Question Video: Rational and Irrational Numbers

Is 0.456 a rational or irrational number?


Video Transcript

Is 0.456 a rational or irrational number?

Let’s begin by recalling the definition of a rational number. A rational number is one that can be written in the form 𝑎 over 𝑏, where 𝑎 and 𝑏 are integers. They’re whole numbers. It follows then that an irrational number is one that can’t be written in this form. So let’s have a look at the number we’ve been given. 0.456 is a terminating decimal. So let’s see if we’re able to write it as a fraction. We spot that the number has four tenths, five hundredths, and six thousandths. This means we can write it as the sum of four-tenths, five hundredths, and six one thousandths. And then we remember that we can add fractions when their denominators are the same.

Let’s create a common denominator of 1000. To achieve this, we’re going to multiply the numerator and denominator of our first fraction by 100 and of our second fraction by 10. Four-tenths is equivalent to four hundred one thousandth. And five one hundredths is equal to fifty one thousandths. And once their denominators are equal, we simply add the numerators. 400 plus 50 plus six is equal to 456. And in turn, 0.456 is equal to four hundred and fifty-six thousandths.

We could also simplify this to 57 over 125. Though, this isn’t entirely necessary. All we really needed to show was that we could write our number as a fraction whose denominator and numerator are both integers. Since we’ve shown that 0.456 can be written as the quotient of two whole numbers, it’s 456 over 1000, we can say that 0.456 must indeed be a rational number. The answer is yes.

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