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.r.qxd 5/4/06 1:45 PM Page 5252 Secrets of Mental MathSimilarly,2 2(z d) z(z 2d) d2To find 77 when z 80 and d 3,2 2 277 (80 3) 80 (80 6) 3 80 74 9 5929Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 53Chapter 3New and Improved Products:Intermediate MultiplicationMathemagics really gets exciting when you perform in front ofan audience.I experienced my first public performance in eighthgrade, at the fairly advanced age of thirteen.Many mathemagi-cians begin even earlier.Zerah Colburn (1804 1839), for exam-ple, reportedly could do lightning calculations before he couldread or write, and he was entertaining audiences by the age ofsix! When I was thirteen, my algebra teacher did a problem on2the board for which the answer was 108.Not content to stopthere, I blurted out,  108 squared is simply 11,664!The teacher did the calculation on the board and arrived atthe same answer.Looking a bit startled, she said,  Yes, that sright.How did you do it? So I told her,  I went down 8 to 100and up 8 to 116.I then multiplied 116 100, which is 11,600,and just added the square of 8, to get 11,664.She had never seen that method before.I was thrilled.Thoughts of  Benjamin s Theorem popped into my head.I actu-ally believed I had discovered something new.When I finallyran across this method a few years later in a book by MartinBenj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 5454 Secrets of Mental MathGardner on recreational math, Mathematical Carnival (1965),it ruined my day! Still, the fact that I had discovered it formyself was very exciting to me.You, too, can impress your friends (or teachers) with somefairly amazing mental multiplication.At the end of the lastchapter you learned how to multiply a two-digit number byitself.In this chapter you will learn how to multiply two differ-ent two-digit numbers, a challenging yet more creative task.Youwill then try your hand or, more accurately, your brain atthree-digit squares.You do not have to know how to do 2-by-2multiplication problems to tackle three-digit squares, so you canlearn either skill first.2-BY-2 MULTIPLICATION PROBLEMSWhen squaring two-digit numbers, the method is always thesame.When multiplying two-digit numbers, however, you canuse lots of different methods to arrive at the same answer.Forme, this is where the fun begins.The first method you will learn is the  addition method,which can be used to solve all 2-by-2 multiplication problems.The Addition MethodTo use the addition method to multiply any two two-digit num-bers, all you need to do is perform two 2-by-1 multiplicationproblems and add the results together.For example:4642 (40 2)40 46 18402 46 921932Benj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 55New and Improved Products: Intermediate Multiplication 55Here you break up 42 into 40 and 2, two numbers that areeasy to multiply.Then you multiply 40 46, which is just 446 with a 0 attached, or 1840.Then you multiply 2 46 92.Finally, you add 1840 92 1932, as diagrammed above.Here s another way to do the same problem:46 (40 6)4240 42 16806 42 2521932The catch here is that multiplying 6 42 is harder to do thanmultiplying 2 46, as in the first problem.Moreover, adding1680 252 is more difficult than adding 1840 92.So howdo you decide which number to break up? I try to choose thenumber that will produce the easier addition problem.In mostcases but not all you will want to break up the number withthe smaller last digit because it usually produces a smaller sec-ond number for you to add.Now try your hand at the following problems:48 81 (80 1)73 (70 3) 5970 48 3360 80 59 47203 48 144 1 59 593504 4779The last problem illustrates why numbers that end in 1 areespecially attractive to break up.If both numbers end in theBenj_0307338401_4p_c03_r1.r.qxd 5/4/06 1:44 PM Page 5656 Secrets of Mental Mathsame digit, you should break up the larger number as illustratedbelow:84 (80 4)3480 34 27204 34 1362856If one number is much larger than the other, it often pays tobreak up the larger number, even if it has a larger last digit.Youwill see what I mean when you try the following problem twodifferent ways:74 (70 4) 7413 13 (10 3)70 13 910 10 74 7404 13 52 3 74 222962 962Did you find the first method to be faster than the second? I did.Here s another exception to the rule of breaking up the num-ber with the smaller last digit [ Pobierz całość w formacie PDF ]