Rational Numbers and Their Decimal Equivalents

Prepared by:

Joseph Malkevitch
Department of Mathematics and Computer Studies
York College (CUNY)
Jamaica, New York 11451

email:

malkevitch@york.cuny.edu

web page:

http://york.cuny.edu/~malk/

A rational number is a number of the form a/b where b is not 0.

Here are some examples: 3/1, 3/4, 6/7, 122/167, 11/17, etc.

Not all numbers are rational as shown by the fact that the diagonal of a square of side 1 would be Ã2. It is not difficult to show this number can not be rational.

Is there an easy way to tell rational numbers from other numbers when one expresses these numbers in the form of decimals. The surpising answer is "yes."

Some examples may help:

1/5 = .2

1/8 = .125

1/3 = .3333333..... (for ever)

2/3 = .666666666..... (for ever)

11/90 = .122222... (for ever)

2/99= .020202....

Sometimes when digits repeat a bar above the digits is used to signify this.

The examples above show that rational numbers sometimes terminate, that is, after a certain point, all the digits to the right of the ones obtained are zero. In other cases, a digit or group of digits repeats over and over again, and sometimes, there is an initial group of digits, and then some digits which repeat over and over again.

The key to what is going on here is the result for expressing the sum of an infinite geometric progression (i.e. a, ar, ar2, ....; each term after the first is gotten from the previous term by multiplying by r):

a + ar + ar2 + ..... (where the absolute value of r is less than 1)

This sum is given by:

a/(1-r).

This result can be derived as follows:

Let S = a + ar + ar2 + .....

Hence rS = ar + ar2 + .....

Subtracting we obtain S -rS = a. Thus, (1-r)S = a from which it follows that S= a/(1-r).

For example, this enables use to see why:

.333333..... is 1/3.

.3333.... can be thought of as the sum of the infinite geometric progression:

.3, .03, .003, etc. Each being gotten from the previous one by multiplying by 1/10.

Thus. the sum of this progression would be: .3/(1-1/10) = .3/(9/10) = (3/10)/(9/10) = 1/3.

More interesting is the example: .122222....

Here we have .1 + .0222222....

The term .02222.... can be thought of as the sum of the geometric progression: .02, .002, .0002, ....

Thus, .12222.... = 1/10 + (.02/(1-1/10) = 1/10 + .02/(9/10) = 1/10 + 1/45 = 11/90

This reasoning leads to a general result.

Theorem:

A number is rational if and only if its decimal equivalent either terminates or is eventually periodic. (That is, some piece of digits after some point repeats over and over again.)

Exercises:

1. Find the rational number which gives rise to the following decimals:

a. .444444....

b. .040404....

c. .13131313....

d. .233333....

e. .0255555555.....

f. .411111.....

g. .602020202......

h. .12052052052.......

2. Is .01001000100001 etc. rational?