10.7: Summary of Key Concepts

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Summary of Key Concepts

Variables and Constants
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 $\pi$ (pi) represents the constant 3.14159 . . . .

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

$A number line with hash marks from -3 to 3.$

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

Types of Real Numbers
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
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
Opposites are numbers that are the same distance from zero on the number line but have opposite signs. The numbers $a$ and $-a$ are opposites.

Double-Negative Property ([link])
$-(-a) = a$

Absolute Value (Geometric)
The absolute value of a number $a$, denoted $|a|$, is the distance from $a$ to 0 on this number line.

Absolute Value (Algebraic) ([link])
$|a| = \begin{cases} a, \text{ if } a \ge 0 \\ -a, \text{ if } a < 0 \end{cases}$

Addition of Signed Numbers
To add two numbers with

like signs, add the absolute values of the numbers and associate with the sum the common sign.
unlike signs, subtract the smaller absolute value from the larger absolute value and associate with the difference the sign of the larger absolute value.

Addition with Zero
$0 + \text{(any number) = that particular number.}$

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

Definition of Subtraction
$a - b = a + (-b)$

Subtraction of Signed Numbers
To perform the subtraction $a - b$, add the opposite of $b$ to $a$, that is, change the sign of $b$ and follow the addition rules .

Multiplication and Division of Signed Numbers ([link])
(+) (+) = (+) $\dfrac{(+)}{(+)} = (+)$ $\dfrac{(+)}{(-)} = (-)$
(-) (-) = (+)
(+) (-) = (-) $\dfrac{(-)}{(-)} = (+)$ $\dfrac{(-)}{(+)} = (-)$
(-) (+) = (-)