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.AsI sum the digits, the only numbers that I hear are the partialsums.That is, when I sum the first (rightmost) column 8 40 7 7 5, I hear 8.12.19.26.31.Then I putdown the 1, carry the 3, and proceed as usual.The next columnwould then sound like 3.5.13.15.22.23.25.Once I have my final answer, I write it down, then check mycomputation by adding the numbers from bottom to top and, Ihope, arrive at the same answer.For instance, the first column would be summed, from bot-tom to top, as 5 7 7 0 4 8 (which in my mind soundslike 5.12.19.23.31).Then I mentally carry the 3,and add 3 2 1 7 2 8 2, and so on.By adding the Benj_0307338401_4p_c06_r1.r.qxd 5/4/06 1:42 PM Page 133Math for the Board: Pencil-and-Paper Math 133numbers in a different order, you are less likely to make thesame mistake twice.Of course, if the answers differ, then at leastone of the calculations must be wrong.MOD SUMSIf I m not sure about my answer, I sometimes check my solutionby a method I call mod sums (because it is based on the elegantmathematics of modular arithmetic).This method also goes bythe names of digital roots and casting out nines.I admit thismethod is not as practical, but it s easy to use.With the mod sums method, you sum the digits of each num-ber until you are left with a single digit.For example, to com-pute the mod sum of 4328, add 4 3 2 8 17.Then addthe digits of 17 to get 1 7 8.Hence the mod sum of 4328 is8.For the previous problem the mod sums of each number arecomputed as follows:4328 17 8884 20 2620 8 81477 19 10 1617 14 514725 58651 2920 1122 Benj_0307338401_4p_c06_r1.r.qxd 5/4/06 1:42 PM Page 134134 Secrets of Mental MathAs illustrated above, the next step is to add all the mod sumstogether (8 2 8 1 5 5).This yields 29, which sums to11, which in turn sums to 2.Note that the mod sum of 8651,your original total of the original digits, is also 2.This is not acoincidence! If you have computed the answer and the modsums correctly, your final mod sums must be the same.If theyare different, you have definitely made a mistake somewhere:there is about a 1 in 9 chance that the mod sums will matchaccidentally.If there is a mistake, then this method will detect it8 times out of 9.The mod sum method is more commonly known to mathe-maticians and accountants as casting out nines because the modsum of a number happens to be equal to the remainder obtainedwhen the number is divided by 9.In the case of the answerabove 8651 the mod sum was 2.If you divide 8651 by 9,the answer is 961 with a remainder of 2.In other words, if youcast out 9 from 8651 a total of 961 times, you ll have a remain-der of 2.There s one small exception to this.Recall that the sumof the digits of any multiple of 9 is also a multiple of 9.Thus, ifa number is a multiple of 9, it will have a mod sum of 9, eventhough it has a remainder of 0.SUBTRACTING ON PAPERYou can t, of course, subtract columns of numbers the same wayyou add them.Rather, you subtract them number by number,which means that all subtraction problems involve just twonumbers.Once again, with pencil and paper at our disposal, itis easier to subtract from right to left.To check your answer, justadd the answer to the second number.If you are correct, thenyou should get the top number.If you want, you can also use mod sums to check your Benj_0307338401_4p_c06_r1.r.qxd 5/4/06 1:42 PM Page 135Math for the Board: Pencil-and-Paper Math 135answer.The key is to subtract the mod sums you arrive at andthen compare that number to the mod sum of your answer:65,717 838,491 727,226 119 10There s one extra twist.If the difference in the mod sums is anegative number or 0, add 9 to it.For instance:42,689 218,764 823,925 6 9 3213PENCIL-AND-PAPER SQUARE ROOTSWith the advent of pocket calculators, the pencil-and-papermethod of calculating square roots has practically become a lostart.You ve already seen how to estimate square roots mentally.Now I ll show you how to do it exactly, using pencil and paper.Remember how in guesstimating square roots you calculatedthe square root of nineteen? Let s look at that problem again, thistime using a method that will give you the exact square root. Benj_0307338401_4p_c06_r1.r.qxd 5/4/06 1:42 PM Page 136136 Secrets of Mental Math4.3 5 800019.00024 168 [ Pobierz całość w formacie PDF ]
