6.3: Terminating and repeating decimals

In the following, “A/B” represents the fraction “A over B.” For example, “2/3” refers to the fraction “2 over 3” or “two-thirds.”

Rational, irrational, terminating, and repeating decimals

Any decimal number is either

• rational, if it can be expressed as a fraction (e.g., 2/5 = 0.4 or 1/3 = 0.333333…),
• or irrational, if not (e.g., √3 = 1.7320508076… or π = 3.141592654…).

The collection of all rational and irrational decimal numbers is called the real numbers. Yes, there are “unreal” numbers too (called imaginary numbers), but we’re not going to go there!

Any rational number can either be expressed as a

• terminating decimal (i.e., a decimal with a finite number of nonzero digits such as 0.4)
• or a repeating decimal (i.e., a decimal with a single digit or a fixed string of digits that repeats forever such as 0.333333…)

Expressing a fraction as a decimal (either terminating or repeating) simply involves long division using the methods in 4.1: Dividing integers. However, expressing a terminating or repeating decimal as a fraction can be more involved:

Express a terminating decimal as a fraction

• Example: Express 0.9876 as a fraction.
• Method: Use a denominator that is a power of 10 and simplify using 1.3: Simplifying fractions.
• 0.9876 = 9,876 ÷ 10,000 = 9,876/10,000.
• 9,876/10,000 = (9,876÷4)/(10,000÷4) = 2,469/2,500.

Express a repeating decimal as a fraction

• Example 1: Express 0.676767… as a fraction.
• Method: Multiply by a power of 10 and subtract so that the result is a whole number.
• Let N = 0.676767…
• Then, 100N = 67.676767… [multiply by a power of 10]
• So, 100N – N = 67.676767… – 0.676767… [subtract first equation from second equation]
• So, 99N = 67. [simplify]
• So, N = 67/99. [divide by the number on the left]

• Example 2: Express 0.916666… as a fraction.
• Let N = 0.916666…
• Then, 100N = 91.6666… [multiply by a power of 10]
• And 1000N = 916.6666… [multiply by a power of 10]
• So, 1000N – 100N = 916.6666… – 91.6666… [subtract second equation from third equation]
• So, 900N = 825. [simplify]
• So, N = 825/900 = 11/12. [divide by the number on the left and simplify]

The video below works through some examples of expressing decimals as fractions.