What are real numbers?

What do they call the study of numbers these days? It used to be called arithmetic. Today it’s called number theory. And mathematicians who like to study our universe of numbers are called number theorists. It sounds more important.

Integers are known as natural numbers. We all understand 1, 2, 3, up to infinity. Whole numbers are everywhere. But natural numbers include negative numbers too. For example, -3, -2, -1, 0, 1, 2, 3. As negative numbers go up in value, they get smaller. So it’s like positive and negative numbers pull in opposite directions from zero. Or you can think of negative numbers as a mirror image of positive numbers. It’s like having a parallel universe either side of the mirror.

But it doesn’t stop there. There are fractions too, which are whole numbers shared. If we think of all the fractions between 0 and 1, we get, 1/2, 1/3, 1/4, 1/5, up to infinity. In fact, there is an infinite world of both integers and fractions.

Oh, and let’s not forget ratios too. A ratio is a different way of thinking about a fraction. So 1/2 becomes 1:2 in ratio format. Our number universe is growing. We have positive and negative whole numbers, or natural numbers, fractions and ratios. Together, we call them rational numbers, and that includes infinity.

Now if you’ve heard of rational numbers, you’ve probably heard of irrational numbers too. But what are they? They are never ending decimal numbers that need rounding, such as the square root of two, or Pi. And there are infinity irrational numbers between consecutive rational numbers? Are you confused yet?

Do you understand square numbers and square roots? Many people don’t. If the square of two is four (two times two), then the square root of four is two. Simple, yes? They are opposites. If the cube of two is eight (two times two times two), then the cube root of eight is two. Squares and square roots are in our number universe.

But I haven’t answered the question, what are real numbers? Real numbers are equal to rational numbers plus irrational numbers. Now you know. What you can deduce is we have an infinite real number universe. This idea was conceived by David Hilbert, a German mathematician around 1900. That’s infinite thinking.