1.Real Numbers a. Rational Numbers i. Natural 1. 1, 2, 3, 4, 5, … ii. Whole 1. 0, 1, 2, 3, 4, … iii. Integers 1. -2, -1, 0, 1, 2, … b. Irrational Numbers i. Cannot be written as a ratio 1. 0.33333333 is rational because it can be written as 2. Pi is irrational because it can’t be written as a fraction ii. Have no pattern of repeat 1. 0.321698715613819617 is irrational 2. 0.125125125125125125 is rational 2. Sets a. This symbol opens a set { this symbol closes it } b. Union i. Union puts two sets together ii. if A = {1, 3, 5} and B = {1, 2, 4} then A ∪ B = {1, 2, 3, 4, 5} c. Intersection i. Intersection finds common numbers between sets 1. If A = {1, 2, 4, 8, 16} and B = {2, 4, 6, 8} then A n B = {2, 4, 8} 3. Exponents Law Example x 1 = x 6 1 = 6 x 0 = 1 7 0 = 1 x -1 = 1/x 4 -1 = 1/4 - x 2 = - (x 2 ) - 4 2 = - (4 2 ) = - (16) = - 16 x m x n = x m+n x 2 x 3 = x 2 + 3 = x 5 x m /x n = x m-n x 6 /x 2 = x 6-2 = x 4 (x m ) n = x m*n (x 2 ) 3 = x 2 * 3 = x 6 (xy) n = x n y n (xy) 3 = x 3 y 3 (x/y) n = x n /y n (x/y) 2 = x 2 / y 2 x -n = 1/x n x -3 = 1/x 3 a. Scientific Notation i. Tells you how many times the decimal is moved ii. If the exponent is positive move decimal right (however many times the exponent says 1. 00,000 iii. If the exponent is negative move decimal left (however many times the exponent says) 1. iv. Number must be between 1 and 10 or [1, 10) 4. Roots (also known as radicals) a. Each square root has a conjugate i. Eg. ii. So √ iii. 2 is the conjugate of -2 and the other way around iv. So if when dealing with equations with square roots we use the symbol so we know the answer can be positive or negative b. Every root (cube root, square root etc.) can be made into an exponent i. Rule says √ 1. So √ c. Simplifying i. Find a perfect square that is a multiple of the number ii. Multiply the perfect square with a number to get original number iii. Separate iv. Simplify perfect square 1. Example a. √ b. √ c. √√ d. √ d. Simplifying fractions i. Take square root of fraction ii. Put square root on numerator and denominator iii. Simplify using steps mentioned above 1. Example a. √ b. √ √ c. e. Adding numbers being multiplied with the same radical i. Take out radical using distributive property ii. Then add two numbers together 1. Example a. √ √ b. √ c. √ d. √ 5. Quadratics a. Rules i. ii. ( iii. ( iv. v. b. FOIL i. Multiply every single number by every number 1. 2. 3. All possible multiplications made now simplify 4. 5. c. Factoring i. Find two numbers whose product equals the middle number and whose quotient equals the last number in the quadratic equation 1. Eg. 2. 3. 4. d. Completing the square i. (Quadratic equation is ) subtract C to the other side. The divide B by 2 and square what you get, add that on both sides. Then factor. Then solve to get solutions 1. 2. 3. 4 divided by 2 equals 2 the square of two is 4 4. 5. √ 6. √ 7. √ √ e. Quadratic formula i. (Quadratic equation is ) simply substitute, nothing else 1. √ f. Vocabulary – The DISCRIMINANT is the number under the root sign i. Random discriminant rules relating to the quadratic equation 6. Polynomials a. Degree = largest exponent b. Polynomials can’t have negative exponents c. Learn long division and synthetic division 7. Imaginary numbers a. The square root of a negative number equals an imaginary number and the square root of the number i. √ √ b. The square root of an imaginary number is – Value of the discriminant Type and number of Solutions Positive Discriminant b² − 4ac > 0 Two Real Solutions If the discriminant is a perfect square the roots are rational. Otherwise, they are irrational. Discriminant is Zero b² − 4ac = 0 One Real Solution Negative Discriminant b² − 4ac < 0 No Real Solutions Two Imaginary Solutions i. 8. Graphs a. Asymptotes i. A line that never touches the line of a function ii. Horizontal 1. Works for fractions 2. If the leading coefficients have the same degree, the divide the coefficients to get what y equals (horizontal line a. the asymptote is iii. Vertical 1. Works for Fractions 2. The thing you cross out is the vertical asymptote a. b. c. is crossed out so x = 8 is the vertical asymptote b. Holes i. A skip in the graph (a literal hole) ii. Works with fractions iii. The thing that is not crossed out is your hole 1. 2. 3. will not be crossed out so x = 3 is the hole c. Logs i. Adding logs means multiply them ii. Dividing logs means divide them iii. Ln means log of e