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• Slide 1
• Slide 2
• Review of Real Numbers and Their Properties Advanced Math Section P.1 Advanced Math Chapter P2
• Slide 3
• 3 Natural numbers {1, 2, 3, 4, }
• Slide 4
• Advanced Math Chapter P4 Whole numbers {0, 1, 2, 3, 4, }
• Slide 5
• Advanced Math Chapter P5 Integers { , -3, -2, -1, 0, 1, 2, 3, }
• Slide 6
• Advanced Math Chapter P6 Rational numbers Can be written as the ratio p/q where q 0 Includes natural, whole, integers, and fractions. The decimal representation of a rational number either terminates (like 0.25) or is repeating.
• Slide 7
• Advanced Math Chapter P7 Irrational numbers Are not rational Have infinite non-repeating decimal representations.
• Slide 8
• Advanced Math Chapter P8 You try Which of the numbers above are Natural numbers? Whole numbers? Integers? Rational numbers? Irrational numbers?
• Slide 9
• Advanced Math Chapter P9 Real numbers Used in everyday life to describe quantities Includes rational and irrational numbers Doesnt include imaginary numbers
• Slide 10
• Advanced Math Chapter P10 Real number line Numbers to the right of origin are positive, numbers to the left are negative Nonnegative numbers are positive or zero Nonpositive numbers are negative or zero
• Slide 11
• Advanced Math Chapter P11 One-to-one correspondence Between real numbers and points on the real number line Every real number corresponds to one point on the number line Every point on the number line corresponds to one real number
• Slide 12
• Advanced Math Chapter P12 Definition of order If a and b are real numbers, a is less than b if b a is positive and on a number line, a is left of b
• Slide 13
• Advanced Math Chapter P13 Bounded intervals Have endpoints Have finite length See chart on page 3
• Slide 14
• Advanced Math Chapter P14 Closed intervals Include endpoints Shown with square brackets or equal to Open intervals dont include endpoints (shown with parentheses)
• Slide 15
• Open intervals Dont include endpoints Shown with parentheses Advanced Math Chapter P15
• Slide 16
• Advanced Math Chapter P16 Example Graph the following on a number line
• Slide 17
• Advanced Math Chapter P17 You try Graph the following on a number line
• Slide 18
• Advanced Math Chapter P18 Unbounded intervals Do not have a finite length See chart on page 4
• Slide 19
• Advanced Math Chapter P19 Example Express the following using inequality notation All x in the interval (2,4]
• Slide 20
• Advanced Math Chapter P20 You try Express the following using inequality notation t is at least 10 and less than 22
• Slide 21
• Advanced Math Chapter P21 Absolute value Magnitude Distance between the origin and the point on the number line
• Slide 22
• Advanced Math Chapter P22 Properties of Absolute values Chart on page 5
• Slide 23
• Advanced Math Chapter P23 Distance between a and b
• Slide 24
• Advanced Math Chapter P24 Variables Letters used to represent numbers
• Slide 25
• Advanced Math Chapter P25 Algebraic expressions Combinations of letters and numbers
• Slide 26
• Advanced Math Chapter P26 Terms Parts of an algebraic expression separated by addition (or subtraction)
• Slide 27
• Advanced Math Chapter P27 Constant term Term that doesnt contain a variable
• Slide 28
• Advanced Math Chapter P28 Evaluating algebraic expressions Substitute numerical values for each of the variables in the expression
• Slide 29
• Advanced Math Chapter P29 You try Evaluate the following for x = 1
• Slide 30
• Advanced Math Chapter P30 Substitution Principle If a = b, then a can be replaced by b in any expression involving a.
• Slide 31
• Advanced Math Chapter P31 Charts Pages 6, 7, and 8
• Slide 32
• Advanced Math Chapter P32 You try Exercises 98 104 even
• Slide 33
• Advanced Math Chapter P33 Factors If a, b, and c are integers such that ab = c, then a and b are factors, or divisors, of c.
