THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

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1 THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do.

2 THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive, egative, or zero. I that sese, a iteger is simply either a atural umber with a positive or egative sig attached to it or the umber 0 which may also cosidered a siged umber. The iteger umber system has five arithmetical operatios. We will study all five i detail i this tutorial. Expoetiatio (a special case of multiplicatio) will also be covered as a separate topic because of its importace i sciece-related applicatios. Siged Numbers Each iteger is said to be a siged umber, which has either a positive sig or a egative sig. Oly the umber 0 may be cosidered as beig either positive or egative.

3 Covetios Ay siged umber be positive. THE INTEGERS + with o visible sig is assumed to For example: Remember that is a exceptio because it may be cosidered as either positive or egative depedig o the cotext. Therefore: The mius sig i frot of a umber may be regarded as the product of (-1) times the umber or: 3. where is a atural umber: For example: - 11 (-1). ( +11) or: - 11 (-1). ( 11) (-1). ( ) The plus sig i frot of a umber may be regarded as the product of (+1) times the umber or: For example: + 11 ( +1). ( 11) + ( +1).( )

4 THE INTEGERS The Absolute Value of a Siged Number The absolute value of a siged umber is the umber itself without the sig. By covetio 1, it is always a positive umber. Sometimes, it is regarded as the positive part of the siged umber. Notatio: where is a atural umber. For example: by covetio 1 Note that the absolute value is always positive.

5 Multiplyig Siged Numbers What to do ad what ot to do

6 THE INTEGERS Multiplyig Siged Numbers What to do The product of two siged umbers is foud by multiplyig the umbers (without the sigs) together ad the applyig the Rule of Sigs to the aswer. The Rule of Sigs for Products Ay product of positive umbers is also a positive umber. The product of a eve umber of egative umbers is a positive umber. The product of a odd umber of egative umbers is a egative umber. EXAMPLE 1 This is the justificatio for covetio 2. ( 4).( -1 ) ( +4). (-1) - ( 4. 1 ) - 4 Each of these umbers is a factor of the etire product.

7 Expoetials of Siged Numbers What to do ad what ot to do

8 THE INTEGERS The Expoetial Operatio of Siged Numbers The expoetial operatio is a special case of the operatio of multiplicatio. A iteger, or a siged umber, whe multiplied together with itself times, is said to be take to the expoet (or power). The otatio for the expoetial operatio is as follows: expoet a : a. a..... a iteger times a What to do atural umber siged umber All rules for multiplyig siged umbers apply to evaluatig expoetials of itegers. Multiplyig Siged Numbers The product of two siged umbers is foud by multiplyig the umbers (without the sigs) together ad the applyig the Rule of Sigs to the aswer.

9 THE INTEGERS The Expoetial Operatio of Siged Numbers EXAMPLE 1 What to do (-4) 3 (-4).(-4).(-4) iteger expoet 3 times - 64 A odd umber of egative factors meas a egative product. EXAMPLE 2 What to do (-4) 4 (-4). (-4). (-4). (-4) iteger expoet 4 times A eve umber of egative factors meas a positive product.

10 Dividig Siged Numbers What to do ad what ot to do

11 THE INTEGERS Dividig Siged Numbers What to do At this iitial stage of divisio, simply determie the sig of the umerator ad the sig of the deomiator separately. The apply the Rule of Sigs for Quotiets. The Rule of Sigs for Quotiets Ay quotiet of positive umbers is also a positive umber. The quotiet of two egative umbers is a positive umber. The quotiet of oe egative umber ad oe positive umber is a egative umber. EXAMPLE 1 11 _ _ 6 Oe egative sig i both umerator ad deomiator meas a egative quotiet.

