Sections 5.2 and 5.3 Signed Numbers; Rational Numbers; Exponents; Order of Operations
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1 Sections 5. nd 5.3 Signed Numbers; Rtionl Numbers; Exponents; Order of Opertions Number Line Negtive numbers Positive numbers Objectives 1. Perform opertions with signed nd rtionl numbers. Evlute exponentil expressions 3. Use squre root to undo squre 4. Use the order of opertions 1/8/011 Section Positive nd negtive numbers re clled signed numbers. 1/8/011 Section 5. Opertions with signed numbers You should know how to dd, subtrct, multiply nd divide signed numbers with your clcultor. Some of the properties of those opertions re given on the following slides. Properties of Addition nd Multipliction For numbers, b, nd c, the following properties hold. Property Commuttive Addition + b = b + Multipliction b = b Associtive ( + b) + c = + (b + c) (b)c = (bc) 1/8/011 Section /8/011 Section 5. 4 Properties of Addition nd Multipliction Absolute Vlue Distributive Property of Multipliction with Respect to Addition (b + c) = b + c (b + c) = b + c The bsolute vlue of rel number cn be defined s the distnce between 0 nd the number on the number line. The symbol for the bsolute vlue of x is x, red the bsolute vlue of x. The bsolute vlue of number is never negtive. 1/8/011 Section /8/011 Section
2 Double Negtive Rule For ny rel number x, ( x) = x. Addition Subtrction Link Subtrction number is the sme s dding the negtive of tht number. 7 6 = 7 + (-6) Exmple: -(-9) = 9 1/8/011 Section /8/011 Section 5. 8 Adding Signed Numbers Like Signs Add two numbers with the sme sign by dding their bsolute vlues. The sign of the sum is the sme s the sign of the two numbers. Unlike Signs Add two numbers with different signs by subtrcting the smller bsolute vlue from the lrger to find the bsolute vlue of the sum. The sum is the sme sign s the number with the lrger bsolute vlue. 1/8/011 Section 5. 9 Exmples Adding Signed Numbers Evlute: Solutions: /8/011 Section Multiplying Signed Numbers Like Signs Multiply two numbers with the sme sign by multiplying their bsolute vlues to find the bsolute vlue of the product. The sign of the product is positive. Unlike Signs Multiply two numbers with different signs by multiplying their bsolute vlues to find the bsolute vlue of the product. The sign of the product is negtive. 1/8/011 Section Dividing Signed Numbers The rules for dividing signed numbers is the sme s the rules for multiplying signed numbers. 1/8/011 Section 5. 1
3 Exmple Multiplying nd Dividing Signed Numbers /(-3) Solutions: Exponentil Expression If is rel number nd n is nturl number, then the exponentil expression n is defined s n =.... n fctors of The number is the bse nd n is the exponent. 1/8/011 Section /8/011 Section Exponentil Nottion Becuse exponents indicte repeted multipliction, rules for multiplying cn be used to evlute exponentil expressions. Exmple Exponentil Nottion Evlute:. (-6) b. -6 Solutions: Terminology: squred 3 cubed n rised to the n or to the n 1/8/011 Section /8/011 Section Exmple Exponentil Nottion The Rdicl Evlute: (c) (-5) 3 (d) (-) 4 The symbol squre root. is clled rdicl or Solution: Definition: If b =, then b is positive, nd b. b = 1/8/011 Section /8/011 Section
4 The Rdicl - Exmple Since 6 is positive, nd 6. 6 = 36 6 = 36 Does the squre of ny other number equl to 36? The Rdicl The rdicl sign undoes squre (lmost). The rdicl of number is lwys positive. When b is positive, When b is negtive, b = b b = b In most pplictions we will be using positive numbers. 1/8/011 Section /8/011 Section 5. 0 Exmple We cn use the rdicl to solve equtions. Find ll numbers tht mke the following equtions true. 1. x = 11. c = 46 Solution: Use the rdicl on your clcultor. 1. x = ±11. c ±6.78 (rounded to the nerest hundredth) 1/8/011 Section 5. 1 Defining the Rtionl Numbers The set of rtionl numbers is the set of ll numbers which cn be expressed in the form, where nd b re signed whole b number nd b is not equl to 0. is clled the numertor. b is clled the denomintor. Exmples: The following re exmples of rtionl numbers: ¼, -½, ¾, 5, 0 1/8/011 Section 5. Rtionl Numbers nd Decimls Any rtionl number cn be expressed s deciml by dividing the denomintor into the numertor. Exmple - Rtionl Numbers nd Decimls Exmple: Express ech rtionl number s deciml b. Solution: b The line bove mens repeting. 1/8/011 Section /8/011 Section
5 Rtionl Numbers nd Decimls The deciml equivlent of ll rtionl numbers either repets or termintes. The deciml equivlent of repet or terminte. is n irrtionl number. does not Reducing Rtionl Number If b is rtionl number nd c is ny number other thn 0, c = b c The rtionl numbers nd b equivlent frctions. b c b c re clled 1/8/011 Section /8/011 Section 5. 6 Exmple Reducing Rtionl Number Find reduced form: ) 10 b) 5 Solution: ) 10 5 = = b) = = Adding Rtionl Numbers To dd or subtrct frctions, you need common denomintor. Exmples: = = = = = Know how to do this on your clcultor. 1/8/011 Section /8/011 Section 5. 8 Multiplying Rtionl Numbers The product of frctions is the product of the numertors divided by the product of the denomintors. You do not need common denomintor. Multiplying Rtionl Numbers Multiplying ny number by frction mens tke tht portion of. Exmple: Multiply 10. ½ Exmple: 3 3 = = Know how to do this on your clcultor. 1/8/011 Section Either on clcultor or by hnd, we find tht = = = = which is ½ of 10. 1/8/011 Section
6 Order of Opertions 1. Perform ll opertions within grouping symbols.. Evlute ll exponentil expressions. 3. Do ll the multiplictions nd divisions in the order in which they occur, working left to right. 4. Finlly, do ll dditions nd subtrctions in the order in which they occur, working left to right. Order of Opertions - PEMDAS We lso use the cronym PEMDAS, prentheses, exponents, multipliction nd division, ddition nd subtrction, for the order of opertions. 1/8/011 Section /8/011 Section 5. 3 Order of Opertions Using PEMDAS Exmple: Simplify Solution: There re no prenthesis. Thus, we begin by evluting exponentil expressions = = = = = 19 then multiply nd divide left to right finlly dd nd subtrct left to right Order of opertions Multipliction or subtrction? 5 7 subtrction 5 (-7) multipliction (-7) 5 subtrction (-7) (-5) multipliction (-5) 3(-7) (-5) (-1) multiply first then subtrct =16 1/8/011 Section /8/011 Section
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