• Slide 34
• Advanced Math Chapter P34 Prime number Integer that has exactly two factors: 1 and itself
• Slide 35
• Advanced Math Chapter P35 Composite Can be written as the product of two or more prime numbers
• Slide 36
• Advanced Math Chapter P36 Fundamental Theorem of Artihmetic Every positive integer greater than 1 can be written as the product of prime numbers in precisely one way Prime factorization
• Slide 37
• Slide 38
• Advanced Math Chapter P38 Exponential notation a to the nth power n is the exponent a is the base
• Slide 39
• Advanced Math Chapter P39 Properties of exponents Chart page 12 Read first two paragraphs on page 13
• Slide 40
• Advanced Math Chapter P40 Examples No calculator
• Slide 41
• Advanced Math Chapter P41 You try No calculator
• Slide 42
• Advanced Math Chapter P42 Example Rewrite with positive exponents and simplify
• Slide 43
• Advanced Math Chapter P43 You try Rewrite with positive exponents and simplify
• Slide 44
• Advanced Math Chapter P44 Scientific notation n is an integer Positive exponents mean large numbers Negative exponents mean small numbers
• Slide 45
• Advanced Math Chapter P45 Examples Write in scientific notation 9,460,000,000,000 0.00003937
• Slide 46
• Advanced Math Chapter P46 You try Write in scientific notation 0.0000899 34,000,000
• Slide 47
• Advanced Math Chapter P47 You try Write in decimal notation 1.6022 10 -19
• Slide 48
• Advanced Math Chapter P48 Definition of nth root Page 15
• Slide 49
• Advanced Math Chapter P49 Principal nth root Page 15
• Slide 50
• Advanced Math Chapter P50 Tables Page 16
• Slide 51
• Advanced Math Chapter P51 Examples No calculators
• Slide 52
• Advanced Math Chapter P52 You try No calculators
• Slide 53
• Advanced Math Chapter P53 A radical is simplified when All possible factors have been removed from the radical All fractions have radical-free denominators The index of the radical is reduced
• Slide 54
• Advanced Math Chapter P54 Examples No calculators
• Slide 55
• Advanced Math Chapter P55 You try No calculators
• Slide 56
• Slide 57
• Advanced Math Chapter P57 Example
• Slide 58
• Advanced Math Chapter P58 You try No calculators
• Slide 59
• Advanced Math Chapter P59 Rationalizing denominators Gets rid of radical in denominator Multiply both numerator and denominator by the conjugate of the denominator
• Slide 60
• Advanced Math Chapter P60 Conjugates
• Slide 61
• Advanced Math Chapter P61 Examples
• Slide 62
• Advanced Math Chapter P62 You try
• Slide 63
• Advanced Math Chapter P63 Rationalizing numerators Sometimes useful Not simplifying radical Multiply numerator and denominator by conjugate of numerator
• Slide 64
• Advanced Math Chapter P64 Rational exponents Definition page 19
• Slide 65
• Advanced Math Chapter P65 You try Change from radical to rational exponent form
• Slide 66
• Advanced Math Chapter P66 You try Change from rational exponent form to radical form
• Slide 67
• You Try Simplify: Advanced Math Chapter P67
• Slide 68
• Advanced Math Chapter P68 Polynomials and Special Products Advanced Math Section P.3
• Slide 69
• Advanced Math Chapter P69 Polynomial a n is the leading coefficient n is the degree of the polynomial A 0 is the constant term
• Slide 70
• Advanced Math Chapter P70 Example Coefficients are 3, 7, 8, and -5 Leading coefficient is 3 Polynomial degree 4
• Slide 71
• Advanced Math Chapter P71 Polynomials in two variables Degree of each term is sum of exponents Degree of polynomial is highest degree of its terms leading coefficient goes with highest-degree term
• Slide 72
• Advanced Math Chapter P72 Standard form Written with descending powers of x, then descending