The topic of decimals, and patterns of decimals, seems to be of slightly greater interest to GMAC in the GMAT OG13e than in previous editions. What decimals terminate? What decimals repeat? In this post, we’ll take a look at these questions.

## Rational Numbers

Integers are positive and negative whole numbers, including zero. Here are the integers:

{ … -3, -2, -1, 0, 1, 2, 3, …}

When we take a ratio of two integers, we get a rational number. A rational number is any number of the form a/b, where a & b are integers, and b ≠ 0. Rational numbers are the set of all fractions made with integer ingredients. Notice that all integers are included in the set of rational numbers, because, for example, 3/1 = 3.

## Rational Numbers as Decimals

When we make a decimal out of a fraction, one of two things happens. It either terminates (comes to an end) or repeats (goes on forever in a pattern). Terminating rational numbers include:

1/2 = 0.5

1/8 = 0.125

3/20 = 0.15

9/160 = 0.05625

Repeating rational numbers include:

1/3 = 0.333333333333333333333333333333333333…

1/7 = 0.142857142857142857142857142857142857…

1/11 = 0.090909090909090909090909090909090909…

1/15 = 0.066666666666666666666666666666666666…

## When Do Rational Number Terminate?

The GMAT won’t give you a complicated fraction like 9/160 and expect you to figure out what its decimal expression is. BUT, the GMAT could give you a fraction like 9/160 and ask whether it terminates or not. How do you know?

Well, first of all, any terminating decimal (like 0.0376) is, essentially, a fraction with a power of ten in the dominator; for example, 0.0376 = 376/10000 = 47/1250. Notice we simplified this fraction, by cancelling a factor of 8 in the numerator. Ten has factors of 2 and 5, so any power of ten will have powers of 2 and powers of 5, and some might be canceled by factors in the numerator , but no other factors will be introduced into the denominator. Thus, if the prime factorization of the denominator of a fraction has only factors of 2 and factors of 5, then it can be written as something over a power of ten, which means its decimal expression will terminate.

If the prime factorization of the denominator of a fraction has only factors of 2 and factors of 5, the decimal expression terminates. If there is any prime factor in the denominator other than 2 or 5, then the decimal expression repeats. Thus,

1/24 repeats (there’s a factor of 3)

1/25 terminates (just powers of 5)

1/28 repeats (there’s a factor of 7)

1/32 terminates (just powers of 2)

1/40 terminates (just powers of 2 and 5)

Notice, as long as the fraction is in lowest terms, the numerator doesn’t matter at all. Since 1/40 terminates, then 7/40, 13/40, or any other integer over 40 also terminates. Since 1/28 repeats, then 5/28 and 15/28 and 25/28 all repeat; notice, though that 7/28 doesn’t repeat, because of the cancellation: 7/28 = 1/4 = 0.25.

## Shortcut Decimals:

There are certain decimals that are good to know as shortcut, both for fraction-to-decimal conversions and for fraction-to-percent conversions. These are

1/2 = 0.5

1/3 = 0.33333333333333333333333333…

2/3 = 0.66666666666666666666666666…

1/4 = 0.25

3/4 = 0.75

1/5 = 0.2 (and times 2, 3, and 4 for other easy decimals)

1/6 = 0.166666666666666666666666666….

5/6 = 0.833333333333333333333333333…

1/8 = 0.125

1/9 = 0.111111111111111111111111111… (and times other digits for other easy decimals)

1/11 = 0.09090909090909090909090909… (and times other digits for other easy decimals)

## Irrationals

There’s another category of decimals that don’t terminate (they go on forever) and they have no repeating pattern. These numbers, the non-terminating non-repeating decimals, are called the irrational numbers. It is impossible to write any one of them as a ratio of two integers. Mr. Pythagoras (c. 570 – c. 495 bce) was the first to prove a number irrational: he proved that the square-root of 2 — — is irrational. We now know: all square-roots of integers that don’t come out evenly are irrational. Another famous irrational number is , or pi, the ratio of a circle’s circumference to its diameter. For example,

= 3.1415926535897932384626433832795028841971693993751058209749445923078164

0628620899862803482534211706798214808651328230664709384460955058223172535940812848111745

0284102701938521105559644622948954930381964428810975665933446128475648233786783165271201

909145648566923460348610454326648213393072602491412737…

That’s the first three hundred digits of pi, and the digits never repeat: they go on forever with no repeating pattern. There are infinitely many irrational numbers: in fact, the infinity of irrational numbers is infinitely bigger than the infinity of the rational numbers, but that gets into some math (http://en.wikipedia.org/wiki/Aleph_number) that is much more advanced than the GMAT.

## Practice Question

1)

(A) 2/27

(B) 3/2

(C) 3/4

(D) 3/8

(E) 9/16

## Practice Question Explanation

1) From our shortcuts, we know 0.166666666666… = 1/6, and 0.444444444444… = 4/9. Therefore (1/6)*(9/4) = 3/8. Answer = **D**

Why do calculators put a 7 as last decimal digit when dividing numbers like 2, 5, 20, 40, 80, 110, 140 and so on by 3? Calculating by hand the result has a never ending 6 as decimal. How do they get the 7?

Hi Zsombi,

Good question!

The calculator is forced to round up by the end of its display, and a repeating 6 decimal would warrant rounding up to 7. But you are right, it is actually a never-ending 6. I hope that makes sense!

Hii

Can u plz help me

I have a question

Is root 6 + root 9

Rational??

