So what's a number? And what does it mean to say there are different types of numbers? Well, to answer these questions we're going to have to start at the beginning with the basics.
The most basic thing you can do in mathematics is count - 1 2 3 4 5. These numbers - these counting numbers - are the simplest number type that we have. They're called the natural numbers. And the natural numbers are great for representing all sorts of stuff, but they're not perfect there's lots of real-life stuff you can't do with only natural numbers so we need more stuff
The first thing that we can do to improve the natural numbers - to add something else - is to add the number zero. This system of the natural numbers together with 0, is called the whole numbers.
This is better, but there's still more that's missing. If all we have are just the whole numbers how do we represent something like debt? How do we describe the elevation of something that's underground? The next thing we can add to this system is the negative numbers which represent these kinds of situations. This system of the negative numbers together with the whole numbers is called the integers.
This is great. It's much better, but it's still not perfect. If I eat half of a pizza how do we represent that with only the integers? We can't. We need something more. So, the next thing we add is all of the fractions. This system of the integers together with all of the fractions is known as the rational numbers.
Let's talk about that for just a second.
The definition of the rational numbers is any number that you can write as a ratio of two integers. This includes numbers like -2/3 but it also includes numbers like five because we can write that as 5/1. It will also include any decimal number that either terminates like 2.5 or repeats like point three repeating (which you might recognize as the decimal for ⅓).
To summarize, we can tell if something is a rational number if it’s an integer, a fraction, a decimal that stops or a decimal that repeats.
Now it might be tempting to think that this is the top -this is everything- but even now we're still missing some stuff. What about decimals that don't terminate and don’t repeat. Stuff like pi or the square root of 2. We call these numbers, the ones that can't be written as a fraction, the irrational numbers.
And if we put those two groups together the rationals and the irrationals we get the real numbers.
And you might think that's it that's the top - that's all of the numbers - what could we possibly be missing?
Well one thing that we can't do with just the real numbers is to take the square root of -1. So mathematicians came up with a new number “i” which is defined to be the square root of -1 and we call these numbers that contain “i” the imaginary numbers. So, 2i, 3i, negative a million “i” are all imaginary.
And when we combine these with real numbers we get complex numbers like 2 + 3i or 10 - 2i.
One of the most important things to understand about these number types is that they form a hierarchy. What does it mean to be a hierarchy? Here’s an example. Take me. I live in an apartment, Which is in my apartment complex which is in Houston which is in Texas which if I combine it with the other states makes up the United States which if I combined it with the other countries makes up the Earth.
So if I’m in my apartment, I’m also on earth. I’m also in Houston. It’s the same way with numbers.
We made these new number types by taking the old ones and adding new stuff so if I have a question like what number type is 5? Well it's a natural number because it's a counting number but it's also a whole number and it's also an integer and it's also a rational number and it's also a real number and it's even also a complex number, but it’s not an irrational number and it’s not an imaginary number.