Density of Numbers Formula

The Formula

For any a < b, there exists c such that a < c < b (e.g., c = \frac{a+b}{2})

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

Quick Example

Between 0.1 and 0.2 are 0.15, 0.11, 0.199, and infinitely more.

Notation

a < c < b means c lies strictly between a and b; (a, b) denotes the open interval of all such numbers

What This Formula Means

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.

Formal View

A set S \subseteq \mathbb{R} is dense in \mathbb{R} if \forall\, a, b \in \mathbb{R} with a < b, \exists\, s \in S such that a < s < b. Both \mathbb{Q} and \mathbb{R} \setminus \mathbb{Q} are dense in \mathbb{R}.

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.

Common Mistakes

  • Thinking there is a 'next' number after 0.5 — there is no smallest number greater than 0.5; you can always find one closer like 0.50001
  • Believing integers are dense — there is no integer between 2 and 3; density applies to rationals and reals, not integers
  • Assuming that two numbers very close together have nothing between them — between 0.999 and 1.000 there are infinitely many numbers like 0.9995

Why This Formula Matters

Fundamental for understanding continuity and the real numbers.

Frequently Asked Questions

What is the Density of Numbers formula?

The property that between any two distinct real numbers, there are infinitely many other real numbers—no two are 'adjacent'.

How do you use the Density of Numbers formula?

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

What do the symbols mean in the Density of Numbers formula?

a < c < b means c lies strictly between a and b; (a, b) denotes the open interval of all such numbers

Why is the Density of Numbers formula important in Math?

Fundamental for understanding continuity and the real numbers.

What do students get wrong about Density of Numbers?

This seems to contradict 'next integer'—density applies to reals, not integers.

What should I learn before the Density of Numbers formula?

Before studying the Density of Numbers formula, you should understand: number line, rational numbers.

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