. Very important facts about complex numbers, like the fact that complex numbers can be written in the form reit, completely fail over even arbitrary fields, never mind arbitrary rings. If you look at extreme examples like the two-element field (which of course has a square root of -1), you'd be hard-pressed to find more similarities than differences.