Repetend
A repetend is the repeating digit or group of digits in a repeating decimal. When you divide and the same remainder recurs, the decimal quotient cycles indefinitely. For example, 1/3 = 0.3333... where 3 is the repetend, written with a bar over it. 1/7 = 0.142857142857... where 142857 is the repetend. Recognizing repetends is a 7th grade math concept from Saxon Math Course 2 that deepens number sense and connects fractions to their decimal representations — a distinction critical for understanding rational numbers.
Key Concepts
Property The repeating digits of a decimal number are called the repetend. We use a bar over the repetend where it first appears to the right of the decimal point to write it concisely.
Examples $0.08333... = 0.08\overline{3}$ $5.14285714... = 5.\overline{142857}$ $454.54545... = 454.\overline{54}$.
Explanation Think of the bar as a crown for the digits that repeat forever! It’s a super handy shortcut, showing which numbers go on an infinite journey, so you don't have to write them out endlessly.
Common Questions
What is a repetend in math?
A repetend is the digit or block of digits that repeats infinitely in a repeating decimal. For 1/3 = 0.333..., the repetend is 3. It is written with a vinculum (bar) over the repeating digits.
Which fractions produce repeating decimals?
Any fraction whose denominator has prime factors other than 2 and 5 will produce a repeating decimal. Denominators like 3, 6, 7, 9, and 11 always give repetends.
How do you identify the repetend?
Perform long division. When the same remainder appears twice, the digits between those two occurrences form the repetend. For 1/7, the remainder cycle has length 6, giving the 6-digit repetend 142857.
What grade learns about repetends?
Repetends are introduced in 7th grade Saxon Math Course 2 as part of understanding the decimal representation of rational numbers and distinguishing terminating from repeating decimals.
How is a repeating decimal written?
Place a bar (vinculum) over the repeating digit or digits. For 0.333..., write 0.3 with a bar over the 3. For 0.142857142857..., place a bar over 142857.
Are all repeating decimals rational numbers?
Yes. Every repeating decimal can be written as a fraction, making it rational. Numbers like pi that never repeat and never terminate are irrational.