Grade 8 Irrational Numbers: Rational vs Irrational

Worked examples

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

1. Example 1

   Write the repeating decimal 0.404040… (the 40 repeats) as a fraction in simplest form.

   1. Let x = 0.404040…; the block "40" has 2 digits, so multiply by 100: 100x = 40.4040…
   2. Subtract x from 100x — the repeating parts cancel: 100x − x = 40.4040… − 0.4040…, so 99x = 40.
   3. Divide by 99: x = 40/99.
   4. So 0.404040… = 40/99.

   Answer: 40/99

2. Example 2

   Which number is irrational? Choices: √16 · 0.5 · 0.1010010001… · √100

   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. √16 = 4 (4 × 4 = 16) — rational.
   3. 0.5 is a decimal that ends — rational.
   4. 0.1010010001… never ends and its digits never settle into a repeating block — irrational.
   5. √100 = 10 (10 × 10 = 100) — rational.
   6. So the irrational number is 0.1010010001…

   Answer: 0.1010010001…

3. Example 3

   Write 0.4111… (only the 1 repeats) as a fraction in simplest form.

   1. Let x = 0.4111…; then 10x = 4.11… and 100x = 41.11…
   2. Subtract — the repeating parts cancel: 100x − 10x = 41.11… − 4.11… = 37, so 90x = 37.
   3. Divide by 90: x = 37/90.
   4. So 0.4111… = 37/90.

   Answer: 37/90