Terminating

Definition

A terminating decimal is a decimal number that has a finite number of digits after the decimal point. It can be expressed as a fraction and eventually stops repeating. Terminating decimals are a type of rational number, as opposed to irrational numbers which have an infinite, non-repeating decimal representation.

5 Must Know Facts For Your Next Test

1. Terminating decimals can always be written as a fraction with a denominator that is a power of 10.
2. The decimal representation of a terminating decimal will eventually stop, whereas the decimal representation of an irrational number goes on forever without repeating.
3. All fractions with a denominator that is a product of 2's and 5's will have a terminating decimal representation.
4. Rational numbers with denominators that are not a product of 2's and 5's will have a repeating decimal representation, not a terminating decimal.
5. Terminating decimals are a subset of rational numbers, and can be converted back and forth between decimal and fraction form.

Review Questions

• Explain how terminating decimals are related to rational numbers.
  • Terminating decimals are a type of rational number, as they can be expressed as a fraction with a finite decimal representation. All terminating decimals are rational numbers, because their decimal expansions eventually stop. This is in contrast to irrational numbers, which have infinite, non-repeating decimal expansions that cannot be written as simple fractions.
• Describe the characteristics that distinguish terminating decimals from repeating decimals.
  • The key distinguishing feature of terminating decimals is that their decimal representation eventually stops, whereas repeating decimals have a digit or set of digits that repeat infinitely. Terminating decimals can always be written as a fraction with a denominator that is a power of 10, while repeating decimals have denominators that are not a product of 2's and 5's. Terminating decimals are a subset of rational numbers, while repeating decimals are also a type of rational number with a more complex decimal expansion.
• Analyze the relationship between the denominator of a fraction and whether its decimal representation will terminate or repeat.
  • The denominator of a fraction determines whether its decimal representation will terminate or repeat. Fractions with denominators that are a product of 2's and 5's will have terminating decimal representations, because the prime factors of the denominator can be cancelled out in the division process. In contrast, fractions with denominators that contain prime factors other than 2 and 5 will have repeating decimal representations that do not terminate. This is a fundamental property that distinguishes terminating decimals, which are a subset of rational numbers, from irrational numbers with non-repeating decimal expansions.

Related terms

Rational Number:

A rational number is a number that can be expressed as a fraction, with a finite or repeating decimal representation.

Irrational Number:

An irrational number is a number that cannot be expressed as a simple fraction, and has an infinite, non-repeating decimal representation.

Decimal Expansion:

The representation of a number as a decimal, with digits to the right of the decimal point.