Fractions and Rational Numbers

• A rational number is a real number which can be represented as a quotient of two integers .
• Rational numbers are exactly those real numbers having decimal expansions which are periodic or
terminating.

Definition. A rational number is a real number which can be written as a/b, where a and b are integers
and b ≠ 0. A real number which is not rational is irrational .

Example. If p is prime, then is irrational. To prove this, suppose to the contrary that is rational. Write = a/b, where a and b are integers
and b ≠ 0. I may assume that (a, b) = 1 — if not, divide out any common factors.

Now

Since p | a^2 and p is prime, p | a. Write a = pc. Then

Now p | b^2, so p | b. Thus, p is a common factor of a and b contradicting my assumption that (a, b) = 1.
It follows that is irrational.
More generally, if are integers, the roots of

are either integers or irrational.

If b is an integer such that b > 1, and a is a real number between 0 and 1 (inclusive), then a can be
written uniquely in the form

This is called the base b expansion of a. Rather than proving this fact, I’ll merely recall the standard
algorithm for computing such an expansion: Subtract from a as many as possible, subtract as many
from what’s left, and so on.

Here is a recursive procedure which generates base b expansions:

To see why this corresponds to the standard algorithm, note that at the first stage I’m trying to find
k ≥ 0 such that

These equations are equivalent to

That is, k = [ba], and a corresponds to x_i.
It’s convenient to arrange the computations in a table, as shown below.

Example. Find 0.4 in base 7.

I fill in the rows from left to right. Starting with an x, multiply by b = 7 to fill in the third column.
Take the greatest integer of the result to fill in the a-column of the next row. Subtract the a- value from the
last bx-value to get the next x, and continue. You can check that this is the algorithm described above.

The expansion clearly repeats after this, since I’m getting 0.4 for x again. Thus,

Definition. The decimal expansion terminates if there is a number N > 0 such that a_n = 0
for n ≥ N.

In this case,

Hence, x is rational.
A decimal expansion is periodic with period k if there is a positive integer N such that
for all n ≥ N.

Periodic expansions also represent rational numbers. Again, I’ll give an example rather than writing
out the unenlightening proof.
The converse is also true: Rational numbers have decimal expansions which are either periodic or
terminating.

Example. Express as a rational number in lowest terms .
Since the number has period 3, I multiply both sides by 10^3:

Next, subtract the first equation from the second :

Example. Express as a (decimal) rational number in lowest terms.
Since the number has period 3, I multiply both sides by

Next, subtract the first equation from the second, being careful about the bases:

Example. Express as a rational number in lowest terms.
Since the number has period 3, I multiply both sides by 10^3

Next, subtract the first equation from the second: