Question

How do you classify real numbers?

Answer 1

Irrational and rational numbers
Rational numbers: integers, whole numbers, counting/natural numbers

Explanation:

Real numbers are either irrational or rational. Rational numbers can be written as fractions (using two integers, such as `4/5` or `-6/3`). Terminating decimals and repeating decimals are examples of rational numbers.

Rational numbers: `3, -9, 12, -777, 0.3bar3, 12/7, 0.46, 0.16bar6`

Irrational numbers: `sqrt2, sqrt3, sqrt5, 2sqrt3, -sqrt13, pi`

There are several different groups of rational numbers. There are integers, whole numbers, and counting/natural numbers. Integers do not have decimals. They can be positive or negative.

Integers: `6, 16, -72, 89, 23, -1, 0`

Whole numbers are all non-negative integers. Examples include `16, 0, 23, 45559`.

Natural/counting numbers are all positive integers. (We don't start counting from zero).

Counting numbers: `1, 2, 3, 4, 5...`

Answer 2

Read below.

Explanation:

Let's start with counting numbers.

These are whole numbers you are familiar with. `(1,2,3,4,...)`

Any opposite numbers to counting numbers are negative whole numbers. `(...,-4,-3,-2,-1)`

All of these numbers and `0` form a group of integers.

Now, any number that can be written as a fraction `a/b` where `a` and `b` are integers and `b` is not equal to zero are called rational numbers. These include all the integers, any finite decimals, or repeating decimals (decimals that have a pattern).

Any number that can't be written as `a/b` are considered irrationals. They can be written in decimal, but they go on forever without any pattern (And there is a way to prove this).

All of the rational numbers and irrational numbers are considered real numbers. Notice here that all of these numbers lie on a straight number line.

Any number that lies outside the number line is a not real number.
For example, we can extend the 1-D line to 2-D plane with complex numbers on it.

You could even extend the plane to 4-D space and even to 8-D space. Interestingly, that is where the extension stops. Note here that there is no 3-D number system nor any number between 3 and 8.