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Operations Basic Arithmetic Operations

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7/9/2013 Operations 2 2 A Subtraction ( – ) Division ( / ) Notation … addition of negatives … multiplication of reciprocals + Addition ( ), Multiplication ( ) Axioms

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7/9/2013 Operations 3 Order of Operations Inverses Additive inverses For every real number x there is a real number -x such that x + (-x) = 0 Examples 1. 7 + (-7) = 0 2. -3 + (-(-3)) = 0 = -3 + 3 Axioms

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7/9/2013 Operations 4 Order of Operations Subtraction The binary operation – is defined by a – b a + (- b ) for any real numbers a and b The operation – is called subtraction … which is just addition of negatives Examples: 1. 7 – 4 = 7 + (-4) = 3 2. 10 – 12 = 10 + (-12) = -2 Axioms

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7/9/2013 Operations 5 Order of Operations Inverses Multiplicative inverses For every real number x there is a real number x –1 such that x x –1 = 0 Examples 1. 7 7 –1 = 1 2. 3 –1 (3 –1 ) –1 = 1 Additive inverse x –1 is also written x 1 Axioms

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7/9/2013 Operations 6 Division The division operation is defined by for any real numbers a and b The operation – is called subtraction … which is just addition of negatives Examples: 1. 7 – 4 = 7 + (-4) = 3 2. 10 – 12 = 10 + (-12) = -2 Order of Operations or / or a b = a / ba / b = ab a b –1 Axioms

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7/9/2013 Operations 7 Order of Operations Addition and Multiplication Multiply first then add Examples 1. 2. 3 5 + 4 7 = 15 + 28 = 43 3 x + 4 5 – 1 7 = 3x + 20 – 7 = 3x + 13

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7/9/2013 Operations 8 Order of Operations Addition and Multiplication Apply distributive property to clear groups Examples 1. 2. 3 ((x + 4) + (x – 1) 7) + 1 = 3 ((x + 4) + 7x – 7) + 1 = 3 (x + 4 + 7x – 7) + 1 = 3x + 12 = 3 (8x – 3) + 1 = 24x – 9 + 1 = 24x – 8 3 (x + 4)

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7/9/2013 Operations 9 9 Intervals Open interval: ( a, b ) The set of all numbers between a and b Closed interval: [ a, b ] The set of all numbers between a and b including a and b Half-open/half-closed: [ a, b ), ( a, b ] [ a, b ) includes a, excludes b ( a, b ] excludes a, includes b Notation { x | a < x < b } = { x | a x b } =

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7/9/2013 Operations 10 Binary Relations Symbols: = Grouping Symbols Symbols: { }, ( ), [ ] Special Symbols ± 10 Notation

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7/9/2013 Operations 11 Finding Averages Add n values, divide by n Call Avg the average of n values of x 11 Calculations with Data Avg = x 1 + x 2 + x 3 + ··· + x n n = n n 1 k = 1k = 1 xkxk

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7/9/2013 Operations 12 Example Average of test scores 73, 85, 14, 92 Works fine for small number of values 12 Calculations with Data 4 1 4 k = 1k = 1 xkxk = 66 = (73 + 85 + 14 + 92) 1 4 = (264) 1 4

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7/9/2013 Operations 13 Example Travel d miles in t hours For d = 200 miles and t = 3 hours Works for large/infinite number of values Calculations with Data Average speed = d t = 200 miles 3 hours = 66.67 mph

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7/9/2013 Operations 14 Calculations with Data Finding percentages A per-100 proportion a is to b as P is to 100 a as a percentage of number b is a b (100) = P … that is a b = 100 P

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7/9/2013 Operations 15 Calculations with Data Example In a chain saw 12 ounces of oil are added per gallon of gasoline What percentage of the mixture is oil ? Note that 1 gal = 128 oz For every 100 ounces of mixture 8.571 ounces are oil % oil 8.571 % 12 128 + 12 = ( 100 ) %

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7/9/2013 Operations 16 Calculations with Data Finding percent change A variable p changes by amount p What percentage of p is p ? Percent change in p is where p is the initial value p p (100) %

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7/9/2013 Operations 17 Calculations with Data Example The price of gasoline increased from $2.56 per gallon to $3.89 per gallon Change is: p = 3.89 – 2.56 = 1.33 Percent change is p p (100) = 1.33 2.56 (100) 51.95 %

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7/9/2013 Operations 18 Calculations with Data Absolute value The unsigned size of a number Definition: a = a, for a 0 – a, for a < 0 For any real number a the absolute value of a, written, is a

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7/9/2013 Operations 19 Calculations with Data Examples 1. 2. 3. 4. 7 = 7 = 3 0 = 0 Question: If a is any real number … … is – a positive or negative ? – 3 so a = ( 3) = 3 – – – Here a = 3 – – x 5 – – + =, for x 5, for x < 5

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7/9/2013 Operations 20 Think about it !

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One step equations Add Subtract Multiply Divide Addition X + 5 = -9 X + 5 - 5 = -9 -5 X = -14 -6 + X = -4 -6 + 6 + X = -4 + 6 X = 2.

One step equations Add Subtract Multiply Divide Addition X + 5 = -9 X + 5 - 5 = -9 -5 X = -14 -6 + X = -4 -6 + 6 + X = -4 + 6 X = 2.

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