Real numbers

The number system is made up of real numbers and imaginary numbers. Real numbers \(\mathbb R\) are all the numbers on a number line. The real number line is infinite, which means it continues forever in both directions. Real numbers are infinite.

Real numbers are made up of all rational \(\mathbb Q\) and all irrational \(\mathbb {Q}^|\) numbers. 

Beginning with the natural numbers, we expand each set to form a larger set of numbers.

We use the following definitions.

The set of natural numbers \(\mathbb N\) includes the numbers used for counting: {\( {1, 2, 3, ...} \)} .

The set of whole numbers \(\mathbb N_o\) is the set of natural numbers plus zero: {\( {0, 1, 2, 3, ...} \)} .

The set of integers  \(\mathbb Z\) adds the negative of the natural numbers to the set of whole numbers 
{\( {..., -3, -2, -1 ,0, 1, 2, 3, ...} \)}. It is important to remember that integral values are not fractions.

The set of rational numbers \(\mathbb Q\) includes fractions that can be written as \( \frac{a}{b} \) where \( a, b \in \mathbb{Z}, b \neq0 \).

The set of irrational numbers \(\mathbb {Q}^|\) are numbers that cannot be written as a fraction with the numerator and denominator as integers. These fractions give us non-repeating and non-terminating (they do not end) decimals.

Test your understanding of real numbers by trying the next activity.