Notation and Symbols Types of Numbers Natural Numbers (Counting Numbers): N = { 1, 2, 3, 4, 5, 6, ... } Whole Numbers: W = { 0, 1, 2, 3, 4, 5, 6, ... } Integers: Z = { ..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ... } a Rational Numbers: A rational number is a number that can be written in the form of where a and b are integers and b b ≠ 0. Irrational Numbers: An irrational number is a number that can be written as an infinite nonrepeating decimal. Real Numbers: The real numbers consist of all rational and irrational numbers. Complex Number: The complex numbers consist of all real numbers and the even roots of negative numbers. Equality and Inequality Symbols = “is equal to” < “is less than” ≤ “is less than or equal to” ≠ “is not equal to” > “is greater than” ≥ “is greater than or equal to” Absolute Value The absolute value of a real number is its distance from 0. Symbolically, aa= if a is a positive number or 0. aa=− if a is a negative number. Principles and Properties Properties of Addition and Multiplication For Addition Name of Property For Multiplication a + b = b + a Commutative Property ab = ba ()ab+ +=ca++()bc Associative Property ab()ca= ()bc a + 0 = 0 + a = a Identity aa⋅=11⋅=a 1 aa+−()= 0 Inverse a⋅=10()a ≠ a Zero-Factor Law: a · 0 = 0 · a = 0 Distributive Property: ab()+ ca=⋅ba+⋅c Addition Principle of Equality Properties of Radicals A = B, A + C = B + C, and A - C = B - C have the If a and b are positive real numbers, n is a positive same solutions (where A, B, and C are algebraic integer, m is any integer, and n a is a real number expressions). then a n a Multiplication (or Division) Principle of Equality 1. nnab abn 2. n = =⋅ b n A B 1 b A = B, AC = BC, and = have the same n C C 3. aa= n m solutions (where A and B are algebraic expressions m 1 1 and C is any nonzero constant, C ≠ 0). 4. aann= ( ) = (am ) n m m or, in radical notation, aan = ()n = n am Rules Divisibility Rules For 2: If the last digit (units digit) of an integer is 0, 2, 4, 6, or 8, then the integer is divisible by 2. For 3: If the sum of the digits of an integer is divisible by 3, then the integer is divisible by 3. For 5: If the last digit of an integer is 0 or 5, then the integer is divisible by 5. For 6: If the integer is divisible by both 2 and 3, then it is divisible by 6. For 9: If the sum of the digits of an integer is divisible by 9, then the integer is divisible by 9. For 10: If the last digit of an integer is 0, then the integer is divisible by 10. Rules for Exponents For nonzero real numbers a and b and integers m and n: 1 The exponent 1: aa= 1 Negative exponents: a−n = an n The exponent 0: a0 = 1 Power rule: ()aam = mn n The product rule: aamn⋅=amn+ Power of a product: ()ab = abnn m n n a mn− a a The quotient rule: = a Power of a quotient: = an b bn Linear Equations Summary of Formulas and Properties of Lines Cartesian Coordinate System Standard Form: y-axis Ax + By = C where A and B do not both equal 0 Slope of a line: Quadrant II Quadrant I (x negative, y positive) (x positive, y positive) yy21− (-, +) (+, +) m = where x1 ≠ x2 (0, 0) xx21− x-axis Origin Slope-intercept form: Quadrant III Quadrant IV y = mx + b with slope m and y-intercept (0,b) (x negative, y negative) (x positive, y negative) Point-slope form: (-, -) (+, -) with slope m and point xy, on the line yy−=11mx( − x ) ( 11) Horizontal line, slope 0: y = b Vertical line, undefined slope: x = a 1. Parallel lines have the same slope. 2. Perpendicular lines have slopes that are negative reciprocals of each other. Factoring Polynomials Special Factoring Techniques 22 33 22 1. xa−=()xa+ ()xa− : difference of two squares 4. xa+=()xa+ ()xa−+xa: sum of two cubes 222 33 22 2. xa++2 xa=+()xa: square of a binomial sum 5. xa−=()xa− ()xa++xa: difference of two cubes 222 3. xa−+2 xa=−()xa: square of a binomial difference Inequalities Linear Inequalities Interval Notation Linear Inequalities have the following forms Algebraic Interval Type of Interval Graph where a, b, and c are real numbers and a ≠ 0: Notation Notation ax + b < c and ax + b ≤ c Open Interval a < x < b ab, ( ) a b ax + b > c and ax + b ≥ c Closed Interval a ≤ x ≤ b ab, Compound Inequalities a b The inequalities c < ax + b < d and c ≤ ax + b ≤ d ax≤<b ab, ) Half-open Interval a b are called compound linear inequalities. ax<≤b ab, ( a b (This includes ca<+xb≤ d and ca≤+xb< d as well.) xa> (a, ∞) Open Interval a xb< −∞, b ( ) b xa≥ a, ∞) Half-open Interval a xb≤ (−∞, b b Systems of Linear Equations Systems of Linear Equations (Two Variables) Consistent Inconsistent Dependent (One solution) (No solution) (Infinite number of solutions) y y y x x x Quadratic Equations Quadratic Formula −±bb2 − 4ac The solutions of the general quadratic equation ax 2 ++bx c = 0, where a ≠ 0, are x = . 