Link: https://www.gregmat.com/quizzes/quiz/prepswift-terminating-versus-non-terminating (Question 1)
Hi,
Other than just “knowing”, how could I test without a calculator whether a fraction is terminating or non-terminating? Even with a calculator, the answer only gives a few places after the decimal point even for numbers I know are non-terminating (like 1/3) so just rely on common knowledge?
Same with roots of numbers. Just because they aren’t an integer, does that necessarily mean they are non-terminating?
If your simplified fraction can be expressed as:
\frac{k}{2^a 5^b}
for some integer k and non-negative integers a and b, then it must be terminating. The reverse is also true: if a rational number cannot be expressed in this form, it is non-terminating.
As for irrational numbers, you will be able to identify them by observation on the GRE. Moreover, keep in mind that any non-zero rational multiple of an irrational number is also irrational. That is more or less all the tools you need.
Yes, the square root of a perfect square is an integer. The square root of any other positive integer is irrational. I’m assuming you know what an irrational number is. Anyhow, you can also extend this notion to the rationals: any rational number whose reduced form has a numerator and denominator that are both perfect squares is itself a perfect square. Otherwise, its square root is irrational.
For example, \frac 14 is a perfect square in the rationals because 1 and 4 are both perfect squares. \frac 12 is not a perfect square (which implies its square root is irrational) because 1 is a perfect square, whereas 2 isn’t.
The same argument holds for cube roots, fourth roots, and so on.