CHAPTER 9. Rational Numbers, Real Numbers, and Algebra


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1 CHAPTER 9 Rtionl Numbers, Rel Numbers, nd Algebr Problem. A mn s boyhood lsted 1 6 of his life, he then plyed soccer for 1 12 of his life, nd he mrried fter 1 8 more of his life. A dughter ws born 9 yers fter his mrrige, nd her birth coincided with the hlfwy point of his life. How old ws the mn when he died? Strtegy 14 Solve n Eqution. This strtegy my be pproprite when A vrible hs been introduced. The words is, is equl to, or equlspper in problem. The stted conditions cn esily be represented with n eqution. Let m = the mn s ge when he died. 1 6 m m m + 9 = 1 2 m Multiply ech side by 24. The mn ws 72 when he died. 4m + 2m + m = 12m 216 = m 72 = m 40
2 9.1. THE RATIONAL NUMBERS The Rtionl Numbers Where we re so fr: Ech rrow represents is subset of. The rtionl numbers re n extension of both the Frctions nd the Integers. The need: We cn solve 5x = using frctions to get x =, but wht bout 5x =? 5 We cn solve x+7 = 0 using integers to get x = 7, but wht bout x+ 2 = 0? So we need to extend both the frctions nd the integers to tke cre of these problems. Two pproches: 1) Focusing on the property tht every number hs n opposite, wew tke the frctions wioth their opposites s the rtionl numbers. Then we note the integers re lso included: 4 = 12 nd 7 = 7, for instnce. 1 2) Focusing on the property tht every nonzero number hs reciprcl, we form ll possible frctions where the numertor is n integer nd the denomintor is nonzero integer. We will follow the second pproch: Definition (Rtionl Numbers). The set of rtionl numbers is the set n o Q = b nd b re integers, b 6= 0.
3 42 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Exmple. Also, 4, 7 2, 5, 6 4, 0 6, = 1 4, 6 7 = 45 7 Ech frction cn be written in this form nd ech integer n cn be written s n 1. We now hve: Definition (Equlity of Rtionl Numbers). Let b nd c d Then b = c d if nd only if d = bc. Recll tht equl refers to the bstrct concept of quntity ttched to number, while equivlent refers to the vrious numerls used for number. 1 2 = 2 4 = 6 = 1 2 = 2 4 = 6 re ll numerls representing the rtion number = 2 = 2 = 6 9 = = 2 2 re ll numerls representing the rtionl number.
4 Theorem. Let b Then 9.1. THE RATIONAL NUMBERS 4 be ny rtionl number nd n ny nonzero integer. b = n bn = n nb. A rtionl number is in simplest form or lowest terms if nd b hve no b common prime fctors nd b is positive. Exmple. 4, 5 7, 0 re in lowest terms. 1 4, 10 15, 0, 0 re not in lowest terms. Exmple = = 8 9 = = = 5 7 = 5 7 Definition (Addition of Rtionl Numbers). Let b nd c d Corollry. be ny rtionl numbers. Then b + c d d + bc =. bd b + c b = + c. b
5 44 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Exmple = ( ) = = = 1 24 " # = = 1 24 Consider so b = b Theorem. Let b since ( )( b) = b. Also, b + b = b = + b b. = 0 b = 0, be ny rtionl number. Then b = b = b. Frctionl Number Line:
6 9.1. THE RATIONAL NUMBERS 45 Properties of Rtionl Number Addition Let b, c d, nd e f be ny rtionl numbers. (Closure) b + c d is rtionl number. (Commuttive) b + c d = c d + b. (Associtive) b d + c + e f = c b + d + e f (Identity) b + 0 = b = 0 + b. 0 = 0 m, m 6= 0. (Additive Inverse) For every rtionl number, there exists unique rtionl b number b such tht Exmple. b + b = 0 = + b b = (mentlly?) = = = 1 5 = = (mentlly?) = = = 17 2
