Repeating & Terminating Decimals

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Numbers are fascinating! The ratio of two numbers can be written as a decimal which either terminates or repeats. In this lesson we delve into this great mystery.

Repeating and Terminating Decimals

I've decided to purchase some floor tiles for my math room. It's great to be inspired by numbers. There it is! A tile pattern with repeating numbers.

A collection of repeating numbers
Sequences of numbers.

These numbers are a reminder of repeating decimals which have a repeating series of numbers going on forever like .123123 · · ·. By the way, 41/333 = .123123 · · ·. Other decimals are terminating decimals and do not have a repeating part like .12 . By the way, 3/25 = .12 .

We can always write repeating and terminating decimals as the ratio of two natural numbers. You see 'ratio' and think rational number. However, there is also another type of decimal that goes on forever but does not have a repeating part. These are called irrational numbers and they cannot be written as a ratio of two numbers. Examples of irrational numbers are π = 3.14159265 · · · and the square root of 2. In this lesson, we are exploring only the decimals which come from rational numbers. This might help me rationalize the floor tile selection.

The Repetend

The repeating series of numbers in a repeating decimal is called the repetend. It's a nice word contracted from 'repetition at the end'. Let's start by identifying repetends. Do you see the repetend in 0.333 … ? Sure, the repetend is 3. How about the repetend of 0.1717 … If you said 17, then you are correct. Ready for some more? What's the repetend of 0.250 ? We might think the repetend of 0.250 is 0 because having a 0 at the end means we can always add another zero and the number won't change. This is a special case: if the repeating part is a zero, we say the decimal number is a terminating decimal. If the repetend is anything other than 0, we say the decimal number is a repeating decimal. How about a question? Which of the following decimal numbers are repeating and which are terminating: 0.25, 0.3, 0.1212 … and 0.123123 … ? Answer: the first two are terminating decimals. The 0.1212 … and the 0.123123 … are repeating decimals.

There is no universally accepted notation for a repeating decimal. Sometimes we will use three dots like in 0.3 … Sometime the three dots will be centered like 0.3 · · · and sometimes a bar will be written above the repetend like 0. 3̅.

Long Division and Factors

Let's only deal with natural numbers: the positive integers greater than zero, like 1, 2, 3, · · · . If we divide one of these natural numbers by a larger natural number, we will always get a fraction less than one. For example, 1/4 is less than one and so is 2500/9999.

The decimal number for these fractions will either be a terminating decimal or a repeating decimal. If we divide 1 by 4 we get 0.25 followed by as many 0's as we'd like. This is a terminating decimal number. If we divide 2500 by 9999 we get 0.25002500· · · . This is a repeating decimal number where the repetend is 2500.

Factoring Denominators

The denominator of the fraction holds some useful information. Going back to 1/4, let's factor the 4. We get 2(2). Going back to 2500/9999 let's factor the 9999. We get 3(3)(11)(101). If the factors of the denominator are only 2's and/or 5's, the decimal number will terminate. If any of the factors are other than 2 or 5, the decimal number will repeat. Amazing! Will this help me in selecting floor tiles? Well, I hope so.

Let's check this out further. What if we consider 3/20 and make a prediction. The factorization of 20 is 2(2)(5). Okay, no factors other than 2 and/or 5. A terminating decimal will result. We do the division and 3/20 and get 0.15 which terminates. Great! What about 4/21 ? Factoring 21 gives us 3(7). Prediction: 4/21 is a repeating decimal. Doing the long division we get 0.190476190476… The repetend is 190476 . Wonderful! Factoring the denominator is very useful.

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