# Order of Real Numbers

## Key Questions

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

• You can either compare their decimal representations or compute the difference to see if it is positive or negative.

Example 1

$\pi = 3.14 \ldots$

$2 \sqrt{3} = 3.46 \ldots$

Hence, $\pi < 2 \sqrt{3}$.

Example 2

$2 \sqrt{2} - e = 0.11 \ldots > 0$

Hence, $2 \sqrt{2} > e$

I hope that this was helpful.

• You can think of a real number as a number that has a decimal representation including the ones having infinitely many digits.

I hope that this was helpful.