CLEP College Mathematics Review

Prime and Composite Numbers

• Prime Number: Whole number that has only 2 factors (1 and itself).

• Composite Number: Whole number that has factors in addition to 1 and itself.

• 0 & 1: Neither prime nor composite.

• Even Numbers: Composite (because they are all divisible by two).

• Numbers that end in 5: Composite (because they are all divisible by 5).

• Prime Numbers up to 20: 2, 3, 5, 7, 11, 13, 17, 19.

Intervals

• Interval: Connected portion of the real line.

• Closed Interval: Illustrated with filled-in circles and denoted by brackets [ ] (e.g., [a, b]).

• Open Interval: Illustrated with open circles and denoted by parentheses ( ) (e.g., (a, b)).

Types of Numbers

• Rational Number: A number that can be written as a ratio/fraction.
    - Example: 8 = 8/1, 3/4, 7,344/123,239 (every whole number is also a rational number).

• Irrational Number: A number that can be represented as a decimal but not a fraction.
    - Examples: $ext{pi} = 3.141592…$, $ext{square root of 2} = 1.414213…$.

• Absolute Value: The distance from 0, denoted as |x|.
    - Example: |3| = 3, |-3| = 3 (absolute value is never negative).

Factors and Prime Factorization

• Factors: Numbers that you multiply to obtain another number.

• Prime Factorization: A list of prime numbers that are factors of a number.
    - Example: Prime factorization of 8 is $2 imes 2 imes 2$.
    - To find: Take the number and divide it by the smallest prime that divides it.

Relations and Functions

• Relation: A relationship between sets of information.

• Domain: Set of all starting points (x-values).

• Range: Set of all ending points (y-values).

• All functions are relations but not all relations are functions.

• Well-behaved Function: A relation that has only one y for each x (no duplications).

• Vertical Line Test: If a vertical line can be drawn that intersects more than one point of the graph, it is NOT a function.

Types of Functions

• Linear Function: $f(x)=x$;
    - Domain: $(- ext{inf}, ext{inf})$
    - Range: $(- ext{inf}, ext{inf})$ (diagonal line).

• Constant Function: $f(x) = c$;
    - Domain: $(- ext{inf}, ext{inf})$
    - Range: $ext{y} | ext{y = c}$ (horizontal line).

• Quadratic Function: $f(x)=x^2$;
    - Domain: $(- ext{inf}, ext{inf})$
    - Range: $[0, ext{inf})$ (smiley face shape, increases quickly).

• Cubic Function: $f(x)=x^3$;
    - Domain: $(- ext{inf}, ext{inf})$
    - Range: $(- ext{inf}, ext{inf})$ (mirror half of quadratic).

• Square Root Function: $f(x)= ext{sq rt}(x)$;
    - Domain: $[0, ext{inf})$
    - Range: $[0, ext{inf})$ (sideways quadratic).

• Absolute Value Function: $f(x) = |x|$;
    - Domain: $(- ext{inf}, ext{inf})$
    - Range: $[0, ext{inf})$ (shaped like a V).

Graphing Functions

1. Find at least 4 ordered pair solutions.

2. Plot points on a graph.

3. Connect the points to form the graph.

• X/Y Intercept: The point where the graph crosses the x-axis or y-axis.

• A function is defined as a relation where no vertical line intersects the graph more than once.

Function Behavior

• Increasing: Function is going up from left to right.

• Decreasing: Function is going down from left to right.

• Constant: A horizontal line.

• Even Function: A function where replacing x with -x does not change the function.

• Odd Function: A function where replacing x with -x changes every term's sign.

Set Theory

• Greatest Integer Function: The greatest integer that is less than or equal to x, denoted as $ext{int}(x)$.

• Function: A relation with no duplicate x's; all x's correspond to unique y's.

• If a value is a fraction, set the denominator to 0 then solve.

Function Operations

• Composition: A function is inserted into another, denoted as the symbol o.

• Inverse: The same points just reversed; to find it:
    1. Draw points.
    2. Draw a reflection line (diagonal through (0,0)).
    3. Reflect points across the line.

• Set: A collection of distinct objects. Sets are only equal if they have the same exact elements.