12 Dividig Siged Numbers EXAMPLE 2-7 _ -4 / THE INTEGERS What ot to do Do t assume that a egative sig outside the fractio affects both umerator ad deomiator.. (-1). (-4) - (-1) (-7) + _ 7 4 The egative sig may be applied to either umerator or deomiator but NOT BOTH. - (-1) (-7) (-7) - _ 7-7 _ -4. (-4) What to do (-1). (-4) 4

13 THE INTEGERS Dividig Siged Numbers What to do Alterate Rule of Sigs for Quotiets If the umber of egative factors i both umerator ad deomiator is eve, the the quotiet is positive. If the umber of egative factors i both umerator ad deomiator is odd, the the quotiet is egative. EXAMPLE 3 Three egative sigs i both umerator ad deomiator meas a egative quotiet _ - _ 7-7 _ -4 (- 1 ). -4 4

14 Addig Siged Numbers What to do ad what ot to do

15 Addig Siged Numbers THE INTEGERS What to do Case 1: whe both umbers have the same sig The sum of two siged umbers with the same sig is foud by addig the umbers (without the sigs) together ad the applyig the Rule of Sigs to the aswer. Case 2: whe the umbers have opposite sigs The sum of two siged umbers havig opposite sigs is foud by subtractig the smaller umber (without the sig) from the larger umber (without the sig) ad the applyig the Rule of Sigs to the aswer. The Rule of Sigs for Sums Ay sum of positive umbers is also a positive umber. Ay sum of egative umbers is also a egative umber. The sum of a positive umber with a egative umber will have the same sig as the larger (i absolute value) of the two umbers.

16 THE INTEGERS Addig Siged Numbers What to do Case 1: whe both umbers have the same sig The sum of two siged umbers with the same sig is foud by addig the umbers (without the sigs) together ad the applyig the Rule of Sigs to the aswer. The Rule of Sigs for Sums Ay sum of positive umbers is also a positive umber. Ay sum of egative umbers is also a egative umber. EXAMPLE 1 ( 4) + ( 17) + ( ) + 21 (-9) + (-9) - ( 9 + 9) - 18 (-3) + (-1) + ( 0) + (-8) -( ) - 12 Each of these umbers is called a term of the etire sum.

17 Addig Siged Numbers THE INTEGERS What to do Case 2: whe the umbers have opposite sigs The sum of two siged umbers havig opposite sigs is foud by subtractig the smaller umber (without the sig) from the larger umber (without the sig) ad the applyig the Rule of Sigs to the aswer. The Rule of Sigs for Sums The sum of a positive umber with a egative umber will have the same sig as the larger (i absolute value) of the two umbers. EXAMPLE ( 3) + (-7) ( 7 ) - 4 (-33) + ( 75 ) +( 75-33) + 42

18 Addig Siged Numbers THE INTEGERS Case 2: whe the umbers have opposite sigs EXAMPLE 3 What to do Whe there are more tha two terms i a sum, add the umbers pairwise from left to right. (-33) + ( 75 ) + (- 43) + (- 43) (- 43)

19 Subtractig Siged Numbers What to do ad what ot to do

20 THE INTEGERS Subtractig Siged Numbers What to do Covert the subtractio problem ito a additio of siged umbers as follows: - ( m) ( ) ( m) + (- ) m where, are siged umbers. The add the resultig siged umbers together accordig to the appropriate rules. EXAMPLE 1 - (-33) ( 75 ) What ot to do - What to do (-33) (- 75) + -( ) After brigig the mius sig ito the bracket of the secod term, do ot forget to place a plus sig i betwee the two terms. Otherwise, the subtractio problem has icorrectly tured ito a multiplicatio problem. / (-33) ( 75 ) (-33) (- 75)

21 Subtractig Siged Numbers THE INTEGERS EXAMPLE 2 What to do Whe there are two subtractios, oe followig the other, subtract the umbers pairwise from left to right. (-33) - ( 75 ) - (-10) (-33) + (- 75) - (-10) (-108) - (-10) (-108)+ ( +10) -( ) - 98

22 THE ARITHMETIC OF FRACTIONS - multiplicatio, divisio - additio, subtractio What to do ad what ot to do.

23 THE RATIONAL NUMBERS REVIEW: Recall that a ratioal umber is simply a well defied fractio, meaig oe havig a o-zero deomiator. Moreover, two or more fractios may have the same umerical value eve though their umerators ad deomiators are ot idetical. For example, 4/8 ad 5/10 are both equal to ½, but their umerators ad deomiators do ot match. I this case, they are said to be equivalet. The Fudametal Law of Fractios The value of a fractio is NOT chaged if BOTH umerator ad deomiator are either multiplied or divided by the same o-zero umber.

