Irrational Numbers Examples in Math

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

An irrational number is a real number that cannot be expressed as a ratio of two integers \frac{p}{q}; its decimal expansion goes on forever without repeating any fixed block of digits.

\pi and \sqrt{2} go on forever without any pattern—you can't write them as a fraction.

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: Some numbers exist on the number line but can't be captured as fractions.

Common stuck point: Accepting that a number can be 'exact' but not writable in finite digits.

Sense of Study hint: Try to write the number as a fraction a/b with whole numbers. If every attempt fails, that's evidence it's irrational.

Worked Examples

Example 1

easy
Classify each number as rational or irrational: \sqrt{16}, \sqrt{5}, 0.75, \pi.

Solution

  1. 1
    \sqrt{16} = 4, which is an integer, so it is rational. 0.75 = \frac{3}{4}, also rational.
  2. 2
    \sqrt{5} is not a perfect square, so \sqrt{5} is irrational. \pi is a well-known irrational number.
  3. 3
    Rational: \sqrt{16}, 0.75. Irrational: \sqrt{5}, \pi.

Answer

\text{Rational: } \sqrt{16},\; 0.75 \qquad \text{Irrational: } \sqrt{5},\; \pi
A number is rational if it can be written as \frac{a}{b} with integers a, b (b \neq 0). Square roots of non-perfect-squares are irrational, and \pi cannot be expressed as any fraction.

Example 2

medium
Show that \sqrt{2} lies between 1.4 and 1.5, then estimate it to one decimal place.

Example 3

medium
Prove that \sqrt{3} is irrational by contradiction.

Practice Problems

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

Example 1

easy
Is \sqrt{49} + \sqrt{3} rational or irrational? Explain.

Example 2

easy
Is 3 + \sqrt{7} rational or irrational?
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Background Knowledge

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

rational numberssquare roots