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: Between any two distinct real numbers you can always squeeze infinitely many more.

Common stuck point: The procedure for density of numbers is the easy part; the trap is asking for the next real number. Asking "Is the question about whether infinitely many numbers fit between two given values, with no smallest gap?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the question about whether infinitely many numbers fit between two given values, with no smallest gap?

Worked Examples

Example 1

medium

Find three rational numbers strictly between

\dfrac{2}{5}

and

\dfrac{3}{5}

Answer

Three rationals between

\dfrac{2}{5}

and

\dfrac{3}{5}

\dfrac{9}{20}

\dfrac{1}{2}

\dfrac{11}{20}

First step

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}

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Example 2

hard

Show that there is an irrational number between

1

and

2

, and find one explicitly.

Example 3

medium

Find a rational number between

\sqrt{3}

and

\sqrt{5}

Example 4

medium

Find a rational between

\frac{1}{3}

and

\frac{2}{5}

Example 5

hard

Find a rational number between

\pi

and

\pi + 0.001

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.

Example 3

easy

Name a number between

\frac{1}{3}

and

\frac{1}{2}

Example 4

easy

Is there a number between

0.5

and

0.6

? Give one.

Example 5

easy

True or false: there is a smallest number greater than

0

Example 6

easy

Is there an integer between

2

and

3

Example 7

easy

Find a number between

1

and

1.0001

Example 8

easy

Average of

\frac{1}{4}

and

\frac{1}{2}

— does it lie between them?

Example 9

easy

Between

0.999

and

1.000

, are there other numbers?

Example 10

easy

Can you find a fraction between

\frac{2}{5}

and

\frac{3}{5}

Example 11

medium

Find two distinct numbers between

\frac{1}{3}

and

\frac{1}{2}

Example 12

medium

Show by averaging that there is a number between

\frac{1}{7}

and

\frac{1}{6}

Example 13

medium

Why is there no 'next' number after

0.5

among the reals? Give a value closer than any you propose.

Example 14

medium

Between

\sqrt{2}\approx1.414

and

\sqrt{3}\approx1.732

, name a rational number.

Example 15

medium

How many numbers lie between

3

and

4

on the real line?

Example 16

medium

Give a number strictly between

-0.01

and

0

Example 17

medium

Find a number between

\frac{99}{100}

and

1

Example 18

medium

Classify: are the whole numbers dense? Justify with an example.

Example 19

medium

Find a number between

-\frac{1}{2}

and

-\frac{1}{3}

Example 20

challenge

Prove that between any two distinct reals

a<b

there are infinitely many reals.

Example 21

challenge

A student claims

\frac{1}{2}

and

\frac{1}{2}+\frac{1}{10^{100}}

are 'adjacent' reals. Refute it.

Example 22

challenge

For which sets

S

is the average of two members always a member:

\{\text{integers}\}

\{\text{rationals}\}

? Explain.

Example 23

easy

Name a number strictly between

\frac{2}{7}

and

\frac{3}{7}

Example 24

easy

Find a decimal strictly between

0.42

and

0.43

Example 25

easy

Find a rational between

0.1

and

0.11

Example 26

easy

Average

0.7

and

0.8

to find a number between them.

Example 27

medium

Find a number between

\frac{5}{8}

and

\frac{3}{4}

by averaging.

Example 28

medium

Find three distinct numbers strictly between

0.2

and

0.3

Example 29

medium

Find an irrational number between

1

and

2

Example 30

medium

Show by averaging that there is a number between

\frac{99}{100}

and

\frac{100}{100}

Example 31

medium

Find a number between

-1.2

and

-1.1

Example 32

medium

Find two rationals strictly between

\frac{1}{10}

and

\frac{1}{9}

Example 33

medium

Is there a real number

x

with

0 < x < 10^{-100}

Example 34

hard

Find an irrational number between

\frac{1}{4}

and

\frac{1}{2}

Example 35

hard

A student claims '

0.4999\dots = 0.5

so there is nothing between them.' True or false?

Example 36

hard

Find a real number strictly greater than every

0.99\dots 9

with finitely many

9

's but strictly less than

1

Example 37

hard

Find a rational number

p/q

between

\frac{22}{7}

and

\pi

Example 38

hard

Show that for any

\epsilon > 0

there is a rational

q

with

|q - \pi| < \epsilon

Example 39

challenge

Prove: between any two distinct rationals

r < s

lies another rational.

Example 40

challenge

Prove the set

\{1/n : n \in \mathbb{N}\}

is NOT dense in

(0, 1)

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Related Concepts

Number Line Rational Numbers Types of Continuity and Discontinuity Limit

Background Knowledge

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

number linerational numbers