Section 1: Number Sets

Generally, Algebraic studies use a set of numbers called the real numbers. However, the real numbers are not the only set of numbers that exist. The other number sets that we will consider are actually proper subsets of the real numbers. By "proper subset" we mean that all of the numbers in the other sets are also in the set of real numbers, but not all of the real numbers are in the other sets. We'll start with a fairly simple set of numbers.

Definition 1: The natural numbers are the numbers we typically think of for counting: 1, 2, 3, 4, 5, 6, and so on.

Then, somewhere along in history, a bright mathematician decided we needed a number to represent nothing. Thus, zero was added to the natural numbers to give us another set.

Definition 2: The whole numbers are the natural numbers with the addition of zero: 0, 1, 2, 3, 4, 5, 6, and so on.

If you've ever tried to subtract 7 from 3 (i.e. 3 -7) you know that you cannot get an answer when you are restricted to the Whole numbers. To do this kind of arithmetic you need negative numbers. When these are added to the Whole numbers, we get a new set of numbers.

Definition 3: The integers are the natural numbers with zero and with the negatives (or additive inverses) of all the natural numbers: ... -4, -3, -2, -1, 0, 1, 2, 3, 4,...

Notice the hierarchy from "smaller" to "larger" sets. All of the Natural numbers are also Whole numbers. Similarly, all of the Whole numbers are also Integers.

While these number sets are sufficient for many, many arithmetic problems you can run into problems when dividing two integers (i.e. a fraction). If the result of that division happens to be an integer, then everything is fine. However, if the result of the division is NOT an integer, then another set of numbers is needed to get an answer.

Definition 4: The rational numbers are numbers that can be labled by fraction notation where both the numerator and the denominator are integers.

The same hierarchy holds, all of the natural numbers, whole numbers, and integers are also rational numbers. However, 1/2 is rational number that does not belong to any of the other sets.

By now, one would think, we must have finally identified all numbers. This is not so. There are numbers out there that cannot be labeled by a fraction of integers. Think about squares and square roots. If I take 2 and multiply it by itself, or square it, then I get 4. Of course, that means that the square root of 4 is 2. Those are all rational numbers (in fact they're natural numbers). But what is the number that you would multiply by itself to get 2? That would be √ 2.   There is no way to write this number as a fraction of two integers. This is an irrational number. Other examples of irrational numbers are Π and the cube root of 5.

This brings us to the real numbers.

Definition 5: The real numbers are all of the rational numbers combined with all of the irrational numbers.

With increasing use of calculators, many people don't think of number sets in terms of fraction notation. They prefer to think about decimal notation. All Rational numbers (remember that this includes the Natural numbers, Whole Numbers, and Integers) will either terminate or repeat in decimal form. To terminate means that there are a finite number of digits after the decimal point. Repeating means that some pattern of digits after the decimal will repeat forever.

Example 1 Terminating 18= .125
Example 2 Repeating 6799 = .676767...

The decimal representation for Irrational Numbers will neither repeat nor terminate. It will go on forever with no stopping and no repeating pattern. As was pointed out before, there is a hierarchy in these numbers sets. Every number set discussed is part of the Real numbers. The Real numbers are divided into the Rational and Irrational numbers, and the Rational numbers are further broken down into the Integers, Whole numbers, and Natural numbers.