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.AsI sum the digits, the only numbers that I hear are the partialsums.That is, when I sum the first (rightmost) column 8 40 7 7 5, I hear 8.12.19.26.31.Then I putdown the 1, carry the 3, and proceed as usual.The next columnwould then sound like 3.5.13.15.22.23.25.Once I have my final answer, I write it down, then check mycomputation by adding the numbers from bottom to top and, Ihope, arrive at the same answer.For instance, the first column would be summed, from bot-tom to top, as 5 7 7 0 4 8 (which in my mind soundslike 5.12.19.23.31).Then I mentally carry the 3,and add 3 2 1 7 2 8 2, and so on.By adding the Benj_0307338401_4p_c06_r1.r.qxd 5/4/06 1:42 PM Page 133Math for the Board: Pencil-and-Paper Math 133numbers in a different order, you are less likely to make thesame mistake twice.Of course, if the answers differ, then at leastone of the calculations must be wrong.MOD SUMSIf I m not sure about my answer, I sometimes check my solutionby a method I call mod sums (because it is based on the elegantmathematics of modular arithmetic).This method also goes bythe names of digital roots and casting out nines.I admit thismethod is not as practical, but it s easy to use.With the mod sums method, you sum the digits of each num-ber until you are left with a single digit.For example, to com-pute the mod sum of 4328, add 4 3 2 8 17.Then addthe digits of 17 to get 1 7 8.Hence the mod sum of 4328 is8.For the previous problem the mod sums of each number arecomputed as follows:4328 17 8884 20 2620 8 81477 19 10 1617 14 514725 58651 2920 1122 Benj_0307338401_4p_c06_r1.r.qxd 5/4/06 1:42 PM Page 134134 Secrets of Mental MathAs illustrated above, the next step is to add all the mod sumstogether (8 2 8 1 5 5).This yields 29, which sums to11, which in turn sums to 2.Note that the mod sum of 8651,your original total of the original digits, is also 2.This is not acoincidence! If you have computed the answer and the modsums correctly, your final mod sums must be the same.If theyare different, you have definitely made a mistake somewhere:there is about a 1 in 9 chance that the mod sums will matchaccidentally.If there is a mistake, then this method will detect it8 times out of 9.The mod sum method is more commonly known to mathe-maticians and accountants as casting out nines because the modsum of a number happens to be equal to the remainder obtainedwhen the number is divided by 9.In the case of the answerabove 8651 the mod sum was 2.If you divide 8651 by 9,the answer is 961 with a remainder of 2.In other words, if youcast out 9 from 8651 a total of 961 times, you ll have a remain-der of 2.There s one small exception to this.Recall that the sumof the digits of any multiple of 9 is also a multiple of 9.Thus, ifa number is a multiple of 9, it will have a mod sum of 9, eventhough it has a remainder of 0.SUBTRACTING ON PAPERYou can t, of course, subtract columns of numbers the same wayyou add them.Rather, you subtract them number by number,which means that all subtraction problems involve just twonumbers.Once again, with pencil and paper at our disposal, itis easier to subtract from right to left.To check your answer, justadd the answer to the second number.If you are correct, thenyou should get the top number.If you want, you can also use mod sums to check your Benj_0307338401_4p_c06_r1.r.qxd 5/4/06 1:42 PM Page 135Math for the Board: Pencil-and-Paper Math 135answer.The key is to subtract the mod sums you arrive at andthen compare that number to the mod sum of your answer:65,717 838,491 727,226 119 10There s one extra twist.If the difference in the mod sums is anegative number or 0, add 9 to it.For instance:42,689 218,764 823,925 6 9 3213PENCIL-AND-PAPER SQUARE ROOTSWith the advent of pocket calculators, the pencil-and-papermethod of calculating square roots has practically become a lostart.You ve already seen how to estimate square roots mentally.Now I ll show you how to do it exactly, using pencil and paper.Remember how in guesstimating square roots you calculatedthe square root of nineteen? Let s look at that problem again, thistime using a method that will give you the exact square root. Benj_0307338401_4p_c06_r1.r.qxd 5/4/06 1:42 PM Page 136136 Secrets of Mental Math4.3 5 800019.00024 168 [ Pobierz całość w formacie PDF ]