2.8: Summary of Key Concepts
Summary of Key Concepts
Variables and Constants
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
A
binary operation
is a process that assigns two numbers to a single number. +,−,×,÷ are binary operations.
Grouping Symbols
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 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: __
Order of Operations
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
The
real number line
allows us to visually display some of the numbers in which we are interested.
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 collection of real numbers has many subcollections. The ones of most interest to us are
- The Natural Numbers: \({1,2,3,…}\)
- The Whole Numbers: \({0,1,2,3,…}\)
- The Integers: \({…,-3,-2,-1,0,1,2,3,…}\)
- 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
- Closure: If \(a\) and \(b\) are real numbers, then \(a+b\) and \(a\cdot b\) are unique real numbers.
- Commutative: \(a + b = b + a\) and \(a \cdot b = b \cdot a\).
- Associative: \(a + (b + c) = (a + b) + c\) and \(a \cdot (b \cdot c) = (a \cdot b) \cdot c\)
- Distributive: \(a(b + c) = a \cdot b. + a \cdot c\)
- Additive Identity: \(0\) is the additive identity. \(a + 0 = a\) and \(0 + a = a\)
- Multiplicative Identity: \(1\) is the multiplicative identity. \(a \cdot 1 = a\) and \(1 \cdot a = a\)
- Additive Inverse: For each real number \(a\) there is exactly one number \(-a\) such that \(a + (-a) = 0\) and \((-a) + a = 0\).
- Multiplicative Inverse: For each nonzero real number there is exactly one nonzero real number
- \(\dfrac{1}{a}\) such that \(a \cdot \dfrac {1}{a} = 1\) and \(\dfrac{1}{a} \cdot a = 1\).
Exponents
Exponents record the number of identical factors that appear in a multiplication.
Rules of Exponents
If \(x\) is a real number and \(n\) and \(m\) are natural numbers, then
- \(x^n \cdot x^m = x^{n + m}\)
- \(\dfrac{x^n}{x^m} = x^{n-m}, x \not = 0\)
- \(x^0 = 1, x \not = 0\)
- \((x^n)^m = x^{n \cdot m}\)
- \((\dfrac{x}{y})^n = \dfrac{x^n}{y^n}, y \not = 0\)