Real Numbers Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Real Numbers.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The complete set of all rational and irrational numbers, filling every point on the continuous number line.

Any number you can point to on an infinitely precise number line.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The reals form a complete, continuous number line with no gaps.

Common stuck point: Students think every number they use is rational. They struggle to accept that most real numbers are irrational — there are infinitely more irrationals than rationals.

Sense of Study hint: Draw a number line and mark a rational (like 1.5) and an irrational (like the square root of 2). Both are points on the same line — that's the reals.

Worked Examples

Example 1

easy
Classify each number as rational or irrational, and state whether it is a real number: -7, \sqrt{9}, \sqrt{7}, \pi.

Solution

  1. 1
    -7 = \frac{-7}{1} is rational. \sqrt{9} = 3 = \frac{3}{1} is rational. Both are real.
  2. 2
    \sqrt{7} is irrational (7 is not a perfect square); \pi is irrational (proven). Both are real.
  3. 3
    All four are real numbers: \mathbb{R} = \mathbb{Q} \cup (\mathbb{R} \setminus \mathbb{Q}).

Answer

\text{All four are real numbers; } {-7,\,\sqrt{9}} \text{ rational; } {\sqrt{7},\,\pi} \text{ irrational}
Every rational and every irrational number is a real number. The real numbers form the complete number line with no gaps. Being 'real' does not mean 'rational' — most real numbers are in fact irrational.

Example 2

medium
Show that between any two distinct real numbers a and b (with a < b), there exists another real number.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Which of the following are NOT real numbers? \sqrt{-4}, -\sqrt{4}, \sqrt[3]{-8}, \frac{1}{0}.

Example 2

medium
Give one example of a real number between \sqrt{2} and \sqrt{3}.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

rational numbersirrational numbers