Algebra Know-It-ALL

Part One 161

Here’s an example of an endless repeating decimal:

43/99= 0.43434343...

Here, the digit sequence 43 repeats forever. We can repeatedly write down the digit pair 43 to
the right of the decimal point, keeping at it for hours, days, or years; but the resulting decimal
expression never reaches the precise value of 43/99. Now let’s look at an example of an endless
nonrepeating decimal:

π= 3.14159265...

The digits go on forever, but there is no pattern to them. We can let a computer grind out
more and more digits, and the resulting decimal expression approaches (but never quite
reaches) the exact value of π.

Question 7-5

How can we convert 356.0056034 into the sum of a whole number and a fraction?

Answer 7-5

We look to the left of the decimal point first. The entire string of numbers here is the integer

356. Now we look to the right of the point. There are seven digits. That means the denomina-
     tor of the fraction should be 10^7 or 10,000,000, and the numerator should be the entire string
     of digits after the decimal point. We get

0056034/10,000,000

for the fractional part. The ciphers at the left in the numerator are useless in the fractional
notation, so we can take them out and add a comma to the digits that remain, getting

56,034/10,000,000

The entire number is the sum of the integer part and the fractional part:

356 + 56,034/10,000,000

Question 7-6

When a number is written in decimal form, is the fractional equivalent (in 10ths, 100ths,
1,000ths, or whatever) in lowest terms?

Answer 7-6

Sometimes, but usually not. Consider 0.55, which converts to 55/100. This is not a fraction
in lowest terms, because it can be reduced to 11/20. But 0.23 converts to 23/100. This is in
lowest terms.

Question 7-7

How can we write down the fractions 1/2, 1/3, 1/4, ..., 1/10 as decimal expressions?