Algebra Know-It-ALL

104 Decimal Fractions

Therefore,

1,000m= ### +m

Subtract m from the expressions on both sides of the equals sign, obtaining

1,000m−m= ### +m−m

This simplifies to

999 m= ###

Finally, we divide each side by 999, getting

m= ###/999

Mission accomplished! Q.E.D.

Conversions

Every rational number can be expressed in two ways: the ratio form as an integer divided by

another integer, and the decimal form as a string of digits with a decimal point somewhere. If

you have a rational number in one form, you can always convert it to the other.

Ratio to decimal

When you see a ratio of integers, you can convert it to decimal form if you have a calculator

that can display enough digits. But that’s the catch! Even a good calculator can fall short in

this respect. If you have a calculator with a 10-digit display and you divide 1 by 7, you will

not even see two full repetitions of the pattern. If you didn’t know better from having seen

the decimal expansion of 1/7 earlier in this chapter, you might not be able to deduce it from

a 10-digit calculator alone. If you have a good computer calculator program, you’re better off.

But even the best calculators can be overwhelmed if you give them a “bad” enough ratio. Try

51/29, for example!

Fortunately, you won’t have to perform ratio-to-decimal conversions very often. When

you come across a problem where you have to do it, the calculator program in any good

personal computer will usually work. In the extreme, you can always resort to old-fashioned,

manual long division. You can also write, or find, a computer program to grind out thousands

of digits and look for patterns.

Terminating decimal to ratio

When you see a terminating decimal expression and you want to convert it to a ratio of

integers, you can do it in steps. Here’s an example. Imagine that you are given this decimal

numeral and are told to put it into ratio form as a quotient of two integers:

3,588. 7601811