24 THE RATIONAL NUMBERS The ratioal umber system has five arithmetical operatios. Four of them - multiplicatio, divisio, additio, ad subtractio - we will study i detail i this tutorial. The fifth, that of expoetiatio (a special case of multiplicatio), will be covered as a topic i its ow right. Divisio ad Subtractio of Ratioal Numbers as Special Cases of Multiplicatio ad Additio We are goig to discover that divisio of fractios is simply a special case of multiplicatio ad that subtractio of fractios, as i the case of whole umbers, may be iterpreted as the additio of siged umbers.

25 Multiplyig Fractios What to do ad what ot to do

26 THE RATIONAL NUMBERS Multiplyig Ratioal Numbers What to do The umerator of the product of two fractios is foud by multiplyig the umerators of each of them together. The deomiator of the product of two fractios is foud by multiplyig the deomiators of each of them together. ( p. q _ ) m. q. p m,,, are, / p m q q 0 where itegers. EXAMPLE 1 ( 4 _ ). -1 _ (-1) _

27 THE RATIONAL NUMBERS Multiplyig Ratioal Numbers ( p. q _ ) m. q. p m,,, are, / p m q q 0 where itegers. The Fudametal Law of Fractios The value of a fractio is NOT chaged if BOTH umerator ad deomiator are either multiplied or divided by the same o-zero umber. EXAMPLE 2 ( _ 5 ). _ I this case, BOTH umerator ad deomiator are divided first by 5 ad the by 4.. (-8).(-5) \ 4 (-8) \.(-5) (-2) (-1) This techique of simplifyig fractios is called cacellig commo factors across the fractio lie. \ \

28 Dividig Fractios What to do ad what ot to do

29 THE RATIONAL NUMBERS Dividig Ratioal Numbers What to do Divisio of two fractios is simply the multiplicatio of the umerator fractio by the reciprocal of the deomiator fractio. ( m _) where p m q, : p q _ p q p q,, are, / m _ q, m 0 p.. ( _ ) m q. m itegers. EXAMPLE 1 11 _ -6 _ : ( 2 ) ( 11 _ ). ( _ 5 ) (-6). (2) 55 -_

30 THE RATIONAL NUMBERS Dividig Ratioal Numbers ( m _ : ) p q _ (_ p ) _ q m The Fudametal Law of Fractios.,,, are,, / p m q q m 0 where itegers. The value of a fractio is NOT chaged if BOTH umerator ad deomiator are either multiplied or divided by the same o-zero umber. EXAMPLE 2 To simplify the aswer, BOTH umerator ad deomiator are divided first by 3 ad the by (-7). _ _ ( _ 1 2 : -14) 3. (-14) 3 \\.(-14) \ (-49). 15 (-49). 15 ( _ ) \ _ 2 35

31 THE RATIONAL NUMBERS Dividig Ratioal Numbers Divisio of two fractios is simply the multiplicatio of the umerator fractio by the reciprocal of the deomiator fractio. EXAMPLE 3 What to do I the case of two (or more) divisios, whe brackets are preset: 1. divisios withi brackets are performed first. _ _ : : ( _ 4 ) _ ( 15 _ ).( _ 9 ) ( _ 3 ). ( (-56) : ) (-14) _ 135 (-56) 1-49 \ \ \ \ _ 8 315