7 46 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Theorem (Additive Cncelltion). Let b, c d, nd e f be ny rtionl numbers. Then If b + e f = c d + e f, then b = c d. Theorem (Opposite of the Opposite). Let be ny rtionl number. Then b = b b. Definition (Subtrction of Rtionl Numbers). Let b nd c d Also, nd b be ny rtionl numbers. Then c b d = b + c b = b + b c b c d = d bd = c b + b c. d = + ( c) b bc bd = d bc. bd = c b Exmple = 2(12) 9( 7) 108 = 24 ( 6) 108 = = = 29 6 = = = 29 6
8 9.1. THE RATIONAL NUMBERS ( )4 7() = = 7(4) = = 24 ( 6) 108 = 28 = = {z} inverse or opposite of 28 Definition (Multipliction of Rtionl Numbers). Let b nd c d be ny rtionl numbers. Then b c d = c bd. Properties of Rtionl Number Multipliction Let b, c d, nd e be ny rtionl numbers. f (Closure) b c is rtionl number. d (Commuttive) b c d = c d b. (Associtive) b c e d f = c b d e. f (Multiplictive Identity) b 1 = b = 1 b 12 + ( 21) 28 1 = m m, m 6= 0. = 28 (Multiplictive Inverse) For every nonzero rtionl number, there exists b unique rtionl number b such tht b b = 1. Note. The multiplictive inverse of rtionl number is lso clled the reciprocl of the number.
9 48 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Distributive Property of Multipliction over Addition Let b, c d, nd e f be ny rtionl numbers. Then c b d + e = f b c d + b e f. Exmple = (mentlly?) = = (mentlly?) h + = = = 2 27 = 2 27 Definition. Let b nd c d be ny rtionl numbers where c d Then b c d = b d c. Common Denomintor Division: b c b = b b c = c, so or c. A third method so b c b = c : since b = b i b c d = b d c = c d c = c d c = b c d = c b d, b c d = c b d. = = 0 is nonzero.
10 Exmple. (1) (2) () 9.1. THE RATIONAL NUMBERS = 9 40 ( 10) = = 47 = = = 9 20.
11 50 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Ordering Rtionl Numbers Three equivlent wys (s we did with frctions): 1) NumberLine Approch: b < c or c d d > if nd only if b b is to the left of c on the rtionl number line. d 2) CommonPositiveDenomintor Approch: b < c if nd only if < c nd b > 0. c )Addition pproch: b < c d if nd only if the is positive rtionl number p q b + p q = c, or equivlently, d Exmple. Compre First wy: b < c d if nd only if c d 5 6 nd b is positive. = = = Then 11 = , so < 5. 6 such tht
12 9.1. THE RATIONAL NUMBERS 51 Second wy: 5 6 = Since 11 < 10 nd 12 > 0, < Cross Multipliction Inequlity 5. 6 Let b nd be ny rtionl numbers, where b > 0 nd d > 0. Then b b < c if nd only if d < bc. d 5 Exmple. Compre 6 nd Third wy: Note tht the denomintors re positive. Since 11 ( 5)12 = 60 nd 6( 11) = 66, nd 66 < 60, 12 < 5 6. Properties of Order for Rtionl Numbers Trnsitive Less thn nd ddition Less thn nd multipliction by positive Less thn nd multipliction by negtive Density Property between ny two rtionl numbers there exists t lest one rtionl number Similr Properties hold for >, pple,.
13 52 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Exmple. Solve 1 x > h 1 1x > 5 2 i x > x < (Multipliction by negtive) 6 (Associtive Property of Multipliction) (Inverse Property of Multipliction) x > 5 2 (Identity Property of Multipliction) 9.2. The Rel Numbers We strted with the whole numbers. x 5 = 0 hs 5 s whole number solution. But consider: 1) x + 5 = 0 hs no whole number solution, but 5 is n integer solution. We extend to the integers. 2) x = 7 hs no whole number solution, but 7 is frctionl solution. We extend to frctions. ) 5x = 2 hs no integer or frctionl solution, but We extend to rtionls. 5 2 is rtionl solution. 4) x 2 = 2 hs no rtionl solution, so we need to extend once gin. How do we know for sure tht there is no rtionl solution to x 2 = 2?
