Terminating and Repeating Decimals
The topic of decimals, and patterns of decimals. 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 \dfrac{a}{b}, where a \And b are integers, and b \not = 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, \dfrac{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:
\dfrac{1}{2} = 0.5
\dfrac{1}{8} = 0.125
\dfrac{3}{20} = 0.15
\dfrac{9}{160} = 0.05625
Repeating rational numbers include:
\dfrac{1}{3} = 0.333333333333333333333333333333333333...
\dfrac{1}{7} = 0.142857142857142857142857142857142857...
\dfrac{1}{11} = 0.090909090909090909090909090909090909...
\dfrac{1}{15} = 0.066666666666666666666666666666666666...
When Do Rational Numbers Terminate?
The QUANT won’t give you a complicated fraction like \dfrac{9}{160} and expect you to figure out what its decimal expression is. BUT, the GMAT could give you a fraction like \dfrac{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 = \dfrac{376}{10000} = \dfrac{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,
\dfrac{1}{24} repeats (there’s a factor of 3)
\dfrac{1}{25} terminates (just powers of 5)
\dfrac{1}{28} repeats (there’s a factor of 7)
\dfrac{1}{32} terminates (just powers of 2)
\dfrac{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 \dfrac{1}{40} terminates, then \dfrac{7}{40}, \dfrac{13}{40}, or any other integer over 40 also terminates. Since \dfrac{1}{28} repeats, then \dfrac{5}{28} and \dfrac{15}{28} and \dfrac{25}{28} all repeat; notice, though that \dfrac{7}{28} doesn’t repeat, because of the cancellation: \dfrac{7}{28} = \dfrac{7}{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
\dfrac{1}{2} = 0.5
\dfrac{1}{3} = 0.33333333333333333333333333...
\dfrac{2}{3} = 0.66666666666666666666666666...
\dfrac{1}{4} = 0.25
\dfrac{3}{4} = 0.75
\dfrac{1}{5} = 0.2 (and times 2, 3, and 4 for other easy decimals)
\dfrac{1}{6} = 0.166666666666666666666666666....
\dfrac{5}{6} = 0.833333333333333333333333333...
\dfrac{1}{8} = 0.125
\dfrac{1}{9} = 0.111111111111111111111111111... (and times other digits for other easy decimals)
\dfrac{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 - \sqrt{2} — is irrational. We now know: all square-roots of integers that don’t come out evenly are irrational. Another famous irrational number is pi, or pi, the ratio of a circle’s circumference to its diameter. For example,
\pi =
3.1415926535897932384626433832795028841
971693993751058209749445923078164062862
089986280348253421170679821480865132823
066470938446095505822317253594081284811
174502841027019385211055596446229489549
303819644288109756659334461284756482337
867831652712019091456485669234603486104
54326648213393072602491412737...
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 Aleph number that is much more advanced than the QUANT.
Practice Question
1) \dfrac{0.16666...}{0.44444...} =
(A) \dfrac{2}{27}
(B) \dfrac{3}{2}
(C) \dfrac{3}{4}
(D) \dfrac{3}{8}
(E) \dfrac{9}{16}
Practice Question Explanation
Q1. From our shortcuts, we know 0.166666666666... = \dfrac{1}{6}, and 0.444444444444... = \dfrac{4}{9}. Therefore \bigg(\dfrac{1}{6}\bigg)*\bigg(\dfrac{9}{4}\bigg) = \dfrac{3}{8}.
Answer = D