# Why do rational numbers repeat?

See explanation...

Suppose

To obtain the decimal expansion of

During the process of long division, you eventually run out of digits to bring down from the dividend

Since there only

For example:

Notice the sequence of remainders:

By signing up, you agree to our Terms of Service and Privacy Policy

Rational numbers repeat because their decimal representations either terminate or form repeating patterns due to the way they are expressed as fractions of two integers. This repetition occurs when the denominator of the fraction has prime factors other than 2 and 5. In such cases, the decimal representation of the rational number repeats because there are only a finite number of remainders possible when dividing the numerator by the denominator. This repetition is a consequence of the arithmetic properties of fractions and the base-10 system used to represent numbers.

By signing up, you agree to our Terms of Service and Privacy Policy

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7