14 9.2. THE REAL NUMBERS 5 Theorem. There is no rtionl number whose squre is 2. Proof. We use indirect resoning. Suppose x is rtionl number whose squre is 2. Then x cn be written in lowest terms s, where is n integer nd b is b positive integer. 2 Since x 2 = 2, = 2, so b 2 b = 2. Then 2 2 = 2b 2, so 2 is even. But then is even, so = 2n for some integer n. Then (2n) 2 = 2b 2, so 4n 2 = 2b 2. Then 2n 2 = b 2, so b 2 is even, nd thus b is even. Then nd b both hve 2 s common fctor, so b cnnot be in lowest terms, contrdiction. Thus x cnnot be rtionl. We hve lerned tht every frction cn be written s repeting deciml, nd vicevers. Then so cn every rtionl number just by tking opposites. Thus the irrtionl numbers, the numbers tht re not rtionl, must hve infinite nonrepeting deciml representtions. Definition (The Rel Numbers). The set of rel numbers, R, is the set of ll numbers tht hve n infinite deciml representtion.
15 54 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Exmple is irrtionl = is rtionl nd nonterminting = = is rtionl nd terminting. We now hve: The Rel Number Line The rel numbers complete the number line, i.e., for ech rel number there is point on the line, nd for ech point on the line there is unique rel number. The numbers cn be ordered by using their infinite decimls.
16 9.2. THE REAL NUMBERS 55 Representing some rel numbers geometriclly. Recll the Pythgoren Theorem. In right tringle whose legs re lengths nd b nd whose hypotenuse (the long side) hs length c, 2 + b 2 = c = c = c 2 2 = c 2 Thus, the length of the hypotenuse c is the positive rel number whose squre is 2. We huve shown this number to be irrtionl nd represent it by p 2. We cn then use compss to find its loction on the rel number line. Just s 4 hs two squre roots, 2 nd 2, so does 2. We use p p 2 to indicte the positive or principl squre root of 2 nd 2 to indicte the other. Definition (Squre Root). Let be nonnegtive rel number. Then the squre root of (i.e., the principl squre root of ), written p, is defined s p = b where b 2 = nd b 0. Since there re infinitely mny primes p, nd p p is lwys irrtionl (why?), there re infinitely mny irrtionls.
17 56 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Exmple. Without using squre root key, pproximte p 7 to the nerest thousndth. Since 2 2 = 4, ( p 7) 2 = 7, nd 2 = 9, so 2 < p 7 <. Since = 6.25, = 6.76, nd = 7.29, so 2.6 < p 7 < 2.7. Since > 7, nd < 7, so 2.64 < p 7 < Since < 7, nd > 7, so < p 7 < So which do we tke, or 2.646? One more step should give us the nswer. Since < 7, nd > 7, so < p 7 < Since both of these round to 2.646, we sy p to gthe nerest thousndth. Note. For ny nonnegtive rel number x, ( p x) 2 = x. is lso n irrtionl number. is the rtio of the circumference to the dimeter of ny circle. 22 nd (.14 nd.1416 re lso often used). 7 Opertion Properties of Rel Numbers Addition Multipliction Closure Closure Commuttive Commuttive Associtive Associtive Identity (0) Identity (1) Inverse ( ) Distributive of Multipliction over Addition Inverse ( 1 for 6= 0)
18 9.2. THE REAL NUMBERS 57 Definition. For ny two rel numbers nd b, < b if nd only if there exists positive rel number p such tht + p = b. Ordering Properties of Rel Numbers (hold for <, >pple, ) Trnsitive Less Thn nd Addition Less Thn nd Multipliction by Positive Less Thn nd Multipliction by Negtive Density Rtionl Exponents Definition (nth Root). Let be rel number nd n positive integer. 1) If 0, then np = b if nd only if b n = nd b 0. 2) If < 0 nd n is odd, then np = b if nd only if b n =. Note. 1) is clled the rdicnd nd n the index. 2) np = b is red the nth root of, nd is cll rdicl. ) For squre roots, we ususlly write p insted of 2 p. 4) n p n =. 5) We cnnot tke even roots of negtive numbers, such s 4p 1, for if b = 4p 1 b 4 = 1, which is impossible. The sme is true for p 1.
