Variables and Constants ((Reference))

A variable is a letter or symbol that represents any member of a set of two or more numbers. A constant is a letter or symbol that represents a specific number. For example, the Greek letter π π (pi) represents the constant 3.14159 . . . .

The Real Number Line ((Reference))

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

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 set of real numbers has many subsets. The ones of most interest to us are:
The natural numbers: {1, 2, 3, 4, . . .}
The whole numbers: {0, 1, 2, 3, 4, . . .}
The integers: {. . . ,-3,-2,-1,0, 1, 2, 3, . . .}
The rational numbers: {All numbers that can be expressed as the quotient of two integers.}

Positive and Negative Numbers ((Reference))

A number is denoted as positive if it is directly preceded by a plus sign (+) or no sign at all. A number is denoted as negative if it is directly preceded by a minus sign (–).

Opposites ((Reference))

Opposites are numbers that are the same distance from zero on the number line but have opposite signs. The numbers aa size 12{a} {} and −a−a size 12{ - a} {} are opposites.

Double-Negative Property ((Reference))

−(−a)=a−(−a)=a size 12{ - \( - a \) =a} {}

Absolute Value (Geometric) ((Reference))

The absolute value of a number aa size 12{a} {}, denoted ∣a∣∣a∣ size 12{ lline a rline } {}, is the distance from aa size 12{a} {} to 0 on the number line.

Absolute Value (Algebraic) ((Reference))

| a | = a , if  a ≥ 0 - a , if  a < 0 |a|= a , if  a ≥ 0 - a , if  a < 0

Addition of Signed Numbers ((Reference))

To add two numbers with

Addition with Zero ((Reference))

0 +(any number)= that particular number0 +(any number)= that particular number size 12{"0 "+ \( "any number" \) =" that particular number"} {}.

Additive Identity ((Reference))

Since adding 0 to any real number leaves that number unchanged, 0 is called the additive identity.

Definition of Subtraction ((Reference))

a−b=a+(−b)a−b=a+(−b) size 12{a - b=a+ \( - b \) } {}

Subtraction of Signed Numbers ((Reference))

To perform the subtraction a−ba−b size 12{a - b} {}, add the opposite of bb size 12{b} {} to aa size 12{a} {}, that is, change the sign of bb size 12{b} {} and follow the addition rules ((Reference)).

Multiplication and Division of Signed Numbers ((Reference))

+ + = + + + = + size 12{ left (+{} right ) left (+{} right )= left (+{} right )} {} + + = + + + = + size 12{ { { left (+{} right )} over { left (+{} right )} } = left (+{} right )} {} + − = − + − = − size 12{ { { left (+{} right )} over { left ( - {} right )} } = left ( - {} right )} {}
− − = + − − = + size 12{ left ( - {} right ) left ( - {} right )= left (+{} right )} {}
+ − = − + − = − size 12{ left (+{} right ) left ( - {} right )= left ( - {} right )} {} − − = + − − = + size 12{ { { left ( - {} right )} over { left ( - {} right )} } = left (+{} right )} {} − + = − − + = − size 12{ { { left ( - {} right )} over { left (+{} right )} } = left ( - {} right )} {}
− + = − − + = − size 12{ left ( - {} right ) left (+{} right )= left ( - {} right )} {}