Set Membership and Operations

• Membership: One set is an element of another. Symbol: sideways U with a line in it.

• Subset: Every member of one set is also a member of another. Symbol: sideways U with a line under it.

• Union: All elements that are members of either A or B. Symbol: U.

• Intersection: All elements that are members of both A AND B. Symbol: upside down U.

• Complements: All elements of A but NOT B. Symbol: .

• Cartesian Product: All ordered pairs of two sets A and B. Symbol: X.
    - Example: {1, 2} X {red, blue} = {1, red}, {1, blue}, {2, red}, {2, blue}.

Logic and Truth Tables

• Truth Table: A table used to compute truth values of logical expressions.

• Boolean Value: Either true or false.

• Boolean N-Tuple: Count of how many pieces are in a set.
    - Example: 1-tuple: (True); (False) … 2-tuple: (T, T); (F, F) …

• Propositional Operator: Rule defined by a truth table.

• Negation Operator: Reverses the value of the input.
    - Symbol: Sideways L.
    - Example: If P is true (e.g., "I am happy"), then Neg P is false (e.g., "I am not happy").

Logical Operations

• Constant True: Always returns true.

• Identity: Always returns the original truth value.

• Constant False: Always returns false.

• Conjunction: Combining of statements. Symbol: ^.
    - Example: If P: I like milk and Q: I like cookies, then P^Q = "I like both milk and cookies."

• Disjunction: Includes and/or to link two statements. Symbol: V.
    - Example: If P: Mary might sneeze, Q: Joe might sneeze, then PVQ = "Mary might sneeze or Joe might sneeze".
    - PVQ is TRUE when P is true, Q is true, or both are true.

• Negation: Turns a statement opposite in value. Symbol: ~.
    - Example: If P = "She is happy", then ~P = "She is not happy".

Implications and Logical Relationships

• Implication: Symbol: =>.
    - Example: If P: X is in NY, then Q: X is in USA, then P=>Q: "If X is in NY, then X is in the USA".
    - Q is TRUE when P is TRUE.

• Conditional Statements: If - Then statements.

• Converse: Reversal of the parts. Ex: P->Q, converse is Q->P.

• Contraposition: Inverted and flipped; if the original statement is true, then the contrapositive is also true, and vice versa.

• Counterexample: A statement that disproves a universal claim.

• Logical Consequence: A relationship between statements that holds true when every statement in the set is true.

• Inverse: Same points as original function, but x's and y's are switched.
    - Example: {(1, 0), (-3, 5), (0, 4)} Inverse: {(0, 1), (5, -3), (4, 0)}.

Factorials and Arrangements

• Factorial: Denoted by the symbol !; calculated by multiplying the number by every number less than that.
    - Example: $5! = (5) imes (4) imes (3) imes (2) imes (1)$.

• Permutation: The number of ways to arrange something where the order is important.
    - Formula: $rac{n!}{(n-r)!}$.
    - Where n = number of objects and r = number of ranks.
    - Example: $0! = 1$.

• Combination: The number of ways to arrange something where the order is not important.
    - Formula: $rac{n!}{(n-r)!r!}$.

Measures of Central Tendency

• Mean: The average of a set of numbers.

• Median: The middle number in a ordered set.

• Mode: The number that occurs most frequently in a set.

• Range: The difference between the highest and lowest numbers in a set.

Probability Concepts

• Experiment: An act with an uncertain outcome.

• Sample Space: Set of all possible outcomes of an experiment.

• Event: Any subset of a sample space (E).

• Empirical Probability: Finding the probability of an event based on observed data.
    - Formula: $rac{ ext{number of times it occurs}}{ ext{total number of observed occurrences}}$.

• Equiprobable Space: All events are equally likely to occur (e.g., tossing a coin).

• Theoretical Probability: Finding the probability of events in an equiprobable sample space.
    - Formula: $rac{ ext{number of outcomes in event}}{ ext{number of outcomes in set}}$.

• Mutually Exclusive Events: Events that have no elements in common.

Properties of Probability

1. 0 < P(E) < 1: Probability will never exceed 1.

2. $P( ext{not } E) = 1 - P(E)$: Probability of not being in event E.

