Algebra Know-It-ALL

100 Decimal Fractions

Don’t be fooled here! These examples don’t represent changes in the order of magnitude.

Instead, they all represent numbers that are very close to each other. The values approach the

last number in either list, which is 0.72941. If that’s as far as you want to go, then adding any

more digits to the right of the 1 will only clutter the page with ciphers (zeros). The following

numerals all represent exactly the same number to a pure mathematician:

0.72941

0.729410

0.7294100

0.72941000

0.729410000

0.7294100000

↓

and so on, forever

Physicists or engineers see the above numbers differently. To them, those extra ciphers are

important, because they represent increasing precision or accuracy. They’re extra significant

figures. Let’s not worry about that right now.

Commas and extra ciphers

In any decimal expression, the number of digits to the left of the decimal point is always finite.

If you “chop off ” the digits to the right of the point, the digits to the left represent an integer.

You might add ciphers to the left-hand end of the digit string without changing the value, but

there’s rarely any reason to do that. You won’t often see a numeral like this:

00,000,004,580,103.7864892022

Instead, it would be written as

4,580,103.7864892022

In a decimal numeral, commas are not customarily inserted to the right of the point, no matter

how many digits there are.

What about lowest terms?

When you see a decimal expression that ends after a certain number of digits to the right of

the point, those digits always express a fraction with a denominator that is some power of

10. This fraction might be in lowest terms, but often it is not. For example, in the decimal
    66.31, the fractional part is 31/100, which is in lowest terms because 31 is prime. However,
    in the decimal 66.35, the fractional part is 35/100. If reduced to lowest terms, that would
    be 7/20.
    When you write or see a decimal expression, you shouldn’t worry about reducing the frac-
    tional part to lowest terms unless the nature of the problem demands it. Those digits to the
    right of the point are always supposed to represent 10ths, 100ths, 1,000ths, and so on.