Numbers

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Numbers

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

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

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 a/b, 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)
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: 3 = 3/1
This means that all the previous sets of numbers (natural numbers, whole numbers, and integers) are subsets of the rational numbers.

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 = .75 Rational (terminates)
2/3 = .66666 Rational (repeats)
Irrational (never repeats or terminates)
Irrational (never repeats or terminates)

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.

An Ordered Set
- The real numbers have the property that they are ordered, which means that given any two different numbers we can always say that one is greater or less than the other. A more formal way of saying this is:
- For any two real numbers a and b, one and only one of the following three statements is true:
- 1. a is less than b, (expressed as a < b)
- 2. a is equal to b, (expressed as a = b)
- 3. a is greater than b, (expressed as a > b)

The ordered nature of the real numbers lets us arrange them along a line (imagine that the line is made up of an infinite number of points all packed so closely together that they form a solid line). The points are ordered so that points to the right are greater than points to the left:

Every real number corresponds to a distance on the number line, starting at the center (zero).

Negative numbers represent distances to the left of zero, and positive numbers are distances to the right.

The arrows on the end indicate that it keeps going forever in both directions.

Absolute Value

When we want to talk about how “large” a number is without regard as to whether it is positive or negative, we use the absolute value function. The absolute value of a number is the distance from that number to the origin (zero) on the number line. That distance is always given as a non-negative number.
In short:
1. If a number is positive (or zero), the absolute value function does nothing to it: /4/ = 4
2. If a number is negative, the absolute value function makes it positive: /-4/ = 4
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