Algebra Know-It-ALL

106 Decimal Fractions

Then write down the part of the expression to the right of the decimal point in this form, with

the point on the extreme left:

.b 1 b 2 b 3 ... bk b 1 b 2 b 3 ... bkb 1 b 2 b 3 ... bk

where b 1 b 2 b 3 ... bkrepresents the sequence of k digits that repeats. (Each b with a subscript

represents a single digit.) The extra spaces after the decimal point, and between each digit, are

there to make the expression easy to read. The fractional part of the expression is

b 1 b 2 b 3 ... bk/999 ... 999

where the denominator has k digits, all 9s. Now you can put back the whole-number part,

getting the number in this form:

a-b 1 b 2 b 3 ... bk/999 ... 999

Here, the dash after the a is there only to separate the whole-number part of the expression

from the fractional part. It is not a minus sign!

Now convert a to a fraction with a denominator consisting of k digits, all 9s. All you have

to do is multiply a by the number 999 ... 999, put the result into the numerator of a fraction,

and then put 999 ... 999 in the denominator, getting

999 ... 999 × a/999 ... 999

Add this to the fraction you got by converting the decimal part of the original expression.

That gives you

(999 ... 999 × a/999 ... 999) + (b 1 b 2 b 3 ... bk/999 ... 999)

You have a common denominator now, so you can easily add to get

[(999 ... 999 × a)+ (b 1 b 2 b 3 ... bk)]/999 ... 999

Remember that the expression 999 ... 999 always stands for a sequence of k digits, all 9s.

Are you confused?

The notation shown above is messy, and it’s easy to “get lost.” Try reading it over a few times and it should

become clearer to you. A specific example, showing the process in “real life action,” can help. Let’s convert

the following expression to a ratio of integers:

23. 860486048604 ...

Again, the extra spaces on either side of the decimal point are there only to make it easy to distinguish

between the whole-number part of the expression and the decimal part.

The decimal portion is a sequence of the digits 8, 6, 0, and 4 that endlessly repeats. We can tell right

away that this is 8,604/9,999.