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.

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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

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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}

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Finding Unions and Intersections of Intervals:

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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.

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

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