Video Transcript
Is 0.456 repeating a rational or an
irrational number?
We can recall that a rational
number can be expressed as a fraction 𝑝 over 𝑞, where 𝑝 and 𝑞 are integers and
𝑞 is not equal to zero. An irrational number is a number
that isn’t rational. So, in order to check if 0.456
repeating is a rational number, we need to check if we can write it as a fraction 𝑝
over 𝑞. Here, we’re going to use a neat
method to write this repeating decimal as a fraction. And it begins by defining a
variable 𝑥 which is equal to 0.456 repeating. We can say that 𝑥 is equal to
0.456456456 and so on. In the next step, we create another
value which has the same decimal digits as 𝑥 does. As we have three digits that
repeat, then if we multiply by 10 to the third power, that’s the same as multiplying
by 1000. And so, we’ll have 1000𝑥 equals
456.456456 and so on.
We now have two values that have
the same decimal digits. And therefore, if we were to
calculate 1000𝑥 subtract 𝑥, this would give us 456 as each decimal digit will be
subtracted from another one of equal value. Continuing our calculation then, we
can write that 999𝑥 is equal to 456. And rearranging by dividing both
sides by 999 will give us that 𝑥 equals 456 over 999. As we’ve already defined 𝑥 to be
0.456 repeating, then we have proved that this decimal can be written as a
fraction. As both the numerator and
denominator are integers and the denominator is not equal to zero, it fits with the
definition of a rational number. So, 0.456 repeating is a rational
number.