19 58 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Exmple. (1) (2) 4p 256 = 4 since 4 4 = p 256 = () (4) 4. p 27 = since = 27. 6p 64 = does not exist. How to define rtionl exponents? Consider 5 1/2. We wnt 5 1/2 5 1/2 = 5 1 = 5. But p 5 p5 = 5. So wht bout 5 1/2 = p 5? Definition (Unit frction Exponent). Let be ny rel number nd n ny positive integer. Then 1/n = np where 1) n is rbitrry when 0, nd 2) n must be odd when < 0.
20 Exmple. 1) ( 64) 1/ = p 64 = 4. 2) 2 1/5 = 5p 2 = THE REAL NUMBERS 59 Definition (Rtionl Exponents). Le be nonnegtive number, nd m n rtionl number in somplest form. Then 1) 2) ) Exmple. Whixh wy seems esier? Properties of Rtionl Exponents m/n = 1/n m = ( m ) 1/n. ( 8) 4/ = ( 8) 1/ 4 = ( 2) 4 = 16, or ( 8) 4/ = ( 8) 4 = / = / = (4 2 ) 1/ = 16 1/ = p / = (64 1/ ) 4 = 4 4 = = Let nd b represent positive rel numbers, nd m nd n rtionl (not necessrily) positive exponents. Then m n = m+n m b m = (b) m ( m ) n = mn m n = m = n m n In dvnced mth one cn lso define rel number exponents which follow the sme properties.
21 60 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Exmple. For 0 nd b 0, p p b = 1/2 b 1/2 = (b) 1/ = p b. Exmple. Simplify p 16 p 48 so tht the rdicnd is s smll s possible. p 16 p 48 = p 16( p 16 ) = ( p 16( p 16 p ) = ( p 16 p 16) p = 16 p. Exmple. Compute nd simplify: p p p = p p p = p p p p p = 2 p p p = Algebr (2 1 + ) p 5 = 4 p 5 Solving Equtions of the Form x + b = cx + d. Method: 1) Add pproprite vlues to ech side to obtin mx = n. 2) Multiply ech side by 1 m (or, equivlently, divide ech side by m). ) The solution is then x = m n.
22 Exmple THE REAL NUMBERS 61 4x + 12 = x + 8 4x x = x + 8 +x 7x + 12 = 8 In trnsposing term from one side to the other, just chnge its sign. Exmple. 4 x 2 7x = x = 4 x = 4 7 = x = 2 5 x = 2 7 (trnsposing) x = x = x = x = x = 24 5
23 62 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Solving Inequlities of the Form x + b < cx + d. (< cn be replced by >, pple,, 6=) Wheres the solution set of the previous problem is inequlities re expressed ssimilr to {x x < 5}. Exmple. 4x + 5 7x 7 4x 7x x x 7 5 n 24 o, solution sets for 5 x 12 1 ( x) pple 1 ( 12) (note the direction chnge) x pple 4 {x x pple 4} Exmple. 2 x 2 < 5 6 x x x < x 5 6 x 2 < x 2 < 1 2 x < x < 7
24 9.2. THE REAL NUMBERS x = 2 7 x < 7 2 n x x < 7 o 2 Exmple. Chd ws the sme ge s Shelly, nd Holly ws 4 yers older thn both of them. Chd s dd ws 20 when Chd ws born, nd the verge ge of the four of them is 9. Wht re their ges? solution. Use strtegy 2 (Use of Vrible) nd Strtegy 14 (Solve n Eqution). Let x = Chd s ge = Shelly s ge x + 4 = Holly s ge x + 20 = Chd s dd s ge 1 (sum of the ges) = (x + x + x x + 20) = (4x + 24) = 9 4 h 1 i 4 4 (4x + 24) = 4 9 4x + 24 = 156 4x = x = (4x) = 1 4 (12) x = Chd nd Shelly re, Holly is 7, nd Chd s dd is 5.
