# How do you classify real numbers?

Apr 16, 2018

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 $\frac{4}{5}$ or $- \frac{6}{3}$). Terminating decimals and repeating decimals are examples of rational numbers.

Rational numbers: $3 , - 9 , 12 , - 777 , 0.3 \overline{3} , \frac{12}{7} , 0.46 , 0.16 \overline{6}$

Irrational numbers: $\sqrt{2} , \sqrt{3} , \sqrt{5} , 2 \sqrt{3} , - \sqrt{13} , \pi$

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: $1 , 2 , 3 , 4 , 5. . .$

Apr 17, 2018

#### Explanation:

These are whole numbers you are familiar with. $\left(1 , 2 , 3 , 4 , \ldots\right)$

Any opposite numbers to counting numbers are negative whole numbers. $\left(\ldots , - 4 , - 3 , - 2 , - 1\right)$

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

Now, any number that can be written as a fraction $\frac{a}{b}$ 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 $\frac{a}{b}$ 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.