Algebra Know-It-ALL

were able to prove that this number, which can be so easily defined in terms of geometry, can-
not be defined as a ratio between two integers!
If you’ve taken any geometry, you know that the circumference of a circle divided by its
diameter is symbolized by the small Greek letter pi (π). Many calculators have a key you can
punch to get π straightaway. You’ll never find any repeating pattern of digits in the decimal
expansion of π. Even if you spend the rest of your life trying, you will fail. That’s because π
is not a rational number. Mathematicians call quantities such as πirrational numbers. You’ll
learn more about them in Chap. 9.

Here’s a challenge!

Imagine a decimal expression that has an endlessly repeating triplet of digits. We can write it down in this
form:

0. ### ### ### ...

where ### represents the sequence of three digits that repeats. The spaces on either side of the decimal
point, and after each triplet of pound signs, are inserted to make the expression clear. Our mission is to
show that this decimal numeral represents the fraction

###/999

Solution

Let’s call the “mystery fraction” m. Our task is to find a fractional expression for m. This process is straight-
forward, but it takes several steps. Follow along closely, and you shouldn’t have any trouble understanding
how it works. We’ve been told that

m= 0. ### ### ### ...

We can break this into a sum of two decimal expressions, one terminating and the other endless, like this:

m= (0. ###) + (0. 000 ### ### ### ...)

Note that the first addend here is ###/1,000. The second addend happens to be the original mystery
number, m,divided by 1,000. Therefore,

m= (###/1,000) + (m/1,000)

The two fractions on the right-hand side of the equals sign have a common denominator, so they’re easy
to add. We get

m= (### +m)/1,000

Now we can multiply each side of this equation by 1,000 and then manipulate the right-hand side, getting

1,000 m= 1,000 (### +m)/1,000

= (### +m) (1,000/1,000)

= ### +m

Endless Decimals 103