25 64 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA 9.. Reltions nd Functions Reltions re the description of reltionships between 2 sets. Definition. A reltion from set A to set B is subset of A B. If A = B, we sy R is reltion on A. Exmple. Let A = {1, 2,, 4, 5, 6} nd the reltion be hs the sme number of fctors s. or R = {(1, 1), (2, 2), (2, ), (2, 5), (, 2), (, ), (, 5), (4, 4), (5, 2), (5, ), (5, 5), (6, 6)} For reltion R on set A: 1) R is reflexive if (, ) 2 R for ll 2 A, i.e., if every element is relted to itself. 2) R is symmetric if whenever (, b) 2 R, then (b, ) 2 R lso, i.e., if is relted to b, then b is relted to. ) R is trnsitive on A if whenever (, b) 2 R nd (b, c) 2 R, then (, c) 2 R, i.e., if is relted to b nd b is relted to c, then hs relted to c. R is n equivlence reltion on set A if it is reflexive, symmetric, nd trnsitive. Exmple. The reltion R is the previous exmple is n equivlence reltion.
26 9.. RELATIONS AND FUNCTIONS 65 An equivlence reltion cretes prtition of the set A s collection of nonempty, pirwise disjoint sets whose union is A. Such prtition cn lso be used to define n equivlence reltion by using is relted to b if they re in the sme subset. Exmple. The set A = {1, 2,, 4, 5, 6, 7, 8} with the reltion hs more fctors thn is trnsitive, but not reflexive or symmetric. The reltion is not equl to on the sme set is symmetic, but not reflexive or trnsitive. Definition. A function is reltion tht mtches ech element of first set to n element of second set in such wy tht no element in the first set is ssigned to two di erent elements in the second set, i.e., is reltion where no two ordered pirs hve the sme first element. A function f tht ssigns n element of set A to n element of set B is written f : A! B. If 2 A, the function nottion for the element in B ssigned to is f(), i.e., (, f()) is n ordered pir of the function (lso reltion) f. A is the domin of f nd B the codomin of f. The set {f() : 2 A} is the rnge of f. The rnge is subset of the codomin. Exmple. 1) None of our previous exmples of reltions were functions. 2) A sequence, list of numbers rrnged in order, clled terms, is function whose domin is the set of whole numbers. 1 is mtched with the first or initil term, 2 with the second term, etc.
27 66 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA, 5, 7, 9, 11,... is n rithmetic sequence with initil term nd common di erence 2, i.e., successive terms di er by 2. The form is, + d, + 2d, + d,.... 2, 6, 18, 54, 162,... is geometric sequence with initil term 2 nd common rtio, the number ech successive term is multiplied by. The form is, r, r 2, r,.... ) f : W! W defined by f(n) = n ssigns ech whole number (the domin) to its cube. Problem (Pge 408 # 8). ) is function for given election, but my not be in generl. b) not function since mny lrge cities, such s Memphis, hve serl zip codes. c) is function (if I understnd my biology). d) is not function since person my hve more thn one pet.
28 9.4. FUNCTIONS AND THEIR GRAPHS Functions nd Their Grphs Review the Crtesin Coordinte System. Note. 1) l nd m re usully x nd y for the x nd yxes. Any letters or nmes my be used. 2) The Romn numerls indicte qudrnts. The qudrnts do not include the xes. ) Ech point in the plne is represented by n ordered pir of numbers (x, y), the coordintes of the point. The xcoordinte of point is the perpendiculr distnce of the point from the yxis. The ycoordinte of point is the perpendiculr distnce of the point from the xxis. 4) The origin is the point (0, 0). 5) We sy we hve coordinte system.
