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Chapter 2 Integers

Chapter 2 Integers. 2.1 Introduction to Integers. The whole numbers { 0, 1, 2, 3, 4,…..} with all negative natural numbers { …., -3, -2, -1} form a set of numbers called the integers Graph integers on a number line.

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Chapter 2 Integers

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  1. Chapter 2 Integers

  2. 2.1 Introduction to Integers The whole numbers { 0, 1, 2, 3, 4,…..} with all negative natural numbers { …., -3, -2, -1} form a set of numbers called the integers Graph integers on a number line -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 2 -3 Use < or > to make a true statement 8 > -7 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 -5> -8 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

  3. Absolute Value Absolute Value : A given number’s distance from zero on a number line • -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 • Absolute value -6 = 6 Absolute value 6= 6 • Absolute value of a number is always positive or zero • The absolute value of a negative number is always positive • The absolute value of a positive number is always positive • The absolute value of zero is zero Example • 9 = 9 21 = 21 • Additive inverse • 6 and -6 are additive inverses and sum = 0 • 6 + (-6) = 0

  4. 2.2 Adding integers + 40 means Moves 40 steps to the right Add integers with like sign 300 + 40 300 340 Starting position Final Position - 300 + (-40)= -340 + (-40) means move 40 steps to the left - 340 -300 Starting position Final Position Add integers with different signs -300 + 80 + 80 means move 80 steps to the right -300 - 220 Starting position Final Position

  5. Examples • 40+ (-23) = -63 24 + (-16) = 8 10 – 25 = -15 Add -15 + 24 + ( -16) + 15 + (-30) 9 + (-16) + 15 + (-30) Add -15 and 24 to get 9 -7 + 15 + (-30) Add -9 and 16 to get -7 8 + (-30) Add -7 and 15 to get 8 -22 Add 8 and – 30 to get -22 Other process -15 + 24 + ( -16) + 15 + (-30) 24 + 15 + (-15) + (-30) ( Rearrange the addends so the positives and negatives are grouped separately) = 39 + (-45) ( Add positives and negatives separately ) = -6 ( Reconcile the positive and negative sums)

  6. No 49( Pg 102) The following table lists the assets and debts for the Smith family. Calculate their net worth. + - Assets Debts Savings = $1498 Credit card balance = $ 1841 Checking = $2148 Mortgage = $ 74,614 Furniture = $18,901 Automobile = $5 488 Jewelry = $3845 Adding all assets and debts 1498 + 2148 + 18901 + 3845 + (-1841) + (-74614) + (-5488) = 26392 + (-81943) = -55551 The net worth = -$55,551

  7. 2.3Subtraction statements as equivalent addition statements 20 – 42= -22 20 + (-42) = -22 20 – 42 = 20 + (-42) = -22 The subtrahend changes to its additive inverse The operation changes from – to + • 10 – 15= -25 • 10 + (-15)= -25 • 10 – 15= -25 • = -10 + (-15) • = -25 The subtrahend changes to its additive inverse The operation changes from – to +

  8. No 48( Pg 111) Gary bought a Fender Stratocaster guitar in 1960 for $ 150. In 2004, he takes the guitar to a guitar show and sells it for $12,000. What was the net ? Is it a profit or loss ? Net = Revenue – cost Revenue = 12000 Cost 150 Net = 12000- 150 = 11850 ; Profit

  9. 2.4 Multiplying, Dividing with integers Rules • When multiplying two numbers that have different signs, the product is Negative 7.( -6) = -42 • When multiplying two numbers that have the same sign, the product is positive - 5 .(-9) = 45 • When multiplying signed numbers, count the total number of negative factors: If there are an even number of negative factors, then the product is positive (-2).(-4).(-5).(-6) = 240 If there are an odd number of negative factors, then the product is negative (-2).(-3).(-4)= -24

  10. Rules • When evaluating an exponential form that has a negative base: If the exponent is even, the product is positive (-6)2 = (-6).(-6) = 36 If the exponent is odd, the product is negative (-4)3 = (-4).(-4).(-4) = -64 • When multiplying or dividing two integers: If they have the same sign, the result is positive 45 divide by 5 = 9 If they have different signs, the result is negative - 45 divide by 5 = -9

  11. Rules • To solve a missing factor, write a related division sentence, dividing the product by the known factor 6x = -36 x= = - 6 Square roots Every positive number has two square roots, a positive root and a negative root means to find the principal square root of n, which is its positive square root. - means to find the additive inverse of the principal square root of n which is the negative square root of n. The square root of a negative number is not an integer. is not an integer

  12. 2.5 Order of Operations Perform Operations in the following order: • Grouping symbols: parenthesis(), brackets[], braces{}, absolute value , radicals , and fraction bars • Exponents • Multiplication or division from left to right, in order as they occur • Addition or subtraction from left to right, in order as they occur

  13. Applications and Problem Solving 1. Solve problems involving net N = R – C, where N represents the net amount, Represents revenue, and C represents cost • Solve problems involving force The formula for force F = ma, F represents force, m represents mass, and a represents acceleration In the American system of measurement Force (F) measured in lbs, mass (m) in slugs, acceleration (a) in ft. sec2 The Earth accelerates all objects downward at the same rate, about -32ft/sec2 In metric system Force (F) in Newtons , mass (m) in kilograms (kg), acceleration (a) in m/sec2 Metric system – 10 meters /second/second • Solve problems involving voltage V = ir V = Voltage (volts), I = current (Ampere) , r = resistance (Ohms) 4. Solve problems involving average rate Distance = rate . Time d = rt Average rate : A measure of the rate at which an object travels a total distance in a total amount of time

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