Terminating Decimal

Definition

A terminating decimal is a decimal number that can be expressed as a fraction with a finite number of digits after the decimal point. This means the decimal representation of the number eventually stops or terminates, rather than repeating infinitely.

5 Must Know Facts For Your Next Test

1. Terminating decimals can always be expressed as fractions with denominators that are powers of 10.
2. Any fraction with a denominator that is a product of 2s and 5s will have a terminating decimal representation.
3. Decimal numbers that cannot be written as fractions with denominators that are powers of 10 will have repeating decimal representations.
4. Rounding a terminating decimal to a certain number of decimal places will not change its essential nature as a terminating decimal.
5. Terminating decimals are a subset of rational numbers, as they can be expressed as the ratio of two integers.

Review Questions

• Explain the key characteristics that distinguish a terminating decimal from a repeating decimal.
  • The primary distinction between a terminating decimal and a repeating decimal is that a terminating decimal can be expressed as a fraction with a denominator that is a power of 10, meaning the decimal representation eventually stops or terminates. In contrast, a repeating decimal has a decimal representation that continues infinitely in a repeating pattern, rather than terminating. Terminating decimals are a subset of rational numbers, while repeating decimals are also rational numbers but have a more complex decimal expansion.
• Describe the relationship between terminating decimals and fractions with denominators that are powers of 10.
  • Terminating decimals can always be expressed as fractions with denominators that are powers of 10. This is because any fraction with a denominator that is a product of 2s and 5s will have a terminating decimal representation. Conversely, decimal numbers that cannot be written as fractions with denominators that are powers of 10 will have repeating decimal representations. This connection between terminating decimals and fractions with powers of 10 denominators is a key characteristic that distinguishes terminating decimals from repeating decimals.
• Analyze the significance of terminating decimals in the context of rational numbers and decimal expansions.
  • Terminating decimals are significant because they are a subset of rational numbers, which are numbers that can be expressed as the ratio of two integers. The fact that terminating decimals can be written as fractions with powers of 10 denominators makes them an important class of rational numbers, as this property allows for easy conversion between decimal and fractional representations. Additionally, the distinction between terminating and repeating decimals is crucial in understanding the decimal expansions of rational numbers, as it highlights the fundamental differences in how certain numbers can be expressed in decimal form.

Related terms

Repeating Decimal: A repeating decimal is a decimal number where the digits after the decimal point repeat infinitely in a pattern, rather than terminating.

Rational Number: A rational number is a number that can be expressed as the ratio of two integers, such as terminating and repeating decimals.

Decimal Expansion: The decimal expansion of a number is the representation of that number in decimal form, which can be either terminating or repeating.