32 THE RATIONAL NUMBERS Dividig Ratioal Numbers Divisio of two fractios is simply the multiplicatio of the umerator fractio by the reciprocal of the deomiator fractio. EXAMPLE 4 What to do I the case of two (or more) divisios with o bracketspreset: 1. perform the divisios i order from left to right. _ 3-49 ( 15 _ ) _ 3-49 ( 15 _ : ) :( _ 4 ) _ 3-49 : : _ ( -14 _ ) ( _ 4 ) _ 2. ( _ : 9 ) \ 4 2 \ _ 9 70

33 Addig Fractios What to do ad what ot to do

34 THE RATIONAL NUMBERS Addig Ratioal Numbers Let s thik a bit about how we collect thigs. If we collect stamps, the i order to add to our collectio, we must have aother stamp - meaig a object of the same type. This is also true i mathematics. I order to add two umbers together they must be of similar type or i some way like each other. This otio of likeess is defied differetly depedig o the cotext. For example, two ratioal umbers may be collected together, or added together, if they have the same deomiators. Why that is comes from the idea that a fractio is the relative magitude of a part of a whole. The deomiator is idicative of the value of the whole.

35 Addig Ratioal Numbers ( p + q _ ) m _ p q _ THE RATIONAL NUMBERS What to do.. q. where p, m, q, are q, / 0 : p + m q formally m _ itegers. Step 1 is to fid a commo deomiator for both fractos. The smallest oe, called the lowest commo deomiator (LCD), is the best choice - but ot the oly choice. q. is oe possible choice for the above two fractios because both deomiators ad are preset. Recall that, to be able to add a uit whole _1 4 + _1 3 q, we would eed Step 2 is to fid equivalet fractios to the origial oes, BOTH havig the chose commo deomiator. _ p. q. NOTE q.. q _ m. q. q

36 Addig Ratioal Numbers ( p + q _ ) m _ The equivalet fractios: p q _ THE RATIONAL NUMBERS What to do.. q. p, m, q, are q / 0 : p + m q formally m _ itegers. Step 3 is to ADD the equivalet fractios together, by ADDING THE NUMERATORS ad placig the sum over the chose commo deomiator. _ p. q. NOTE Performig the additio: q.. q _ m. q. q p _ + p. m. )( m _) q q (_ p. ) q. + (_ m.q q ) +. q.. NOTE q. q

37 THE RATIONAL NUMBERS Addig Ratioal Numbers ( p + q _ ) m _ : p + m q. q.,,, are, / p m q q 0 where. itegers. EXAMPLE _ _ 9 What to do Step 1 is to fid a commo deomiator. The smallest oe, called the lowest commo deomiator (LCD), is the best choice - but ot the oly choice.. is oe possible choice for the above two fractios because both deomiators ad are preset To choose as the commo deomiator would be tatamout to dividig the pie ito pieces. 6 9 However, both ad fit ito a much smaller umber tha. I fact, both fit ito (or divide ito). Is this the smallest?

38 THE RATIONAL NUMBERS EXAMPLE cotiued What is the smallest umber ito which both 6 ad 9 fit? Fidig LCD s To fid the lowest commo deomiator (LCD) from we proceed as follows: 6 ad 9 1. Factor both deomiators ito primes (i.e. util they ca t be factored further) Collect all of the distict factors across both deomiators ad take the maximum umbers of each which appear. distict factors i both 2, 3 maximum umber of each: oe factor of 2 ; two factors of 3 3. Multiply together the maximum umbers of those distict factors which appear. This is the LCD. _5 6 LCD for ad is:.. -1 _

39 THE RATIONAL NUMBERS EXAMPLE cotiued _5 6 LCD for ad is:.. -1 _ Step 2 is to fid equivalet fractios to the origial oes, BOTH havig the chose commo deomiator. 5 _ 6 _ _ 9 _(-1) Step 3 is to ADD the equivalet fractios together, by ADDING THE NUMERATORS ad placig the sum over the chose commo deomiator. _ _ 9 _ _(-1) (-2)

40 THE RATIONAL NUMBERS Addig Ratioal Numbers EXAMPLE _ _ 5 What ot to do _ _ 5 \ 1 _ 5 4 \ + -3 _ 5 1 DO NOT cacel commo factors across a sig. + \ \ 5 + (-3) DO NOT add deomiators ad DO NOT add umerators without a commo deomiator.

