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.
