Density of Numbers Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Density of 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 property that between any two distinct real numbers, there are infinitely many other real numbers—no two are 'adjacent'.

No matter how close two numbers are, you can always find one between them.

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 real number line has no gaps anywhere—you can always find another number between any two given numbers.

Common stuck point: This seems to contradict 'next integer'—density applies to reals, not integers.

Sense of Study hint: Try averaging the two numbers: add them and divide by 2. The result is always between them, and you can repeat this forever.

Worked Examples

Example 1

medium
Find three rational numbers strictly between \dfrac{2}{5} and \dfrac{3}{5}.

Solution

  1. 1
    Method 1 (mediant / averaging): Average of \dfrac{2}{5} and \dfrac{3}{5}: \dfrac{1}{2}\left(\dfrac{2}{5}+\dfrac{3}{5}\right) = \dfrac{1}{2} = 0.5. First number: \dfrac{1}{2}.
  2. 2
    Average of \dfrac{2}{5} and \dfrac{1}{2}: \dfrac{1}{2}\left(\dfrac{4}{10}+\dfrac{5}{10}\right) = \dfrac{9}{20}. Second number: \dfrac{9}{20}.
  3. 3
    Average of \dfrac{1}{2} and \dfrac{3}{5}: \dfrac{1}{2}\left(\dfrac{5}{10}+\dfrac{6}{10}\right) = \dfrac{11}{20}. Third number: \dfrac{11}{20}.
  4. 4
    Check order: \dfrac{2}{5} = \dfrac{8}{20} < \dfrac{9}{20} < \dfrac{10}{20} < \dfrac{11}{20} < \dfrac{12}{20} = \dfrac{3}{5}. ✓

Answer

Three rationals between \dfrac{2}{5} and \dfrac{3}{5}: \dfrac{9}{20}, \dfrac{1}{2}, \dfrac{11}{20}.
The rationals are dense: between any two distinct rationals there are infinitely many more. Repeated averaging is a simple constructive method. This property — density — distinguishes rationals and reals from integers.

Example 2

hard
Show that there is an irrational number between 1 and 2, and find one explicitly.

Practice Problems

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

Example 1

easy
Find a rational number between 0.3 and 0.4 by averaging, and another by finding a decimal with more places.

Example 2

medium
Are there integers between 3 and 4? Are there rationals? Explain what this says about the density of integers vs. rationals.
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Background Knowledge

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

number linerational numbers