Practice Density of Numbers in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

medium

Show by averaging that there is a number between

\frac{1}{7}

and

\frac{1}{6}

Example 2

easy

Can you find a fraction between

\frac{2}{5}

and

\frac{3}{5}

Example 3

medium

Find a number between

\frac{99}{100}

and

1

Example 4

medium

Find an irrational number between

1

and

2

Example 5

medium

Why is there no 'next' number after

0.5

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

Example 6

medium

Are the natural numbers

\{1, 2, 3, \dots\}

dense?

Example 7

easy

Average

0.7

and

0.8

to find a number between them.

Example 8

hard

Find a rational number

p/q

between

\frac{22}{7}

and

\pi

Example 9

medium

Show by averaging that there is a number between

\frac{99}{100}

and

\frac{100}{100}

Example 10

medium

Are there integers between

3

and

4

? Are there rationals? Explain what this says about the density of integers vs. rationals.

Example 11

easy

Between

0

and

1

, how many rational numbers are there?

Example 12

challenge

Prove the set

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

is NOT dense in

(0, 1)

Example 13

hard

Show that for any

\epsilon > 0

there is a rational

q

with

|q - \pi| < \epsilon

Example 14

hard

Find a rational number between

\pi

and

\pi + 0.001

Example 15

easy

Find a number between

1

and

1.0001

Example 16

easy

Name a number strictly between

\frac{2}{7}

and

\frac{3}{7}

Example 17

hard

Show that there is an irrational number between

1

and

2

, and find one explicitly.

Example 18

medium

Find a number between

-1.2

and

-1.1

Example 19

hard

Find an irrational number between

\frac{1}{4}

and

\frac{1}{2}

Example 20

medium

Find two distinct numbers between

\frac{1}{3}

and

\frac{1}{2}

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Related Concepts

Number Line Rational Numbers Types of Continuity and Discontinuity Limit