Part 6: Number Theory

# 6.3: Terminating and repeating decimals

Notes

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.

Video Tips

Practice Exercises

Do the following exercises to practice expressing decimals as fractions.