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What Is Decimal Expansion? Your Quick and Easy Guide

Value of pi written on a blackboard

Decimal expansion may sound complicated, but it only means turning a fraction or a whole number into its decimal representation. This is helpful if you are working with a calculator or want to perform operations on numbers in a decimal format.

Here, we'll show you how to use decimal expansion and the difference between finite and infinite decimal expansion.

But first, a quick review: The decimal place marks different units such as tenths, hundredths, thousandths, and so on.

And a quick note: Irrational numbers, such as Euler's number, pi, or the golden ratio, cannot be expressed as a quotient of a numerator and denominator (a fraction).

Finite Decimal Expansion vs. Infinite Decimal Expansion

There are two types of decimal expansion: finite decimal expansion and infinite decimal expansion (sometimes called periodic decimal expansion).

A finite decimal expansion has a clear end. Take the fraction 1/2. Its decimal expansion is 0.5, which is non-repeating. The terminating decimal is the 5 in the tenths place digit.

An infinite decimal expansion does not have a clear end — it goes on in an infinite sequence. An easy example is 1/3. Its decimal expansion is 0.3333, or Example of an infinite decimal expansion, in which the bar represents that the digit repeats forever.

Another example of an infinite decimal expansion is 1/11, which expands to 0.0909 or Example of an infinite decimal expansion, (meaning the .09 repeats forever). The period, or the group of digits that repeat, is 09. The number of digits in the decimal expansion goes on infinitely.

When To Use Decimal Expansion

Remember that finding a number’s decimal expansion is just a fancy way of putting it into decimal form. Keep track of the decimals, and put a bar over the digits that repeat on forever.

Sometimes, you will need to convert between fractions and decimals. Other times, you just need to recognize when a number has a repeating decimal expansion or a terminating decimal expansion.

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