The Real Number System
The real numbers can be thought of as all the numbers on the number line. This includes
several subsets of numbers:
●​ Natural numbers (N): These are the counting numbers: {1,2,3,...}.
●​ Whole numbers (W): This set includes the natural numbers and zero: {0,1,2,3,...}.
●​ Integers (Z): This set includes all whole numbers and their negative counterparts:
{...,−3,−2,−1,0,1,2,3,...}.
●​ Rational numbers (Q): These are numbers that can be written as a fraction qp​, where p
and q are integers and q=0. Examples include 21​, −3 (which can be written as −13​), and
0.75 (which is 43​).
●​ Irrational numbers (I): These are numbers that cannot be written as a simple fraction.
Their decimal representations go on forever without repeating. Examples include
Key Properties of Real Numbers
●​ Commutative Property: The order of the numbers doesn't change the result.
○​ Addition: a+b=b+a (e.g., 3+5=5+3)
○​ Multiplication: a⋅b=b⋅a (e.g., 4⋅6=6⋅4)
●​ Associative Property: The grouping of the numbers doesn't change the result.
○​ Addition: (a+b)+c=a+(b+c) (e.g., (2+3)+4=2+(3+4))
○​ Multiplication: (a⋅b)⋅c=a⋅(b⋅c) (e.g., (2⋅3)⋅4=2⋅(3⋅4))
●​ Distributive Property: This property links addition and multiplication.
○​ a(b+c)=ab+ac (e.g., 2(3+4)=2⋅3+2⋅4=6+8=14)
Order of Operations (PEMDAS)
PEMDAS is a mnemonic to remember the correct order to perform mathematical operations:
●​
●​
●​
●​
Parentheses (or any grouping symbols like brackets or braces)
Exponents
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)
Example: 10−2⋅(3+1)2÷8
1.​ Parentheses: 3+1=4 10−2⋅(4)2÷8
2.​ Exponents: (4)2=16 10−2⋅16÷8
3.​ Multiplication/ Division (left to right): 2⋅16=32 10−32÷8 32÷8=4 10−4
4.​ Addition/ Subtraction: 10−4=6 The final answer is 6.
Solving Linear Equations
The goal is to isolate the variable (e.g., x) on one side of the equation. To do this, use inverse
operations. Remember, whatever you do to one side of the equation, you must do to the other.
●​ The inverse of addition is subtraction.
●​ The inverse of multiplication is division.
Example: Solve for x in the equation 3x+5=17.
1.​ Subtract 5 from both sides to undo the addition: 3x+5−5=17−5 3x=12
2.​ Divide both sides by 3 to undo the multiplication: 33x​=312​x=4
3.​ Check your solution by plugging the answer back into the original equation:
3(4)+5=12+5=17. The solution is correct.
Graphing Linear Equations
A linear equation is an equation that forms a straight line when graphed on a coordinate plane.
The standard form is y=mx+b.
●​ m is the slope, which represents the steepness of the line. Slope is "rise over run":
change in x / change in y​.
●​ b is the y-intercept, which is the point where the line crosses the y-axis (when x=0).
Example: Graph the equation y=2x+1.
1.​ Identify the y-intercept. Here, b=1, so the line crosses the y-axis at the point (0,1). Plot
this point.
2.​ Identify the slope. Here, m=2, which can be written as 12​. This means for every 1 unit
you move to the right (run), you move 2 units up (rise).
3.​ From the y-intercept (0,1), move 1 unit to the right and 2 units up. This brings you to the
point (1,3). Plot this point.
4.​ Draw a straight line through the two plotted points. This is the graph of the equation.