Grade 8 Irrational Numbers: Approximating Irrationals

Worked examples

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

1. Example 1

   √87 is between two consecutive whole numbers. What is the smaller one?

   1. Find the perfect squares on each side of 87: 9² = 81 and 10² = 100.
   2. 81 < 87 < 100, so 9 < √87 < 10.
   3. The smaller one is 9.

   Answer: 9

2. Example 2

   Estimate √106 to the nearest whole number.

   1. 10² = 100 and 11² = 121, so √106 is between 10 and 11.
   2. Halfway between them is 10.5, and 10.5² = 110.25. 106 is less than 110.25, so √106 is closer to 10.
   3. So √106 ≈ 10 to the nearest whole number.

   Answer: 10

3. Example 3

   Estimate √83 to the nearest tenth.

   1. 9² = 81 and 10² = 100, so √83 is between 9 and 10.
   2. Try tenths: 9.1² = 82.81 and 9.2² = 84.64, so √83 is between 9.1 and 9.2.
   3. Halfway is 9.15, and 9.15² = 83.7225. 83 is less than 83.7225, so √83 is closer to 9.1.
   4. So √83 ≈ 9.1 to the nearest tenth.

   Answer: 9.1