Definition. A rational number is a real number which can be written as , where a and b are integers and . 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 , where a and b are integers and . I may assume that --- if not, divide out any common factors.
Now
Since and p is prime, . Write . Then
Now , so . Thus, p is a common factor of a and b contradicting my assumption that .
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 , 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 's as possible, subtract as many 's 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 such that
These equations are equivalent to
That is, , and a corresponds to .
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 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 -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 such that for .
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 .
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 :
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 :
Next, subtract the first equation from the second:
Copyright 2008 by Bruce Ikenaga