Question

Is 0.2525252525252525 a rational or irrational

Answer 1

Answer: Yes

Explanation:

A repeating decimal can be a rational number.

Answer 2

Rational

Explanation:

Rational and irrational numbers are 2 of the broadest categories within real numbers.

Writing the Number as a Fraction

Rational numbers are defined as numbers that can be written as simple fractions. Simple fractions have integers for the numerator and denominator. This means neither the numerator nor denominator is a fraction or decimal. The number above is 0.25 repeated. When 2 digits are repeated you can place that number over 99 to create an equivalent fraction.

• 0.2525252525252525 =
  `\displaystyle (25)/(99)`

Since this number can be written as a fraction, it must be a rational number.

Bar Notation

Additionally, repeated decimals are always rational. In the example above, the digits .25 are repeated. If you don't want to write it as a fraction, you can use bar notation. Bar notation helps to shorten repeated decimals and show that they are rational. In this notation, you place a bar over the digits that are repeated.

• 0.2525252525252525 =
  `0.\overline{25}`