Algebra Know-It-ALL

102 Decimal Fractions

A good way to see the difference between a terminating decimal and an endless repeating

decimal is to use the “1/x” key on your calculator and start with 2 for x. Then try it with 3, 4,

5, and so on, watching the results:

1/2 = 0.5

1/3 = 0.333333333333...

1/4 = 0.25

1/5 = 0.2

1/6 = 0.166666666666...

1/7 = 0.142857142857...

1/8 = 0.125

1/9 = 0.111111111111...

The fractions 1/2, 1/4, 1/5, and 1/8 all work out as terminating decimals. The fractions 1/3,

1/6, 1/7, and 1/9 divide out as endless repeating decimals. The presence of an ellipsis (three

periods) indicates that the pattern continues forever. Note the uniqueness of 1/7, which goes

through a repeating cycle of the six digits 142857.

Are you confused?

When you divide an integer by another integer, and if the two integers are large enough, your calculator

display might not show enough digits to let you see the pattern of repetition. Take a calculator that can

show 10 digits, and divide out this fraction:

138,297,004,792/999,999,999,999

Now suppose you show your calculator display to a friend, tell her it’s the quotient of two integers you

entered, and then ask her what fraction you put in. Even if she has a Ph.D. in math, she will not be able

to figure it out. The pattern here is too big for the display.

Once in a while you’ll come across a situation where an integer is divided by another integer and you

can’t see the pattern in the decimal expression because the repeating sequence has too many digits. But

there is always a pattern whether you can see it or not. That’s because any rational number can be expressed

as either a terminating decimal or an endless repeating decimal.

Endless nonrepeating decimals

You might wonder whether there are any decimal numbers that go on forever with digits to

the right of the decimal point, but that don’t produce a repeating sequence. The answer is

“Yes.” Examples are easy to find.

Consider the circumference of a perfect circle divided by its diameter. This value is always

the same, no matter how large or small the circle happens to be. In ancient times, people

knew that the circumference of a circle is slightly more than 3 times its diameter. They tried

to define it as a fractional ratio—that is, as a quotient of two integers—but the best they could

do was to come close. For a long time they thought it was 22/7. Eventually, mathematicians