41 Subtractig Fractios What to do ad what ot to do

42 p q _ - m _ THE RATIONAL NUMBERS Subtractig Ratioal Numbers p q _ : p - m q formally What to do. -. q. where p, m, q, are q, / 0 m _ itegers. Step 1 is to fid a commo deomiator for both fractos. The smallest oe, called the lowest commo deomiator (LCD), is the best choice - but ot the oly choice. q. is oe possible choice for the above two fractios because both deomiators ad are preset. Recall that, to be able to subtract a uit whole _1 3 - q _1 4, we eeded Step 2 is to fid equivalet fractios to the origial oes, BOTH havig the chose commo deomiator. _ p. q. NOTE q.. q _ m. q. q

43 p q _ THE RATIONAL NUMBERS Subtractig Ratioal Numbers ( m _) : p - - m q The equivalet fractios: p q _ formally. What to do -. q. p, m, q, are q / 0 Step 3 is to SUBTRACT the equivalet fractios together, by SUBTRACTING THE NUMERATORS ad placig the differece over the chose commo deomiator. _ p. q. NOTE Performig the subtractio: q.. q m m. q. q itegers. -. _ m.q q p. - m. - q p _ ) m q ( p ).. q NOTE q.. q q.

44 THE RATIONAL NUMBERS Subtractig Ratioal Numbers ( m _) : p - - m q p q _ formally. - q..,,, are / p m q q 0 itegers. EXAMPLE _ _ 7 What to do Step 1 is to fid a commo deomiator. The smallest oe, called the lowest commo deomiator (LCD), is the best choice - but ot the oly choice is oe possible choice for the above two fractios because both deomiators ad are preset Is this the lowest commo deomiator?

45 THE RATIONAL NUMBERS EXAMPLE cotiued What is the smallest umber ito which both 10 ad 7 fit? Fidig LCD s To fid the lowest commo deomiator (LCD) from we proceed as follows: 10 ad 7 1. Factor both deomiators ito primes (i.e. util they ca t be factored further) distict factors i both, 2 5 maximum umber of each: oe factor of oe factor of _3 10 LCD for ad is: 7-4 _ is a prime umber. 2. Collect all of the distict factors across both deomiators ad take the maximum umbers of each which appear., 2 ; oe factor of 5 3. Multiply together the maximum umbers of those distict factors which appear. This is the LCD. 7 ;

46 THE RATIONAL NUMBERS EXAMPLE cotiued _3 10 LCD for ad is: -4 _ Step 2 is to fid equivalet fractios to the origial oes, BOTH havig the chose commo deomiator. (_ 3 ) (_ 3. 7 ) -4 _ _(-4) Step 3 is to ADD the equivalet fractios together, by ADDING THE NUMERATORS ad placig the sum over the chose commo deomiator. (_ 3 ) 10 -( -4 _) 7 _ _(-4) (-40)

47 THE RATIONAL NUMBERS Subtractig Ratioal Numbers EXAMPLE _ _ 13 What ot to do _ _ 13 \ _ _ 13 \ \ - DO NOT cacel commo factors - across a sig. \ DO NOT subtract deomiators ad DO NOT subtract umerators without a commo deomiator.

48 THE RATIONAL NUMBERS Addedum: Mixed Numbers Mixed umbers are combiatios (actually additios) of whole umbers ad fractios. EXAMPLES 1 _1 is read as: 2 oe ad a half is iterpreted as: 1 + _1 2 is equal to: 1 + _1 _ 1 + _1 1 _ 2 + _1 _ _5 is read as: 8 mius (or egative) three ad five eighths - is iterpreted as: 3 + _5 8 is equal to: - ( _ _5 ) - ( _ 3 + ) - (_ _5 ) - _

49

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