How to Tell That a Number Is Rational

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The easiest way to tell if a number is rational or not is to attempt to express it as a fraction. If you can, then the number is rational. If not, then the number is irrational. According to Math Is Fun, the formal definition of a rational number is "a number that can be in the form p/q, where p and q are integers and q is not equal to zero." All integers are rational numbers, because they can be written as a fraction (for example, the integer 8 = 8/1). For decimals, though, the process takes a few steps.

  • Use your knowledge of decimal-to-fraction conversions to determine if your decimal is a rational number. For example, you can write the decimal 0.5 as the fraction 1/2. So, if your number is 42.5, multiply 42*2 and add one for the 0.5, and you have 85/2. This matches the p/q test, and so it is rational. Other common decimal-to-fraction conversions include 0.25 = 1/4, 0.16666 (repeating sixes) = 1/6, 0.2 = 1/5, 0.25 = 1/4, 0.33333 (repeating threes) = 1/3, and 0.5 = 1/2.

    If the decimal doesn't match a fraction you already know, move on to Step 2.

  • Access the Internet and visit a decimal-to-fraction converter like the one hosted by WebMath (see the link under "Resources" below). Enter the decimal number into the field provided and then click on the button that will convert it to a fraction.

  • Classify radicals that cannot be converted to an even fraction and transcendental numbers, such as pi or e, as irrational numbers. While the square root of four is two, which is a rational integer, and the square root of nine is three, another rational integer, the square root of 12 is not an integer. You can break down 12 into its factors (223), but it is still not rational. Uneven square roots and transcendental numbers do not resolve to p/q fractions.

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