For fraction p/q to be a terminating decimal, the numerator

[esledge]

This rule took a while for me to internalize. It's tough to picture a decimal terminating when the denominator is so huge, such as DWG's example of 43/256. I found it helped me to think about the basic patterns:

1/2^1 = 0.5
1/2^2 = 0.25
1/2^3 = 0.125
1/2^4 = 0.0625
1/2^5 = 0.03125
1/2^6 = 0.015625
1/2^7 = 0.0078125

1/5^1 = 0.2
1/5^2 = 0.04
1/5^3 = 0.008
1/5^4 = 0.0016
1/5^5 = 0.00032
1/5^6 = 0.000064
1/5^7 = 0.0000128

Every one of these terminates, and the pattern indicates that would continue to be true for higher powers. The number of decimal places increases along with the powers of 2 or 5, but the number of decimal places will always be finite.

In contrast, any factors other than 2 or 5 in the denominator can quickly be shown to be non-terminating, even for the most basic case (exponent of 1). Higher powers would be even messier:
1/3 = 0.33333(3 repeating)
1/6 = 0.16666(6 repeating)
1/7 = 0.142857(142857 repeating)
1/9 = 0.11111(1 repeating)

[victoria.babin]

I have a couple of follow up questions about this -- I reviewed the above and I don't think they were covered.

1) The definition of a terminating decimal is that ANY integer is in the numerator, correct? At one point in the explanation above it appeared that the numerator had to be a non-negative power of 2. Just looking for clarification.

2) If it can be any integer in the numerator, I'm confused about the necessary power of 2 or power of 5 constraint in the denominator. What about a terminating decimal like (9)/(3)=3? The original question on the CAT exam used the example of a whole number, 36, in its definition of a terminating decimal.

Thanks for the help.

[esledge]

1) The definition of a terminating decimal is that ANY integer is in the numerator, correct? At one point in the explanation above it appeared that the numerator had to be a non-negative power of 2. Just looking for clarification.

The numerator can be anything. For determining whether a decimal terminates, we only care about the factors that remain in the denominator after the fraction has been reduced as much as possible.

2) If it can be any integer in the numerator, I'm confused about the necessary power of 2 or power of 5 constraint in the denominator. What about a terminating decimal like (9)/(3)=3? The original question on the CAT exam used the example of a whole number, 36, in its definition of a terminating decimal.

After you reduce 9/3 as much as possible, you end up with 3/1. Thus, the denominator only has powers of 2 and 5 (i.e. 2^0 or 5^0)--no other factors.

For another example: 18/30 is a terminating decimal. Even though there is a "problematic" factor of 3 in the denominator of 30, it is cancelled by a 3 in the numerator.

18/30 = 3/5 = 3/(5^1) = 0.6

[bocu.alina]

Harish, I believe you mean "denominator" in the following part of the explanation:

1) The fraction should have a numerator of 2, 4, 8, 16 etc. OR all non-negative powers of 2. (Ex: 2^2 = 4, 2^3 = 8)
Ex: 1/2 = 0.5, 5/4 = 1.25, 18/8 = 2.25

[bocu.alina]

Harish, I believe you mean "denominator" in the following part of the explanation:

1) The fraction should have a numerator of 2, 4, 8, 16 etc. OR all non-negative powers of 2. (Ex: 2^2 = 4, 2^3 = 8)
Ex: 1/2 = 0.5, 5/4 = 1.25, 18/8 = 2.25

[tim]

Alina,
Thanks for pointing this out. Of course, given that this thread is two years old, I’m guessing Harish is no longer around to see either of your responses.. :)