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Basic Properties of Real Numbers: Summary of Key Concepts

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

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Summary: This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret. This module provides a summary of the key concepts of the chapter "Basic Properties of Real Numbers".

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Summary of Key Concepts

Variables and Constants ((Reference))

A variable is a letter or symbol that represents any member of a collection of two or more numbers. A constant is a letter or symbol that represents a specific number.

Binary Operation ((Reference))

A binary operation is a process that assigns two numbers to a single number. +, −, ×, ÷ +, −, ×, ÷ are binary operations.

Grouping Symbols ((Reference))

Grouping symbols are used to indicate that a particular collection of numbers and meaningful operations is to be considered as a single number ( 5÷0 5÷0 is not meaningful). Grouping symbols can also direct us in operations when more than two operations are to be performed. Common algebraic grouping symbols are

Parentheses: ( ) Brackets: [ ] Braces: { } Bar:    ¯ Parentheses: ( ) Brackets: [ ] Braces: { } Bar:    ¯

Order of Operations ((Reference), (Reference))

When two or more operations are to be performed on a collection of numbers, the correct value can be obtained only by using the correct order of operations.

The Real Number Line ((Reference))

The real number line allows us to visually display some of the numbers in which we are interested.

A real number line with arrows on each end, labeled from negative four to four in increments of one.

Coordinate and Graph ((Reference))

The number associated with a point on the number line is called the coordinate of the point. The point associated with a number is called the graph of the number.

Real Number ((Reference))

A real number is any number that is the coordinate of a point on the real number line.

Types of Real Numbers ((Reference))

The collection of real numbers has many subcollections. The ones of most interest to us are

  • the natural numbers : { 1 , 2 , 3 , … } {1,2,3,…}
  • the whole numbers : { 0 , 1 , 2 , 3 , … } {0,1,2,3,…}
  • the integers : { … , -3 , -2 , -1 , 0 , 1 , 2 , 3 , … } {…,-3,-2,-1,0,1,2,3,…}
  • the rational numbers : {all numbers that can be expressed as the quotient of two integers}
  • the irrational numbers : {all numbers that have nonending and nonrepeating decimal representations}

Properties of Real Numbers ((Reference))

  • Closure : If aa and bb are real numbers, then a+ba+b and a⋅ba⋅b are unique real numbers.
  • Commutative : a+b = b+a a+b=b+a and a⋅b = b⋅a a⋅b=b⋅a
  • Associative : a+(b+c) = (a+b)+c a+(b+c)=(a+b)+c and a⋅(b⋅c) = (a⋅b)⋅c a⋅(b⋅c)=(a⋅b)⋅c
  • Distributive : a( b+c ) = a⋅b + a⋅c a(b+c)=a⋅b+a⋅c
  • Additive identity : 0 is the additive identity. a+0 =a a+0=a and 0+a =a 0+a=a.
  • Multiplicative identity : 1 is the multiplicative identity. a⋅1 =a a⋅1=a and 1⋅a =a 1⋅a=a.
  • Additive inverse : For each real number aa there is exactly one number -a-a such that a+ (-a) =0 a+(-a)=0and (-a) +a =0 (-a)+a=0.
  • Multiplicative inverse : For each nonzero real number aa there is exactly one nonzero real number 1 a 1 a such that a ⋅ 1 a = 1 a⋅ 1 a =1 and 1 a ⋅ a = 1 1 a ⋅a=1.

Exponents ((Reference))

Exponents record the number of identical factors that appear in a multiplication.

x ⋅ x ⋅ x ⋅  . . .  ⋅ x ︸ n factors of x = x n x ⋅ x ⋅ x ⋅  . . .  ⋅ x ︸ n factors of x = x n

Rules of Exponents ((Reference), (Reference))

If x x is a real number and n n and m m are natural numbers, then

  • x n ⋅ x m = x n+m x n ⋅ x m = x n+m
  • x n x m = x n−m x n x m = x n−m , x≠0 x≠0
  • x 0 = 1 x 0 =1, x≠0 x≠0
  • ( x n ) m = x n⋅m ( x n ) m = x n⋅m
  • ( x y ) n = x n y n ( x y ) n = x n y n , y≠0 y≠0

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