Terminating decimal

Definition

A terminating decimal is a decimal representation of a number that has a finite number of digits after the decimal point. This means that the decimal expansion comes to an end, rather than continuing infinitely. Terminating decimals are a specific subset of rational numbers, which can be expressed as the quotient of two integers.

5 Must Know Facts For Your Next Test

1. Terminating decimals can always be expressed as fractions with denominators that are powers of 10, such as 10, 100, or 1000.
2. Examples of terminating decimals include numbers like 0.5, 1.25, and 2.75, each of which has a clear endpoint in their decimal form.
3. Any terminating decimal can be converted into a fraction by identifying the place value of the last digit.
4. The set of terminating decimals is a subset of rational numbers, which means all terminating decimals are rational but not all rational numbers are terminating decimals.
5. The number of digits in a terminating decimal directly relates to its fractional representation; for instance, the decimal 0.75 has two digits after the decimal point, corresponding to a denominator of 100.

Review Questions

• How do you determine if a decimal is terminating or not?
  • To determine if a decimal is terminating, you need to check its fractional form. If the denominator (after simplifying) has only prime factors of 2 and/or 5, then it will result in a terminating decimal. For example, the fraction 3/8 simplifies to a decimal 0.375, which terminates. Conversely, if any prime factor other than 2 or 5 exists in the denominator, like in 1/3 (which results in 0.333...), it will not terminate.
• Compare and contrast terminating decimals and repeating decimals with examples.
  • Terminating decimals and repeating decimals are both types of decimal representations, but they differ in their structure. A terminating decimal ends after a finite number of digits, like 0.75, while a repeating decimal continues infinitely with a repeating pattern, such as 0.666... This distinction arises from their mathematical properties; terminating decimals can be expressed as fractions with denominators made up solely of the factors 2 and 5, while repeating decimals involve other prime factors in their denominators.
• Evaluate how understanding terminating decimals impacts your comprehension of rational numbers as a whole.
  • Understanding terminating decimals is crucial for grasping the broader concept of rational numbers because it highlights one specific case where numbers can be represented both as fractions and as decimals. Recognizing that all terminating decimals are rational reinforces the idea that any decimal number can often be represented in fractional form. This comprehension lays the groundwork for exploring more complex relationships among rational numbers, including those involving repeating decimals and irrational numbers, deepening overall mathematical literacy.

Related terms

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

Repeating Decimal: A repeating decimal is a decimal representation that continues infinitely with a repeating sequence of digits after the decimal point.

Fraction: A fraction is a numerical expression that represents the division of one integer by another, often expressed in the form a/b.