Real Numbers - Algebra II Honors - Algebra II

Properties of Real Numbers: Objectives to graph and order real numbers to identify and use properties of real numbers ● to evaluate algebraic equations ● to simplify algebraic expressions Subsets of Real Numbers: ● ● natural numbers - numbers used for counting; starts with number 1 and goes up by 1's. 1,2,3,4.... whole numbers - natural numbers + zero. 0,1,2,3,4,... integers - natural numbers (positive integers), zero, and the negative integers ...-4,-3,-2,-1,0,1,2,3,4... - Each negative integer is the opposite, or additive inverse, of a positive integer Z is the standard symbol for the set of integers. rational numbers - all the numbers that can be written as quotients of integers. Examples: 7/5, -3/2, -4/5, 0, 0.3, -1.2, 9 ● ● ● Each quotient must have a nonzero denominator. Some rational numbers can be written as terminating decimals. Ex. 1/8=0.125 All other rational numbers can be written as repeating decimals. Ex. 1/3 or 0.33. Use Q for the quotient. irrational numbers - all the numbers that cannot be written as quotients of integers. ● ● their decimal representations neither terminate nor repeat. if a positive rational number is not a perfect square such as 25 or 4/9, then its square root is irrational. Properties of Real Numbers Let a, b, and c represent real numbers. Property Closure Commutativ Associative Identity Addition a + b is a real number. a+b= b+a (a + b) + c = a + (b+ c) a + 0= a, 0 + a = a Multiplication ab is a real number ab=ba (ab)c = a(bc) a · 1 = a, 1 · a = a Inverse Distributive + (-a) = 0 (b + c) = ab + bc a · 1/a = 1 Vocabulary: absolute value - the difference a number is from zero. coefficient - (for now) a number multiplied against something, typically a power of a variable. For ex.: 3x (the coefficient is 3). equation - two expressions connected by equality (=). For example: x2 + 2x - 1 = 5x - 2 term -product of a coefficient and one or more variables to various powers. For example, 7x2, 3y, 2x3y. A plain number can also be a term. - like terms - terms that are identical with respect to variables (differ only in coefficients). For ex.: 3x and 4x are like, 3y and 2x are not. variable - a symbol, usually a lowercase letter, that represents one or more numbers. Ex. x, y, t. Properties for Simplifying Algebraic Expressions Let a, b, and c represent real numbers. Definition of Subtraction a-b=a+(-b) Definition of Division a ÷ b = a · 1/b, b ± 0 Distributive Property for Subtraction Multiplication by 0 a (b - c) = ab - ac 0·a=0 Multiplication by - 1 -1·a=-a Opposite of a Sum - (a + b) = - a + (- b) Opposite of a Difference - ( a - b) = b - a Opposite of a Product - (ab) = - a · b = a · (- b ) Opposite of an Opposite - (- a) = a Solving Equalities and Inequalities. Solving Absolute Value Equations and Inequations Objectives: ● ● ● ● to solve equations to solve and graph inequalities to solve absolute value equations to solve absolute value inequalities Properties of Equality Reflexive a=a Symmetric Transitive Addition subtraction multiplication division substitution if a = b, then b = if a = b and b = c, then a = c if a = b, then a + c = b + c if a = b, then a - c = b - c if a = b, then ac = bc if a = b and c≠0, then a/c = b/c if a = b, then b may be substituted for a in any expression to obtain an equivalent expression. Properties of Inequalities let a, b, and c represent all real numbers Transitive if a≤b and b≤c, then a≤c addition if a≤b then a+c≤b+c subtraction if a≤b then a-c≤b-c if a≤b and c>0, then ac≤bc multiplication if a≤b and c<0 then ac≥bc if a≤b and c>o then a/c≤b/c division if a≤b and c<0 then a/c≥b/c Vocabulary ● ● ● Compound inequality is a pair of inequalities joined by and or or To solve a compound inequality joined by and, find all values of the variable that make both inequalities true to solve a compound inequality joined by or, find all values of the variable that makes at least one of the inequalities true. Vocabulary ● ● ● ● ● Algebraic definition of Absolute Value if x>0, then lxl=x if x<0 then lxl=-x when solving an absolute value equation, remember that there are two solutions, a positive and a negative. extraneous solution is a solution of an equation derived from an original equation that is not a solution of the original equation.