There are types of numbers that make up the real number system. The natural numbers:

1, 2, 3, 4, . . .

The integers consist of the natural numbers together with their negatives and 0:

. . . , -3, -2, -1, 0, 1, 2, 3, 4, . . .

the rational numbers by taking ratios of integers. Thus any rational number r can be expressed as

r = m÷n

There are also real numbers, such as !2, that cannot be expressed as a ratio of integers and are therefore called irrational numbers. It can be shown, with varying degrees of difficulty, that these numbers are also irrational:

 √3 √5 ∜2 ℼ √5÷ℼ 2

The set of all real numbers is usually denoted by the symbol ʀ.

 Properties of Real Numbers

The Distributive Property:

 Addition and Subtraction

Multiplication and Division

Using the LCD to Add Fractions:

The Real Line

The real numbers can be represented by points on a line.


Sets and Intervals

A set is a collection of objects, and these objects are called the elements of the set. If S is a set, the notation a ⊂ S means that a is an element of S, and b o S means that b is not an element of S. For example, if Z represents the set of integers, then -3 ⊂ Z but ℼ ⊄ Z. Some sets can be described by listing their elements within braces. 


Union and Intersection of Sets


Certain sets of real numbers, called intervals, occur frequently in calculus and correspond geometrically to line segments. If a  b, then the open interval from a to b consists of all numbers between a and b and is denoted 1a, b2 . The closed interval from a to b includes the endpoints and is denoted 3a, b4 . Using set-builder notation, we can write

{a, b} = {x | a < x < b} [a, b] = {x | a <= x <= b}


Finding Unions and Intersections of Intervals:


Absolute Value and Distance

The absolute value of a number a, denoted by | a | , is the distance from a to 0 on the real number line. Distance is always positive or zero, so we have | a | >= 0 for every number a. Remembering that -a is positive when a is negative, we have the following definition.



What is the distance on the real line between the numbers -2 and 11?




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