Rational and irrational numbers worksheet

Grade 8 math, Common Core 8.NS.A.1. 10 problems with an answer key and worked steps. Free to print, and you can make new problems any time.

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Rational and irrational numbers

Grade 8 math, Common Core 8.NS.A.1

Answer each question about rational and irrational numbers.

1.
Is √4 rational or irrational?
2.
Between which two whole numbers is √62?
3.
Order from least to greatest: √8, 3.05, 72, 2.9
4.
Is 0.2936… (the digits repeat) rational or irrational?
5.
Between which two whole numbers is √91?
6.
Order from least to greatest: 3.05, 2.9, 72, π
7.
Between which two whole numbers is √6?
8.
Is √202 rational or irrational?
9.
Between which two whole numbers is √63?
10.
Order from least to greatest: √10, √8, 72, 3.05

Answer key: Rational and irrational numbers

Grade 8 math. Each answer has a few worked steps, to help check the method as well as the answer.

1. Rational (it equals 2)
2. 7 and 8
7² = 49 and 8² = 64
3. √8, 2.9, 3.05, 72
√8 ≈ 2.83, 72 ≈ 3.5
4. Rational
5. 9 and 10
9² = 81 and 10² = 100
6. 2.9, 3.05, π, 72
π ≈ 3.14, 72 ≈ 3.5
7. 2 and 3
2² = 4 and 3² = 9
8. Irrational
9. 7 and 8
7² = 49 and 8² = 64
10. √8, 3.05, √10, 72
√8 ≈ 2.83, √10 ≈ 3.16, 72 ≈ 3.5

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About rational and irrational numbers

A rational number can be written as a fraction of two integers: 0.25, −8, 23 and 0.121212… (a repeating decimal) are all rational. An irrational number can’t: its decimal goes on forever without repeating, like π or √2. The square root of a whole number that isn’t a perfect square is irrational, and you can estimate it between two whole numbers.

Worked example 1

Between which two whole numbers is √46?

Step 1.6² = 36 and 7² = 49

Answer: 6 and 7

Worked example 2

Order from least to greatest: 3.2, √10, π, 103

Step 1.π ≈ 3.14, √10 ≈ 3.16, 103 ≈ 3.33

Answer: π, √10, 3.2, 103

Example 3

Is √64 rational or irrational?

Answer: Rational (it equals 8)

Watch out for

√49 is rational because it equals 7. √50 is irrational.
Decimals that stop or repeat are rational.
To estimate √40, find the perfect squares around 40: 36 and 49, so √40 is between 6 and 7.

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