Numbers are Mental Tools
Sometimes we think numbers are just numbers, but really, there are many different kinds of them. Numbers are mental tools. We need different tools for solving different kinds of problems. That leads us to …
When you count imaginary sheep at night to go to sleep, you start with one, proceed to two, and so forth. You never have zero sheep, because you start counting when you imagine the first one. Counting things like sheep uses the counting numbers.
Although the number zero seems an obvious concept to us, that seems to be the result of our modern upbringing. Of course, ancient people were plenty familiar with not having any of something, but from the counting number perspective, if you don’t have any of something, you can’t count it, so you can’t describe it with a number. It took a long time before we started thinking it was OK to have an actual number to represent not having anything.
Although ideas related to zero came up in different places and times, the mathematician who first defined zero in the modern mathematical sense was Brahmagupta in his book Brahmasphutasiddhanta, written in the year 628, when he was 30 years old. It also describes rules for doing arithmetic with negative numbers, and, amazingly to me, was written entirely in poetry! What would you think if your next math book was entirely poetry? Nowadays, when we include the number zero with the counting numbers, we call that the natural numbers.
It’s hard to imagine what a negative sheep would be. However, negative numbers are really useful for some kinds of things.
For instance, suppose that I’m a math tutor (as you might have guessed, I really am one). Usually my students pay me before the tutoring, sometimes a whole month at once, sometimes only for one lesson, and so forth. I keep track of how much money they’ve given me and how much I’ve charged them to make sure that they don’t overpay or underpay me. Occasionally, somebody forgets a check, but I tutor them anyway. Since they now owe me money, I just write that as a negative number in their account, and when they finally pay me, I add a positive number to the negative one, and their account (usually) goes positive again. Since these numbers are all whole numbers, but are sometimes positive and sometimes negative, they are integers.
The natural numbers are all integers, but the negative whole numbers are also integers.
What kind of number is -1? or -4? or -9?
They are integers.
Math people generally agree on the definitions of counting numbers, natural numbers, and integers. However, they aren’t really so consistent with the term whole numbers. Whole numbers sort of emphasize the fact that fractions aren’t allowed, without getting very much into the issue of whether negative numbers or zero are permitted. Since in talking about prime and composite numbers we repeatedly emphasized that prime numbers and composite numbers must be greater than one, zero and the negative numbers are not an issue. This website is mostly focused on prime numbers, so we decided to go with whole numbers most of the time, because it’s the most self-explanatory choice.
Rational numbers mean fractions. In almost every lesson about rational numbers, there’s going to be pizza! First of all, most people really like pizza, and it’s an excuse to talk about it. But also, most pizzas are too big for one person to eat by themselves, and we cut them into pieces. Unlike some foods that get spooned into portions or cut into blocks, when you look at a slice of pizza, it’s usually easy to tell how many slices the whole pizza was cut into. When you talk about breaking things into equal pieces and then grouping those pieces, you have entered the world of fractions. Rational numbers is just the formal mathematical word for talking about numbers with fractions.
It turns out that there are ways of breaking numbers up that can never be done by breaking them into equal pieces and then grouping these pieces together (no matter how invisibly tiny the crumbs are made to be). That can be hard to believe, but there are some pretty important and common numbers like that, and they are called irrational numbers. Math people love proving hard-to-believe things, and some of the proofs that numbers are irrational are really clever. Will I prove that some important number (like pi, or the square root of 2) is not a rational number? I don’t want to get too distracted doing that here, but if you’re one of the people that wish I did, I think you are a mathematician!
Irrational numbers, by themselves, aren’t as useful for general problem solving as other kinds of numbers that we’ve been talking about. So far, each group of numbers that we’ve talked about includes the one before it. For instance, the natural numbers included the counting numbers, the integers included the natural numbers, and the rational numbers included the integers. But the irrational numbers don’t include the rational numbers. In particular, the irrational numbers are missing some elements essential for most kinds of work, like the numbers zero and one.
There’s two common things that get done with irrational numbers. One is to prove that this number or that number is irrational. The other is to combine the rational and irrational numbers together to create the real numbers.
Here’s something else interesting about real numbers. Not only are they the combination of the rational and irrational numbers, but they’re also the combination of the algebraic numbers and the transcendental numbers. We won’t really get into what algebraic and transcendental numbers are here (it takes a lot of explaining), but it’s interesting to know that there’s more than one useful way to divvy up the real numbers.
The real numbers, which include all the kinds of numbers we’ve mentioned so far, are hugely important in solving real life math problems in lots of fields, especially science and engineering.
When you multiply two positive numbers by each other, you get a positive number. When you multiply two negative numbers by each other you get a positive number. So, is there any number you can multiply by itself to get a negative number? Yes, and no. There is no such real number, but there is a made-up number called i, that when multiplied by itself is negative one. Since it’s not a real number, we call it an imaginary number (it’s sort of a little math joke).
When you mix up real numbers with imaginary numbers, for instance by adding them together, you get an amazing kind of numbers called complex numbers. Why are they amazing? For one thing, there is the Fundamental Theorem of Algebra, which is true for complex numbers just as the Fundamental Theorem of Arithmetic is true for whole numbers. A major branch of mathematics, called complex analysis, explores properties of complex numbers.
Do imaginary or complex numbers exist in the real world? Well, really, no numbers exist in the real world. We use numbers to describe things, and yes, there are things in the real world we describe with complex numbers. Probably the most famous is electrical circuits; without using complex numbers we wouldn’t be able to design the modern electronics that are at the heart of so much of today’s technology. Another field in which complex numbers is important is acoustics, which is the scientific study of sound.
More Kinds of Numbers!
There are lots more kinds of numbers. Different kinds of numbers are each interesting in their own ways. Some are useful for solving real world problems. Others aren’t. They may or may not have properties that we assume “normal” numbers have. For instance we usually say that “multiplication is commutative”. That means that you can change which factor comes first without changing the result, so 2 x 3 is 6 and 3 x 2 is 6. That’s true for all factors, not just 2, 3, and 6. However, although multiplication is commutative for, say, real numbers, there are kinds of numbers where it is not.
Inventing and figuring out how some of those kinds of numbers work is fun. Abstract Algebra is the study of which fundamental properties of numbers lead to which other important properties. In that study, there’s many kinds of numbers, very different from the ones we’re used to. Studying some of these weird numbers feels like learning about strange creatures that live at the bottom of the ocean. They can be really interesting!
Next Topic: The Sieve of Eratosthenes