Grade 8 Irrational Numbers: Unit Review

Worked examples

Real problems from the practice worksheets, solved one step at a time.

1. Example 1

   Estimate √20 to the nearest tenth.

   1. 4² = 16 and 5² = 25, so √20 is between 4 and 5.
   2. Try tenths: 4.4² = 19.36 and 4.5² = 20.25, so √20 is between 4.4 and 4.5.
   3. Halfway is 4.45, and 4.45² = 19.8025. 20 is more than 19.8025, so √20 is closer to 4.5.
   4. So √20 ≈ 4.5 to the nearest tenth.

   Answer: 4.5

2. Example 2

   How many of these numbers are irrational? √82 · 7/10 · √144 · 7.25 · √13

   1. A rational number can be written as a fraction of two integers; its decimal ends or repeats. An irrational number cannot — its decimal goes on forever without repeating.
   2. 82 is not a perfect square (9² = 81, 10² = 100), so √82 is irrational.
   3. 7/10 is a fraction of two integers — rational.
   4. √144 = 12 (12 × 12 = 144) — rational.
   5. 7.25 is a decimal that ends — rational.
   6. 13 is not a perfect square (3² = 9, 4² = 16), so √13 is irrational.
   7. The irrational ones are √82, √13 — that is 2 numbers.

   Answer: 2

3. Example 3

   Points are marked on a number line at 9.9, 10.4, 10.9 and 11.4. Which point is closest to √107?

   1. Square the points around it: 9.9² = 98.01 and 10.4² = 108.16, so √107 is between 9.9 and 10.4.
   2. Halfway between them is 10.15, and 10.15² = 103.0225. 107 is more than that, so √107 is closer to 10.4.
   3. The closest point is 10.4.

   Answer: 10.4