2a y General Information on Quadratic Functions Line of Symmetry For the quadratic function ya=+xb2 xc+ 1. If a > 0, the parabola “opens upward.” 2. If a < 0, the parabola “opens downward.” b 3. x =− is the line of symmetry. 2a b x 4. The vertex (turning point) occurs where x =− . 2a Vertex Substitute this value for x in the function to find the y-value of the vertex. The vertex is the lowest point on the curve if the parabola opens upward and it is the highest point on the curve if the parabola opens downward. Relationships Between Measurements in the U.S. Customary and Metric Metric System Equivalents Length U.S. to Metric Metric to U.S. 1 millimeter (mm) = 0.001 meter 1 m = 1000 mm Length 1 centimeter (cm) = 0.01 meter 1 m = 100 cm 12in.c= .54 m ()exact 10cm = .394 in. 1 decimeter (dm) = 0.1 meter 1 m = 10 dm 10ft = .305 m 13mf= .28 t 1 meter (m) = 1.0 meter 10yd = .914 m 11my= .09 d 10km = .62 mi 1 dekameter (dam) = 10 meters 11mi = .61 km 1 hectometer (hm) = 100 meters Area 1 kilometer (km) = 1000 meters 16in.c22= .45 m 10cm 22= ..155 in Mass 10ft 22= .093 m 11mf22= 0.764 t 1 milligram (mg) = 0.001 gram 1 g = 1000 mg 10yd 22= .836 m 11my22= .196 d 1 centigram (cg) = 0.01 gram 10acre = .405 ha 12ha = .47 acres 1 decigram (dg) = 0.1 gram Volume 1 gram (g) = 1.0 gram 3333 11in..= 6 387 cm 10cm = ..06 in 1 dekagram (dag) = 10 grams 10ft 33= .028 m 13mf33= 5.315 t 1 hectogram (hg) = 100 grams 10qt = .946 L 11Lq= .06 t 1 kilogram (kg) = 1000 grams 1 g = 0.001 kg 13gal = .785 L 10L = .264 gal 1 metric ton (t) = 1000 kilograms 1 kg = 0.001 t Mass 1t = 1000 kg = 1,000,000 g = 1,000,000,000 mg 12oz = 83. 5 g 10go= .035 z Area 10lb = .454 kg 12kg = .205 lb Small Area Land Area 22 2 Temperature Equivalents 1 cm = 100 mm 1 am= 100 2 Celsius Fahrenheit 1 dm 22==100 cm 10 000 mm 2 1 ha ==100 am10 000 22 2 2 100° Water boils 212° 1 md==100 mc10 000 m = 1 000 000 mmm 90° 194° Volume 80° 176° 1 cm3 = 1000 mm3 70° 158° 1 dm3 = 1000 cm3 = 1 000 000 mm3 60° 140° 1 m3 = 1000 dm3 = 1 000 000 cm3 = 1 000 000 000 mm3 50° 122° Liquid Volume 40° 104° 1 milliliter (mL) = 0.001 liter = 1 cm3 30° Comfort 86° 1 liter (L) = 1.0 liter = 1 dm3 = 1000 mL 20° range 68° 1 hectoliter (hL) = 100 liters 10° 50° 1 kiloliter (kL) = 1000 liters = 10 hL = 1 m3 0° Water freezes 32° Relationships Between Measurements in the U.S. Customary System Length Area Liquid Volume 12 inches (in.) = 1 foot (ft) 1 ft 22= 144 in. 1 pint (pt) = 16 fluid ounces (fl oz) 36 inches = 1 yard (yd) 19yd 22= ft 1 quart (qt) = 2 pt = 32 fl oz 3 feet = 1 yard (yd) 1 acre 4840 yd 2243,560 ft 5280 feet = 1 mile (mi) == 1 gallon (gal) = 4 qt Formulas and Definitions Terms Related to the Basic Equation R · B = A R = rate or percent (as a decimal or fraction) B = base (number we are finding the percent of) A = amount (a part of the base) Terms Related to Profit Profit: The difference between selling price and cost. profit= selling price - cost Percent of Profit: profit 1. Percent of profitbased on cost: cost profit 2. Percent of profitbased on selling price: selling price Formulas for Calculating Interest Simple Interest Compound Interest nt r I = P ⋅ r ⋅ t, AP=+1 n where I = interest (earned or paid) P = principal (the amount invested or borrowed) r = annual rate of interest in decimal form t = time (in years or fraction of a year) A = the future value of the investment n = the number of compounding periods in 1 year Formulas for Calculating Inflation and Depreciation Inflation Depreciation t t AP=+()1 r VP=−()1 r where P = the original value r = the annual rate of depreciation or appreciation/inflation in decimal form t = time (in years or fraction of a year) List of Common Formulas Formula Meaning 5 5 CF=−()32 Temperature in degrees Celsius C equals times the difference between the Fahrenheit 9 9 temperature F and 32.