Why do rational numbers repeat?

Question

Genesis Amaya · Accepted Answer

See explanation...

Suppose #p/q# is a rational number, where #p# and #q# are both integers and #q > 0#.

To obtain the decimal expansion of #p/q# you can long divide #p# by #q#.

During the process of long division, you eventually run out of digits to bring down from the dividend #p#. From that point on, the digits of the quotient are determined purely by the sequence of values of the running remainder, which is always in the range #0# to #q-1#.

Since there only #q# different possible values for the running remainder, it will eventually repeat, and so will the digits of the quotient from that point.

For example: #186/7# ...

Notice the sequence of remainders: #4, color(blue)(4), 5, 1, 3, 2, 6, color(blue)(4), 5# which starts to repeat again.

Nova Allen · Answer

undefined 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.