Practice Infinity Intuition in Math

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

Worksheet PDF Free Answer-key PDF Family

Quick Recap

The concept of endlessness or unboundedness—a process that goes on forever with no final stopping point.

Numbers never stop—there's always a bigger one. Infinity isn't a number, it's a direction.

Showing a random 20 of 50 problems.

Example 1

challenge
Two runners: A covers 1,2,3,…1,2,3,\ldots meters in successive minutes; B covers 2,4,6,…2,4,6,\ldots meters. After forever, who has gone farther, or is it the same?

Example 2

medium
Does the sequence 1,2,3,…1,2,3,\ldots have a largest term? What does that tell you about the set?

Example 3

medium
A student writes ∞−∞=0\infty-\infty=0. Why is this wrong?

Example 4

easy
Is the number of grains of sand on Earth finite or infinite?

Example 5

medium
Explain why there are infinitely many fractions between 00 and 11.

Example 6

hard
Use a bijection to show that {0,1,2,…}\{0, 1, 2, \ldots\} and {5,6,7,…}\{5, 6, 7, \ldots\} have the same size.

Example 7

medium
A bug starts at 00, jumps half the remaining distance to 11 each step. Does it ever reach 11?

Example 8

easy
Is every set with no last element infinite?

Example 9

hard
Evaluate ∑n=1∞12n\displaystyle\sum_{n=1}^{\infty} \frac{1}{2^n} and explain why an infinite sum can have a finite value.

Example 10

medium
Can a finite line segment contain infinitely many points?

Example 11

easy
Can you list ALL the even numbers on paper?

Example 12

hard
True or false: there are infinitely many prime numbers.

Example 13

medium
Are there more counting numbers 1,2,3,…1,2,3,\ldots or more even numbers 2,4,6,…2,4,6,\ldots?

Example 14

medium
Compare: is there 'more' natural numbers or 'more' integers? Explain using a bijection.

Example 15

medium
True or false: the set of integers {…,−2,−1,0,1,2,…}\{\ldots,-2,-1,0,1,2,\ldots\} is twice as big as the natural numbers.

Example 16

medium
Show that the set of even positive integers can be put in one-to-one correspondence with the set of all positive integers, even though the evens seem 'smaller'.

Example 17

easy
How many fractions are there between 00 and 11?

Example 18

easy
Does 0.999…0.999\ldots (repeating forever) reach an end?

Example 19

easy
Is the set of counting numbers 1,2,3,…1,2,3,\ldots finite or infinite?

Example 20

challenge
True or false: there are 'more' real numbers between 00 and 11 than there are natural numbers.
← Back to Infinity Intuition Worked Examples Formula Explained