terminating decimal

maria · Posted: Thu Jun 05, 2008 6:47 pm Post subject: terminating decimal

any decimal that has only a finite number of nonzero digits is a terminating decimal. for example, 24, 0.82, and 5.096 are three terminating numbers. If r and s are positive integers and the ratio is r/s is expressed as a decimal, is r/s a terminating decimal?
1. 90<r< 100
2. s = 4

Thanks

chidcguy · Posted: Thu Jun 05, 2008 7:52 pm Post subject:

Answer is B

If we can express the denominator in the form of 2^x X 5^y the decimal will terminate.

4 is 2 ^ 2 X 5 ^ 0

phumbert · Posted: Thu Aug 14, 2008 4:32 pm Post subject:

rhymes_with_luck · Posted: Thu Aug 14, 2008 6:15 pm Post subject:

man think of 1/4, 2/4, 3/4. 4/4 ....etc, have you ever seen decimals recurring for this fractions ?

Hopefully, got the point.

pepeprepa · Posted: Fri Aug 15, 2008 1:21 am Post subject:

With any number divided by 4 the remainder will be 0, 1, 2, or 3. I think we agree on that. So their quotients are 0.0, 0.25, 0.5, 0.75
Any number divided by 4 always has terminating decimals.

Ian Stewart · Posted: Fri Aug 15, 2008 3:58 am Post subject:

sumithshah · Posted: Sat Aug 16, 2008 11:08 am Post subject:

Im not sure if this rule applies here ( im too sleepy to solve this ) but anything that is divided by a power of 5 or 2 in the denominator will ALWAYS terminate.

pepeprepa · Posted: Sat Aug 16, 2008 11:19 am Post subject:

Can you explain further your tip because
1/(3^2)= 0.1111111111

sumithshah · Posted: Sat Aug 16, 2008 11:36 am Post subject:

When I say a power of 5 or 2, I mean 2^X or 5^Y for all x, y E integers && X, Y != 0

so basically A/(2^X) would be terminating and B/(5^X) would be terminating.

Ian Stewart · Posted: Sun Aug 17, 2008 6:55 am Post subject:

sumithshah · Posted: Sun Aug 17, 2008 7:23 am Post subject:

ah! reduced was the key word. So basically if you have a fraction

ABC / XYZ, reduce it and now, if we have p/q and q is solely a power of 5 or 2 or both ( and nothing else - so eg p/15 wont make a terminating decimal) then its terminating.

Gurus - correct me if im wrong

pepeprepa · Posted: Sun Aug 17, 2008 8:05 am Post subject:

Yep Ian I am ok that's a non terminating decimal.
It was a counter-example to what I understood wrongly of sumithshah property.
Indeed what sumithsha told was: A/(2^X) would be terminating and B/(5^X) would be terminating.

Finally, any reduced fraction which has a prime at the denominator is a non terminating, am I right?
For example,
1/3
2/3
1/7
8/9

4meonly · Posted: Sun Aug 17, 2008 8:48 am Post subject:

pepeprepa · Posted: Sun Aug 17, 2008 8:53 am Post subject:

Ok for your principle:
"if denumerator can be expressed in prime factorisation of 2 and 5 it is always terminal dicimal"

But I don't understand what you mean by 6/3 and 14/7 I talk about reduced fractions...

Ian Stewart · Posted: Sun Aug 17, 2008 9:11 am Post subject:

I'll try to state the rule as unambiguously as possible. If you need to know whether a fraction represents a terminating decimal:

1) Reduce the fraction completely;
2) Prime factorize the denominator;
3) Look at this prime factorization, and ignore the exponents. If there is a prime besides 2 or 5 in the denominator, the fraction represents a recurring (non-terminating) decimal. If the only primes in the denominator are 2, 5, or both, the fraction represents a terminating decimal.

So 9/125, 3/32, 7/80 and 3/30 are all terminating decimals (when you look at 3/30, you must reduce the fraction first, of course), while 7/121, 5/33, 7/60 and 19/99 all represent recurring decimals.