This types of numbers worksheet has 40 questions on the real number system: counting and whole numbers, integers, rational numbers and irrational numbers. Each answer says why the number fits.
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Three sets
Which sets? gives a number such as −4, 0.333… or √16 and asks every set it belongs to. Pick the number finds the irrational one or the odd one out. True or false checks how the sets fit inside each other.
Practice or print
Practice brings a missed question back until you get it right. The sheet on screen is what prints: remove a question, add more up to all 40, and tick With answer key.
Simplify first: √16 is 4 and 12/3 is 4, so both are counting numbers, and so whole, integer, rational and real too.
A decimal that stops or repeats is rational; one that does neither is irrational.
Questions and answers
What are the types of numbers?
Counting (natural) numbers 1, 2, 3…; whole numbers, which add 0; integers, which add the negatives; rational numbers, any ratio of integers; and irrational numbers such as √2 and π. Rational and irrational together make the real numbers.
Is 0 a whole number or a natural number?
0 is a whole number, an integer and a rational number. Most school books start the natural (counting) numbers at 1, though some mathematicians include 0.
How do you know if a square root is rational?
The square root of a counting number is rational only if the number is a perfect square: √49 = 7 is rational, √50 is irrational.
All 40 questions with answers
Which sets? 18
Which sets does 0 belong to?Answer: Whole, integer, rational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which sets does 0.25 belong to?Answer: Rational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which sets does 0.333… belong to?Answer: Rational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which sets does 1.414213…, never repeating belong to?Answer: Irrational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which sets does 3/4 belong to?Answer: Rational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which sets does 7 belong to?Answer: Counting (natural), whole, integer, rational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which sets does 12/3 belong to?Answer: Counting (natural), whole, integer, rational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which sets does 100 belong to?Answer: Counting (natural), whole, integer, rational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Show all 40 questions (32 more)
Which sets?, continued
Which sets does π belong to?Answer: Irrational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which sets does −2.5 belong to?Answer: Rational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which sets does −4 belong to?Answer: Integer, rational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which sets does −7/8 belong to?Answer: Rational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which sets does −15 belong to?Answer: Integer, rational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which sets does −√9 belong to?Answer: Integer, rational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which sets does √2 belong to?Answer: Irrational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which sets does √10 belong to?Answer: Irrational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which sets does √16 belong to?Answer: Counting (natural), whole, integer, rational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which sets does √49 belong to?Answer: Counting (natural), whole, integer, rational, real. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Pick the number 12
Which of these is a whole number but not a counting number?Answer: 0. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which of these is an integer but not a whole number?Answer: −6. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which of these is rational but not an integer?Answer: 2/3. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which of these numbers is irrational: 0.121212…, √8, 1/9 or √36?Answer: √8. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which of these numbers is irrational: 1/100, 0.1001, 0.101001000100001… or 0.101010…?Answer: 0.101001000100001…. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which of these numbers is irrational: 22/7, 3.14, π or −3?Answer: π. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which of these numbers is irrational: −1/2, −√2, −2 or −0.2?Answer: −√2. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which of these numbers is irrational: √3, −√4, 4/1 or 0.5?Answer: √3. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which of these numbers is irrational: √4, √1, √2 or √0?Answer: √2. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which of these numbers is irrational: √5, −3, 0.6 or √25?Answer: √5. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which of these numbers is irrational: √81, √100, √121 or √50?Answer: √50. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
Which of these square roots is rational?Answer: √64. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
True or false 10
True or false: A decimal that repeats forever is irrational.Answer: False. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
True or false: A number can be both rational and irrational.Answer: False. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
True or false: Every counting number is a whole number.Answer: True. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
True or false: Every integer is a rational number.Answer: True. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
True or false: Every integer is a whole number.Answer: False. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
True or false: Every irrational number is a real number.Answer: True. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
True or false: Every rational number is an integer.Answer: False. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
True or false: Every whole number is an integer.Answer: True. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
True or false: The square root of every counting number is irrational.Answer: False. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
True or false: Zero is a rational number.Answer: True. Source: OpenStax Prealgebra 2e, section 7.1 Rational and Irrational Numbers. Read 2026-09-23.
How this list was compiled: Questions, wrong choices and explanations were written for this site. The definitions follow OpenStax Prealgebra 2e: counting (natural) numbers 1, 2, 3…, whole numbers 0, 1, 2, 3… (section 1.1, https://openstax.org/books/prealgebra-2e/pages/1-1-introduction-to-whole-numbers), integers, rational numbers as ratios of integers whose decimals stop or repeat, and irrational numbers whose decimals do neither (section 7.1, https://openstax.org/books/prealgebra-2e/pages/7-1-rational-and-irrational-numbers), both read 2026-09-23. That square roots of non-square counting numbers are irrational was checked on Wikipedia's Irrational number the same day. Licence: own work