Terminating Decimal

Definition

A terminating decimal is a decimal number that can be expressed as a fraction with a finite number of digits in the denominator. It is a decimal representation that eventually ends or 'terminates' after a finite number of digits.

5 Must Know Facts For Your Next Test

1. Terminating decimals can always be written as a ratio of two integers, making them rational numbers.
2. The decimal representation of a terminating decimal will eventually end, rather than repeating endlessly like a repeating decimal.
3. Dividing any integer by another integer will always result in a terminating decimal, as long as the denominator is not a factor of 10.
4. Fractions with denominators that are powers of 10 (e.g., 1/10, 1/100, 1/1000) will always have terminating decimal representations.
5. Terminating decimals can be converted to fractions by writing the decimal as a ratio with the appropriate power of 10 in the denominator.

Review Questions

• Explain how terminating decimals are related to rational numbers and their decimal representations.
  • Terminating decimals are a type of rational number, as they can be expressed as a ratio of two integers. This means that the decimal representation of a terminating decimal will eventually end, rather than repeating endlessly like a repeating decimal. Terminating decimals can always be converted to a fraction by writing the decimal as a ratio with the appropriate power of 10 in the denominator. This connection between terminating decimals, rational numbers, and their decimal expansions is an important concept in understanding the properties of decimal representations.
• Describe the process of converting a terminating decimal to a fraction.
  • To convert a terminating decimal to a fraction, you can write the decimal as a ratio with the appropriate power of 10 in the denominator. For example, the terminating decimal 0.375 can be written as the fraction 375/1000, since the decimal representation ends after three digits. Similarly, the terminating decimal 0.25 can be written as the fraction 25/100, as the decimal representation ends after two digits. This process of converting terminating decimals to fractions is a useful skill in understanding the relationship between decimal and fractional representations of numbers.
• Analyze the factors that determine whether a decimal representation will be terminating or repeating.
  • The key factor that determines whether a decimal representation will be terminating or repeating is the denominator of the fraction being represented. If the denominator is a factor of 10 (e.g., 2, 5, 10, 20, 50, 100, etc.), then the decimal representation will be terminating. This is because the powers of 10 in the denominator can be canceled out, resulting in a finite decimal expansion. Conversely, if the denominator is not a factor of 10, the decimal representation will be repeating, as the division process will continue endlessly without a finite pattern. Understanding this relationship between the denominator and the type of decimal representation is crucial in working with rational numbers and their decimal expansions.

Related terms

Repeating Decimal: A repeating decimal is a decimal number where the digits after the decimal point repeat endlessly in a pattern.

Rational Number: A rational number is a number that can be expressed as a ratio of two integers, where the denominator is not zero.

Decimal Expansion: The decimal expansion of a number is the representation of that number in decimal form, using a finite or infinite sequence of digits.