Friday, April 17, 2009

Rational Numbers

We grouped the negative numbers with the natural numbers to get the integers. Now we increase our set of numbers further to get the rational numbers.

The rational numbers include all of the fractions made by dividing one integer by another, except that you can't divide by zero. So positive and negative fractions, fractions smaller than one, fractions larger than one, and all the integers (you can have 1 on the bottom your fraction) make up the rational numbers.

You can remember that the RATIOnals are made up from RATIOs.

Rational numbers do not have to be written as a fraction, but can be in decimal form - they are still rational.

Some rational numbers:
1/2, 6, -4/5, 10/3, 54 3/4, 1.75, -0.33333

With the inclusion of fractions in the rational numbers we are better off for division that we were with the integers. We can now divide any two numbers and get another rational number, with only one exception. We cannot divide by zero.

Like the integers, we can add, subtract and multiply without restriction. But we still have to be careful about exponentiation.

In most cases we can only raise rationals to the power of an integer. 32 is OK, and, unlike the integers, we can now do 3-2, but we cannot do 31/2 as that is outside of the rational numbers. We can only raise numbers to a fraction when the answer lies in the rational numbers. For example 25(1/2) = 5 or (8/27)(1/3) = 2/3.

The rational numbers are given the symbol: ℚ

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