**THE REAL NUMBER SYSTEM**

The real number system evolved over time by expanding the notion of what we mean by the word “number.” At first, “number” meant something you could count, like how many sheep a farmer owns. These are called the natural numbers, or sometimes the counting numbers.

Natural Numbers

or “Counting Numbers”

1, 2, 3, 4, 5, . . .

The use of three dots at the end of the list is a common mathematical notation to indicate that the list keeps going forever.

At some point, the idea of “zero” came to be considered as a number. If the farmer does not have any sheep, then the number of sheep that the farmer owns is zero. We call the set of natural numbers plus the number zero the whole numbers.

**Whole Numbers**

Natural Numbers together with “zero”

0, 1, 2, 3, 4, 5, . . .

**Integers**

Whole numbers plus negatives

. . . –4, –3, –2, –1, 0, 1, 2, 3, 4, . . .

**Rational Numbers**

All numbers of the form , where a and b are integers (but b cannot be zero)

Rational numbers include what we usually call fractions

RESTRICTION: The denominator cannot be zero! (But the numerator can)

If the numerator is zero, then the whole fraction is just equal to zero. If I have zero thirds or zero fourths, than I don’t have anything. However, it makes no sense at all to talk about a fraction measured in “zeroths.”

Fractions can be numbers smaller than 1, like 1/2 or 3/4 (called proper fractions), or they can be numbers bigger than 1 (called improper fractions), like two-and-a-half, which we could also write as 5/2

All integers can also be thought of as rational numbers, with a denominator of 1:

This means that all the previous sets of numbers (natural numbers, whole numbers, and integers) are subsets of the rational numbers.

Now it might seem as though the set of rational numbers would cover every possible case, but that is not so. There are numbers that cannot be expressed as a fraction, and these numbers are called irrational because they are not rational.

Irrational Numbers

Cannot be expressed as a ratio of integers.

As decimals they never repeat or terminate (rationals always do one or the other)

Examples:

3/4 Rational (terminates)

2/3 Rational (repeats)

5/11 Rational (repeats)

5/7 Rational (repeats)

square root of 2: Irrational (never repeats or terminates)

3.1416 Irrational (never repeats or terminates)

**The Real Numbers**

Rationals + Irrationals

All points on the number line

Or all possible distances on the number line

When we put the irrational numbers together with the rational numbers, we finally have the complete set of real numbers. Any number that represents an amount of something, such as a weight, a volume, or the distance between two points, will always be a real number. The following diagram illustrates the relationships of the sets that make up the real numbers.