Try our Free Online Math Solver!

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 acolumn of the next row.
Subtract the avalue 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:
Prev  Next 