Video Transcript
Is 0.4 recurring a rational or an
irrational number?
In this question, we are given a
number and asked to determine if the number is rational or irrational.
To answer this question, we can
start by noting that the number we have been given has a horizontal bar over the
decimal digit four. We can recall that a horizontal bar
or a dot over decimal digits mean that those digits repeat. Therefore, this number can be
written as 0.444 and this expansion continues indefinitely.
To determine if this number is
rational or irrational, we can recall that rational numbers are the quotient of
integers. That is, numbers of the form 𝑝
over 𝑞, where 𝑝 and 𝑞 are integers and 𝑞 is nonzero. An irrational number is any
nonrational number, so they are numbers that cannot be written in this way.
It is difficult to show directly if
we can write this number as the quotient of two integers. So instead, we can recall that any
number with a finite or repeating decimal expansion is a rational number and any
number with an infinite nonrepeating decimal expansion is irrational.
We can see that the given number
has a repeating decimal expansion, since the digit of four has a horizontal bar over
it. We can write this as 0.4 recurring
is an element of the rational numbers, 𝑞, or we can just say that it is a rational
number.