3. $P(E) = 1 - P( ext{not } E)$: Two opposing probabilities.

4. $P(E ext{ or } F) = P(E) + P(F)$: Only for mutually exclusive events.

5. $P(E ext{ or } F) = P(E) + P(F) - P(E ext{ and } F)$: For events that are not mutually exclusive.

Independent Events and Distribution

• Independent Events: The outcome of one event does not affect the other event.
    - Formula: $P(E ext{ and } F) = P(E) imes P(F)$.

Statistics and Dispersion

• Standard Deviation: A measure of how spread out the numbers are; symbol: Theta.
    1. Find the mean.
    2. Subtract each number from the mean and square it.
    3. Average all squared differences.

Exponential and Logarithmic Functions

• Logarithm: The inverse of exponentials; it undoes exponentials.
    - If $y = b^x$, then $ext{log}_{b}(y) = x$.

Geometry Formulas

• Pythagorean Theorem: $a^2 + b^2 = c^2$; the sum of the squares of the two legs equals the square of the hypotenuse.

• Area of Triangle: $rac{1}{2} imes (b imes h)$ or $rac{1}{2}b(c imes ext{sin}(A))$ or $rac{1}{2}a(c imes ext{sin}(B))$ or $rac{1}{2}a(b imes ext{sin}(C))$.

• Perimeter of Triangle: $a + b + c$.

• Perimeter of Rectangle: $2L + 2W$.

• Area of Rectangle: $L imes W$.

• Perimeter of Parallelogram: $B imes H$.

• Area of Trapezoid: $rac{1}{2} imes (A + B) imes H$.

• Circumference of Circle: $2 ext{pi} imes R$.

• Area of Circle: $ext{pi} imes R^2$.

• Arc Length of Circular Sector: $s = rt$.

• Area of Circular Sector: $rac{1}{2} imes (R^2 imes T)$.

• Volume of Sphere: $rac{4}{3} imes ext{pi} imes R^3$.

• Surface Area of Sphere: $4 imes ext{pi} imes R^2$.

• Volume of Cylinder: $ext{pi} imes R^2 imes H$.

• Surface Area of Cylinder: $2 imes ext{pi} imes R imes H$.

• Volume of Cone: $rac{1}{3} imes ext{pi} imes R^2 imes H$.

• Surface Area of Cone: $ext{pi} imes r imes ext{sq rt}(r^2 + h^2)$.

Algebra and Complex Numbers

• Fundamental Theorem of Algebra: Every polynomial can be factored using complex numbers.

• Imaginary Unit (i): Defined as the square root of -1, with the property that $i^2 = -1$.
    - Simplify square root of -1 as i and $i^2$ as -1.

• Complex Number: Can be expressed in the form $a + bi$.
    - Two complex numbers are equal if both their real and imaginary parts are the same (i.e., $a = c$ and $b = d$).
    - They can be added and subtracted accordingly.

• Adding/Subtracting Complex Numbers:
    1. Add or subtract the real number parts.
    2. Do the same for the imaginary parts.
    3. Write the result in standard form (a + bi).
    - Example: $(7 + 2i) + (9 - 5i) = (7 + 9) + (2 - 5)i = 16 - 3i$.

• Multiplying Complex Numbers:
    1. Simplify using $i^2 = -1$ where necessary.
    2. Write the final answer in standard form (a + bi).
    - Example: $-2i(5 + 3i) = -10i - 6i^2 = -10i + 6 = 6 - 10i$.

• Dividing Complex Numbers:
    1. Find the conjugate of the denominator (change the sign).
    2. Multiply both the numerator and denominator by the conjugate.
    3. Simplify, replacing any occurrence of $-i^2$ with -1.
    4. Write the final answer in standard form (a + bi).

Square Roots of Negative Numbers

• Square root of a negative: Expressed in terms of i; specifically, $ext{Square root of } (-b) = i ext{Square root of } b$.

Polynomial and Factor Theorems

• Remainder Theorem: When dividing a polynomial by a linear factor, the remainder must be smaller than what it was divided by.

• Factor Theorem: If a polynomial is divided by a factor of that polynomial, then the remainder is zero; this is the reverse of the Remainder Theorem.