← Back to the Irrational Numbers worksheet
Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Standards: 8.NS.A.1, 8.NS.A.2 · Level: Grade 8
Problem 1 Classify the numbers
0.75 = 34 is rational. √81 = 9 is rational. √35 is not a perfect square, so it is irrational. −79 is rational. π is irrational. 0.121212… repeats, so it is rational.
Irrational: √35 and π. The other four are rational.
Problem 2 Perfect squares and roots
√16 = 4, √144 = 12 and √0.25 = 0.5 are whole numbers or terminating decimals. √50 and √6 lie between consecutive integers and are not perfect squares.
Rational: √16, √144, √0.25. Irrational: √50, √6.
Problem 3 Terminating decimals as fractions
(a) 0.375 = 3751,000. Divide top and bottom by 125 to get 38.
(b) 2.6 = 2610 = 135.
(a) 38 (b) 135
Problem 4 Single-digit repeating decimals
A one-digit block over 9 works. (a) Let x = 0.888…; 10x − x = 8, so x = 89.
(b) 10x − x = 2, so x = 29.
(a) 89 (b) 29
Problem 5 Fractions as decimals
7 ÷ 8 = 0.875, which stops. 5 ÷ 11 = 0.454545…, a repeating block of 45.
78 = 0.875; 511 = 0.4545… repeats.
Problem 6 Between two integers
6² = 36 and 7² = 49, and 36 < 40 < 49. So 6 < √40 < 7. The point at about 6.3 agrees, since 6.3² = 39.69.
√40 is between 6 and 7.
Problem 7 Two-digit repeating blocks
Multiply by 100 so the block lines up. (a) 100x − x = 23, so x = 2399.
(b) 100x − x = 7, so x = 799.
(a) 2399 (b) 799
Problem 8 Three-digit repeating block
Let x = 0.405405…; 1,000x = 405.405405…. Subtract: 999x = 405, so x = 405999. Divide by 27: 1537. Check: 15 ÷ 37 = 0.405405…
1537
Problem 9 A block that starts late
Let x = 0.1666…. Then 10x = 1.666… and 100x = 16.666…. Subtract: 100x − 10x = 15, so 90x = 15 and x = 1590 = 16.
16
Problem 10 Square root to the nearest tenth
5.2² = 27.04 and 5.3² = 28.09. Since 28 is much closer to 28.09, √28 is close to 5.3. Check 5.29² = 27.9841, so √28 ≈ 5.29, which rounds to 5.3.
√28 ≈ 5.3
Problem 11 Match roots to points
Squares: 2.2² = 4.84, 2.3² = 5.29, so √5 ≈ 2.24. 2.6² = 6.76, 2.7² = 7.29, so √7 ≈ 2.65. 3.6² = 12.96, so √13 ≈ 3.61. Order the points left to right.
A = √5, B = √7, C = √13
Problem 12 Root versus decimal
(a) 5.5² = 30.25, which is more than 30, so √30 < 5.5. (b) 7² = 49, which is less than 50, so √50 > 7.
(a) √30 < 5.5 (b) √50 > 7
Problem 13 Order from least to greatest
√17: 4.1² = 16.81 and 4.2² = 17.64, so √17 ≈ 4.123. 1 + π ≈ 4.1416. 418 = 4.125. And 4.2 is exact.
4.123 < 4.125 < 4.1416 < 4.2.
√17, 418, 1 + π, 4.2
Problem 14 Rational or irrational results
(a) Adding a rational number to √2 leaves a never-ending, nonrepeating decimal: irrational. (b) √2 · √2 = 2, which is rational. (c) A nonzero rational times π stays irrational.
(a) irrational (b) rational (c) irrational
Problem 15 Square garden
Side = √75. Since 8² = 64 and 9² = 81, the side is between 8 and 9 feet. 8.6² = 73.96 and 8.7² = 75.69, and 75 is closer to 75.69 than to 73.96. Check 8.66² = 74.9956.
About 8.7 feet
Problem 16 Lap fraction
Let x = 0.6363…. Multiply by 100: 100x = 63.6363…. Subtract: 99x = 63, so x = 6399 = 711.
711 mile
Problem 17 Is 0.7272… the same as 0.72?
0.7272… = 7299 = 811. And 0.72 = 72100 = 1825. Common denominator 275: 811 = 200275 and 1825 = 198275.
Since 200 > 198, the repeating decimal is larger. Ravi is wrong; 0.7272… > 0.72.
Problem 18 Tile side length
12² = 144 and 13² = 169, so the side is between 12 and 13. Then 12.2² = 148.84 and 12.3² = 151.29. Since 150 is between these, √150 is between 12.2 and 12.3. The midpoint 12.25² = 150.0625 is above 150, so √150 < 12.25 and rounds down.
About 12.2 inches
Problem 19 Is 3.14 equal to π?
3.14 is a terminating decimal, so it is rational. π is irrational, so it cannot equal any terminating or repeating decimal. Using π = 3.14159…, we get 3.14 < π < 3.15.
3.14 is only an approximation; 3.14 < π < 3.15.
Problem 20 Close call
6.7² = 44.89, which is less than 45, so √45 > 6.7. Then 6.71² = 45.0241 is more than 45, so 6.70 < √45 < 6.71. The difference is less than 0.01 and about 0.008, which rounds to 0.01.
√45 is greater, by about 0.01.