29 68 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Grphs of Liner Functions Liner Functions re functions whose grphs re lines. Recll: A function is reltion ech element of first set, clled the domin, to n element of second set, clled the rnge,in such wy tht no element of the first set is ssigned to more thn one element in the second set. Exmple. The following tble displys the number of cricket chirps per minute t vrious tempertures: We grph the points (n, T ): cricket chirps per minute, n temperture, T ( F) We note the points (n, T ) fll on line. Thus we hve the grph of liner function. A liner function hs the form f(x) = x + b,
30 9.4. FUNCTIONS AND THEIR GRAPHS 69 where nd b re constnts. For our exmple, the formul would be T (n) = n + b. Cn you find nd b? (Hint: Find b first!) Look t the leftwrd extension of the grph to n = 0. It looks s though T = 40. If T (0) = 0 + b = b, then b = 40. So we hve T (n) = n To find, we try ny other point on the grph, such s (20, 45) or T (20) = 45. Then T (20) = = 45 {z } 20 = 5 Our formul is thus T (n) = 1 4 n = 1 4 if you check, this formul works for ll 5 of our points. We ll just look t (60, 55). T (60) = = = 55 F. 4 We cn lso use the formul to predict other vlues, such s the temperture when cricket chirps 90 times per minute. T (90) = = = 62.5 F. 4 Wht is the domin here, the possible vlues of n? n must be: 1) nonnegtive. 2) whole number. ) not too lrge (for biologists to determine).
31 70 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA Grphs of Qudrtic Functions Exmple. A bll is tossed up verticlly t velocity of 72 feet per second from point 10 feet bove the ground. It is known from physics tht the height of the bll bove the ground, in feet, is given by the position function p(t) = 16t t + 10, where t is the time in seconds. At wht time t is the bll t its highest point? solution. A qudrtic function is function of the form f(x) = x 2 + bx + c where, b, nd c re constnts nd 6= 0. Grphs of qudrtic functions re prbols. Our function p is qudrtic function. We form tble of vlues nd its grph. t (sec) p(t) (ft) We plot our points nd drw smooth curve through them. The bll is t its highest point between 2 nd seconds.
32 9.4. FUNCTIONS AND THEIR GRAPHS 71 We cn use grphing clcultor to find the highest point. We cn lso find tht f(4.5) = 10. Since we hve two tcoordintes with the sme height, 10, the time of the highest point is hlfwy between 0 nd 4.5, nmely t = Grphs of Exponentil Functions An exponentil function is of the form f(x) = b x where 6= 0 nd b > 0, but b 6= 1. Exmple. How long does it tke to double your money when the interest rte is 2% compunded nnully? Assume $ is deposited. solution. It is known tht for n initil principl P 0 nd n interest rte of 100r%, compounded nnully, nd time t in yers, the mount of principl fter t yers is given by the formul P (t) = P 0 (1 + r) t. For our exmple, the formul is
33 72 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA P (t) = 100(1 +.02) t = 100(1.02) t. The grph is below in red. To nswer the question sked, we drw horizontl line (blue) t P (t)=200. Where this line meets the grph, we drw verticl line to the txis. It looks s though this line meets the txis just to the right of 5. We find P (5) = nd P (6) = Keeping in mind tht interest is only pid t the end of time period, it will tke 6 yers to (t lest) double our money.
34 Step Functions 9.4. FUNCTIONS AND THEIR GRAPHS 7 The grph pictured below is grph of step function, since its vlues re pictured in series of line segments or steps. A postge function is nother exmple of step function. If the gretest integer function f(x) = [x] is defined to be the gretest integer less thn or equl to x, the formul for the bove grph is h 1 i F (x) = 2 x + 2. ) f( 1) = f(2) = f(.75) = 1
35 74 9. RATIONAL NUMBERS, REAL NUMBERS, AND ALGEBRA b) Wht re the domin nd rnge? c) f(x) = when f(x) = 1.5 when domin = {x 2 pple x < 6}, rnge = {1, 2,, 4} 2 pple x < 4 never occurs Exmple. Wter is poured t constnt rte into the three continers shown below. Which grph corresponds to which continer? The height would rise t constnt rte in (b), which mens grph with constnt slope. This is (i). In (), the wter rises quickly t first due to the nrrow bottom, but then slows down. This would be like grph (ii). Since (c) goes from wider to nrrower, the wter rises slowly t first, but then fster s time goes on. This is grph (iii).
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