The set of real numbers

Real Numbers

A Comprehensive Guide to Understanding Real Numbers and Their Subsets

Published: July 8, 2025

Real Numbers

The set of Real Numbers is the set that consists of the union of the set of rational numbers and the set of irrational numbers, and it is symbolized by R.

$R = Q \cup Q'$ where $Q \cap Q' = \emptyset$

Example

Fill in the blank box with the appropriate symbol $ \in $ or $ \notin $:

1

$0.63$

Q'

2

$\sqrt{5}$

R

3

$|-4|$

N

4

$\sqrt[3]{3\frac{3}{8}}$

Z

5

$\sqrt[3]{3}$

Q

6

$\sqrt{-1}$

R

Solution

1

$$ \begin{aligned} &\because 0.63 \text{ can be written as } \frac{63}{100}, \text{ which is a rational number} \\ &\therefore 0.63 \notin \text{Q'} \text{ (Irrational Numbers)} \end{aligned} $$

2

$$ \begin{aligned} &\because \sqrt{5} \text{ is an irrational number} \\ &\because \text{Real numbers include both rational and irrational numbers} \\ &\therefore \sqrt{5} \in \text{R} \end{aligned} $$

3

$$ \begin{aligned} &\because |-4| = 4 \\ &\therefore 4 \in \text{N} \end{aligned} $$

4

$$ \begin{aligned} &\because \sqrt[3]{3\frac{3}{8}} = \sqrt[3]{\frac{3 \times 8 + 3}{8}} = \sqrt[3]{\frac{24+3}{8}} = \sqrt[3]{\frac{27}{8}} = \frac{\sqrt[3]{27}}{\sqrt[3]{8}} = \frac{3}{2} \\ &\therefore \frac{3}{2} \in \text{Q} \\ &\because \text{However, } \frac{3}{2} \notin \text{Z} \text{ (Integers)} \\ &\therefore \sqrt[3]{3\frac{3}{8}} \notin \text{Z} \end{aligned} $$

5

$$ \begin{aligned} &\because \sqrt[3]{3} \text{ cannot be expressed as a simple fraction} \\ &\because \text{It is an irrational number} \\ &\therefore \sqrt[3]{3} \notin \text{Q} \text{ (Rational Numbers)} \end{aligned} $$

6

$$ \begin{aligned} &\because \sqrt{-1} \text{ is an imaginary number (not a real number)} \\ &\therefore \sqrt{-1} \notin \text{R} \text{ (Real Numbers)} \end{aligned} $$

• Previous Information: There is no square root for a negative rational number.

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