Session 37 Introduction to Integers. How may we compare the values of the following two accounts?

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1 Session 7 Introduction to Integers How my we compre the vlues of the following two ccounts? Sm hd two ccounts, svings ccount nd lon. She hd certificte of deposit worth $4,60 nd cr lon of $8,290. Which ccount hs the gretest vlue? The nswer to this question depends on how we think bout the sitution. If we re sking which sitution involves more dollrs, then $8,290 involves greter dollr mount thn $4,60. However, if we re sking which vlue is worth more to Sm, obviously the certificte of deposit, which is money she owns is worth more thn the lon, which is money she owes. In mthemticl terms, Sm s CD hs positive vlue nd the lon hs negtive vlue. In mthemtics, when we look t just the number of dollrs nd not whether the vlue is positive or negtive we re tlking bout the bsolute vlues. We consider two different models to help us understnd the properties of positive nd negtive numbers. First we consider the number line model nd then we consider the chip model (set model). The Set of Integers using the Number Line Model Consider the two exmples below. On this first number line the vlue 1 is distnce of one unit from zero to the right. We use vector of length one pointing to the right to represent positive one, 1. 1 The vlue 1 is distnce of one unit from zero in the opposite direction. For this reson the vlue 1 is sometimes referred to s the opposite of 1. We use vector of length one pointing to the left to represent negtive one,. We could write the vlue positive 1 s +1, but we normlly leve out the positive sign. A vlue with no sign is ssumed to be positive vlue. Exmples: () One step forwrd, +1. (b) One step bckwre,. (c) A temperture of one degree bove zero, +1. (d) A temperture of one degree below zero,. Definition. The set of integers is the set {,,,, 0, 1, 2,, }. Notice tht this is the set of whole numbers {0, 1, 2,, 4,... } together with ll of the opposites. (The opposite of zero is zero.) In the number line model, integers represent vlues s directed distnces. To find loction on the number line we need to know both the distnce (number of units) nd the direction (whether we should count those units moving left or moving right). MDEV 102 p. 16

2 Another term for directed distnce is vector. A vector hs both distnce nd direction. In this context n rrow is used to indicte the vector s direction. The rrow used to represent vector is different from ry. In geometry, ry represents n infinite set of points. For geometric ry, the rrow indictes the direction in which the ry extends indefinitely. A vector is not set of points. The length of the vector indictes specific, finite distnce nd the rrow prt only indictes the direction. The rrow prt of vector does not indicte tht the distnce continues forever. Here is one lst, importnt point. Distnce by itself is non-directed nd is lwys zero or positive. Exmples: directed distnce of 4 to the left of zero undirected distnce of 4 1 directed distnce of 1 to the right of zero The integer lbels on the number line re clled the coordintes of the points. Since the number line coordintes re mesured from zero, zero is clled the origin. Absolute Vlue Absolute vlue refers to the distnce from zero without reference to direction. Absolute vlue, since it is non-directed distnce, is lwys zero or positive. The nottion for bsolute vlue is to put stright verticl lines round vlue. Exmple: Suppose Pt wlked steps forwrd nd Kim wlked 6 steps bckwrd. Who took more steps? Even though we would represent Pt s steps s nd Kim s steps s 6, we would sy tht Kim took more steps. red s the bsolute vlue of is since the (non-directed) distnce is five units from 0 (the origin) red s the bsolute vlue of 6 is 6 since the (non-directed) distnce is six units from 0 (the origin). We would write = < 6 = 6. Pt took fewer steps thn did Kim. This problem nd the opening problem for this session both motivte the concept of bsolute vlue. Exmple: red s bsolute vlue of equls becuse the (non-directed) distnce is. MDEV 102 p. 164

3 The Set of Integers Using the Chip Model (Set Model) The chip model uses colored bens or chips to represent negtive integers nd positive integers. A green ben or shded circle,, will be used to represent 1. A red ben or open circle,, will be used to represent. Exmples: () A deposit of $1 my be represented by. (b) A withdrwl of $1 my be represented by. (c) Receiving cndy brs my be represented s. (d) Giving wy cndy brs my be represented s. Another importnt concept with the chip model is tht there re mny different wys to represent n mount of zero. Exmples: () No chips t ll would represent n mount of zero. (b) would represent n mount of zero, e.g., deposit of $1 nd withdrwl of $1 leves no chnge in the blnce. (c) would represent n mount of zero, e.g., receive cndy brs nd give wy cndy brs leves no chnge in the mount of cndy brs. In other words, two chips of opposite color cncel ech other out, so both chips cn be removed. This exmple illustrtes the property tht 1 + () = + 1 = 0. Additive Inverse We generlize the reltionship where opposites of the sme bsolute vlue cncel ech other out. We further illustrte with exmples with both models. + () = 0 + = 0 Beginning t zero, in the mesurement model, move of three units to the right followed by three units to the left brings us bck to zero. Beginning t zero, in the mesurement model, move of five units to the left followed by five units to the right brings us bck to zero. MDEV 102 p. 16

4 We notice tht the result of combining number nd its opposite is zero. This motivtes the following forml definition. Additive Inverse: For every integer n, there is unique integer m such tht n + m = m + n = 0. The integer m is clled the dditive inverse of n. This property of integers is clled the inverse property for integer ddition. Exmples: Find the dditive inverse for ech of the following integers. Integer Additive Inverse If we consider the negtive sign s representing the opposite, the bove exmple illustrtes tht () = nd (7) = 7, i.e. the opposite of n integer is equivlent to the dditive inverse of the integer. This reltionship llows us to give n lgebric definition for bsolute vlue. Algebric Definition of Absolute Vlue: The bsolute vlue of is defined by if 0. if 0 Exmples: () = since > 0. (b) = () = since < 0. (c) 18 = 18 since 18 > 0. (d) 4 = (4) = 24 since 4 < 0. Compring the Two Models The number line is good model for the expressing positions reltive to some specific vlue, such s grphing on coordinte plnes (reltive to the origin), tempertures on thermometers (reltive to zero degrees or to freezing), nd elevtion (reltive to se level). It is lso good when we need to express frctionl prts of the whole, since we cn find these on the number line, but it would be difficult to do with bens or chips. The chip model is good model for smll mounts (with no frctionl prts) nd for items tht hve tomic chrges such s protons, electrons, nd neutrons. It is lso helpful model for understnding how integer rithmetic works. Extending the Concept of Positive nd Negtive to Rtionl Numbers Definition. The set of rtionl numbers is the set Q : is n integer nd b is nturl number b. Exmples: () is rtionl number since 1. MDEV 102 p. 166

5 2 2 (b) 2 is rtionl number since 2 (c).1 is rtionl number since Wht, if ny, re the reltionships between the sets of nturl numbers (N), whole numbers (W), integers (Z), nd rtionl numbers (Q)? Q Z W N The Euler digrm ( vrition of Venn digrm) illustrtes tht there is subset reltionship between the sets. N W Z Q Note tht they re proper subsets. Why? MDEV 102 p. 167