Hi Gauri,

Happy to help! The sum of a irrational number and rational number is irrational. In this case, we have

sqrt(6) + sqrt(9) = sqrt(6) + 3

Sqrt(6) is an irrational number and is 2.4494… w

So, the sum is

2.4494… + 3 = 5.4494…

The three dots represent the irrational decimal part of sqrt(6). Since the decimal does not terminate, the decimal in the sum will not terminate either. Therefore sqrt(6) + sqrt(9) results in a irrational number.

Hope this helps

What if the number is 3.067 but only the 7 has a repeating line over the number. Is that rational or irrational?

Hi Jason,

A number can be a rational number and not have the first decimal place repeating. Consider some common repeating decimals:

1/6 = 0.1666…

5/6 = 0.8333…

1/12 = 0.08333…

5/12 = 0.41666…

I do not know off hand what the fraction representation of your suggested 3.0677… would be, but presumably we could find a ratio of two integers that would create that number, making it a rational number.

No. On a number line 1 is farther away from 0 than .75

Would .75 be bigger than 1?

No it would be smaller because 1 is a whole number and .75 is 3/4 of a whole so 1 is bigger

Thanks

Great article Mike!!!

Dear Hamza,

I’m glad you found this helpful! Best of luck to you!

Mike

Hi Mike definitely great explanation about rationals and irrationals.

I have a question.

Is there some sort of formula to determine when a decimal number is Irrational? so it can be implemented by code.

Thank you.

Matt Hall,

Matt,

I’m happy to respond. Everything about the relationship of rationals and irrationals defies all attempts to encapsulate it in a formula. For simple GMAT purposes, the GMAT will give you a decimal with, say, 10 or 12 places after the decimal showing. Either there will be a simple repeating pattern or not. If there’s a repeating pattern, the number is rational. For GMAT purposes, if the decimals shown contain no repeating pattern, then we can assume the decimal is irrational. Technically, the GMAT never asks about rational or irrational anyway: that’s already between the GMAT.

Now, in the bigger picture, it’s certainly true that there are decimals that have repeating patterns that consist, say, of a string of some large number of decimals. For example, the decimal pattern of 1/29 repeats a pattern that is 28-decimal-places long:

1/29 =

0.03448275862068965517241379310344827586206896551724137931

03448275862068965517241379310344827586206896551724137931

03448275862068965517241379310344827586206896551724137931 . . .

(courtesy of Wolfram Alpha)

As I am sure you appreciate, if you have only, say, the first 12 decimals places, there are an infinity of possible rational numbers and another, larger infinity of possible irrational numbers that start with those initial 12 decimals. Having 12 decimal places, while extremely precise, is known mathematically as a

decimal approximation, a sharply curtailed approximation of the infinite decimal. From a mathematical point of view, if we have 12 decimal places, we have essentially nothing. Mathematician regularly examine decimals to millions and billions of decimal places. For example, in 2014, pi was calculated to 13,300,000,000,000 decimals. If we are looking at any number that can be printed on a single sheet of paper, on a single line of text, then, from a purely mathematical point of view, that’s kiddie pool stuff.Basically, there’s no code as efficient as the human mind on this. However many decimals we have, the question is simple: do we see a repeating pattern or not? The human brain is a better pattern-detecting and pattern-matching machine than any computer.

Does all this answer your question?

Mike

Example of 9/160 at top of the page is wrong. You forgot to add 0 after decimal point.

Dear “John”,

Very perceptive! Thank you very much for pointing out that typo. I just corrected it. Best of luck to you, my friend.

Mike

Can you please give an example of the division problem of 1/8 converted into a decimal and can you tell me if the answer is terminating or if it’s repeating?

If a denominator have prime factors of only 2 or only or both then the number terminates ( because any denominator will be in the form ten’s , so factors of 10 is 2* 5 )

If the denominator factors are other than 2 and 5 then the decimal repeats

But taking an ex 1/14 ( factors are 2 and 7 ) it has a repeating decimal.

It’s very useful,thank you.Is my understanding is Right?

Dear Anusha,

Your understanding is correct. For example, 1/35 has 5 & 7 in the denominator, so this would be a repeating non-terminating decimals.

1/35 = 0.0285714285714285714 …

But, notice, if we put something divisible by 7 in the numerator, then the sevens would cancel, and the fraction would terminate:

21/35 = 3/5 = 0.6

Does all this make sense?

Mike

If the number is itself 2 & 5. And if multiplies it with 5/2 and gives answers in which digit in ones place is 0 then that number will terminates always for example

3/4 it will terminates

4*5/2=10 ad we got 0 at ones digit we can easily identify it.

Dear “anonmoyous” or “anonymous”,

Yes, I believe if I understand you correctly, we are saying very much the same thing. You seem to understand this issue well, which is great. Best of luck to you.

Mike

Give me some reasons why 1/6 is irrational number?

Hi Fariha,

Happy to help! Can you tell me why you think 1/6 is an irrational number? 1/6 is actually not an irrational number because we can express it as a ratio of integers or a fraction! I hope that helps.

Hi Mike…

Thanks a ton for this explanation..I just gave a Practice Test from GMAC ( the new Exam pack) and got two questions to do with terminating decimals…Have my GMAT in 36 hours from now..I have absolutely loved the Magoosh product…Hope to get a good score on Monday and write a good testimonial for Magoosh

Dear Nishant,

You are quite welcome. I am very glad you found this blog article helpful, and I’m glad you like the Magoosh product. Best of luck to you my friend!

Mike

Found what i was looking for!

Thanks this really helped!

Dear Saketh,

I’m glad it helped you. Best of luck to you!

Mike

I didn’t find what I was looking for, but it is very useful!

Thank You for spreading your knowledge with others.

Sincerely,

Sara

Well, thank you. I hope you find that for which you seek. Best of luck to you.

Mike