Chapter 12 - How To Use Short Cuts

In this chapter we shift gears entirely, and learn how to build on our simplified arithmetic a different, but very useful, system of conversion.

This second section does not ignore the first. Indeed, the swift and confident working of these new principles depends very much on a smooth and automatic handling of your basic number combinations—the practical, fast, fully integrated system of speed arithmetic based on the phenomenal simplicity of the modern Japanese abacus.

Before going on to the short cuts, let your mind range back over what you have learned.

In complement addition, you have learned to add from left to right. You have discarded the twenty hardest digit combinations by using a complement instead of adding over ten. You have learned to record and forget tens as you go along, picking them up at the end of each column.

In complement subtraction, you have learned never to subtract a larger digit from a smaller, but instead to use a complement. You have learned to avoid the cumbersome, confusing system of "borrowing," by accomplishing the same thing in the answer itself. This lets you work from left to right.

In no-carry multiplication, you have learned to work from left to right too. You also now have the technique for producing the answer swiftly and easily without carrying—without juggling a number of digits in your mind at the same time or stopping to make notes of them as you work.

In all three operations, your answer takes form naturally in your head or on the paper—just as it takes form naturally on the abacus.

In shorthand division, you have learned to combine the techniques of no-carry multiplication and complement subtraction with the European "shorthand" method of long division, shorten it still further in several respects, and get your answer with half the pencil work needed in traditional long division. You have also learned a simple but effective secret for producing your next answer digit in a flash by dividing with only the first digit of the divider, raised by one, and adjusting it later if you need to. This, also, has been adapted from abacus theory.
This entire system is a remarkably fast and easy approach to numbers, once you fully master it. But it also lends itself so beautifully to speeding up still further the better standard short cuts that you can compound your new handling of figures by carrying the "abacus" system into short cuts.

The approaches of the two methods are very different.

Our simplified arithmetic is an integrated system. Standard short cuts are not. They consist of a variety of tricks, mainly in converting problems to simpler forms, that apply to a wide variety of problems but not to all of them. Further, they have never before been assembled into anything resembling a unified whole.

I hope to show before this book is finished that the more useful of the standard short cuts can be learned in relation to each other, so that you can reach for the most effective in any particular case. More than on any other single foundation, this "integrated" approach to the short cuts will rest on the firm base of your number sense.

Perhaps this is the appropriate time to explode some of the fables about number short cuts. Valuable short cuts there are, but not one of them will enable you to multiply 38,657 by 49,956 in ten seconds in your head, without effort, and with perfect accuracy. History has been made by a number of mathematics geniuses, but a thorough sifting of the evidence exposes the single secret possessed by every one of them. It is a secret, alas, beyond most of us.

Jedidiah Buxton, the illiterate son of an English schoolteacher, was able to calculate entirely in his head the problem: "Multiply two times two 140 times, then assume that this answer is in quarter-pennies and reduce it to pounds, shillings, and pence." His answer was in 39 digits. Oddly enough, however, Buxton did not know very much about arithmetic. His methods were almost unbelievably crude. Instead of multiplying by 300, for instance, he would multiply by 5, then 20 (in effect multiplying by 100, which he could have done by adding two zeros), then multiply the result by 3. Buxton's one secret was the basic secret of all the "mental calculators" of history—a staggeringly complete memory for figures. This enabled him to handle immense numbers in his head, even doing calculations that took days or weeks, remembering every digit as if returning to a mental blackboard when he resumed work on a problem.

Thomas Fuller was a slave who showed no signs of his unusual gift until he was 70, a little after the American Revolution. Then he gave demonstrations on the order of finding the number of seconds in 70 years, 17 days, in just a minute and a half. Zerah Colburn, the son of a Vermont farmer, started at eight. His father took him on an exhibition tour, and he brought a skeptical academic audience in England to the verge of tears by giving the 16th power of 8 (8 used as a multiplier 16 times) faster than the answer could be written; the answer, incidentally, is 281,474,976,710,656.

The point to these stories is simply this: the secret possessed by every mental calculator is nothing more or less than a prodigious memory for numbers. Some of them became real mathematicians; others never learned to apply their mental oddity to anything more serious than number stunts to impress paid audiences. In some cases, their actual understanding of numbers, as such, was ludicrous.

For most of us who have trouble remembering what to "carry" or whether or not we "borrowed" because we had to deal with another number in the meantime, the most fruitful approach is to strip down our methods to the simplest, fastest, and easiest techniques.

In a number of cases, however, there is a way to "see through" a seemingly complex problem—see through the ap-parantly unrelated figures and reduce them to a simple, sensible relationship that we can almost recognize at a glance. It cannot produce an exact answer to every problem; instead, it picks and chooses, from the many problems we must solve, the one-half to three-quarters, roughly, that can be solved in less time if we convert them to other forms first.

The classic short cuts are really nothing more or less than methods of conversion—conversion from one form into another form entirely, where the relationships can more easily be seen or solved.

For instance, if faced with the problem 15 / 45, you might or might not recognize the answer at once as 3. If not, and if you were trained and alert to conversion possibilities, you would note that 15 doubled is 30—a far simpler divider. Being knowledgeable about short-cut methods, you would double the 15 and the 45 as well (to keep the relationship identical) and see the problem as 30 / 90. This can be nothing but 3.

Most short cuts are basically as simple as that, which is an example of "proportionate change."

Another example we have mentioned before is the shortcut method of figuring a 15% tip. The 15, you will note, is the same as the 15 in the last paragraph. But in this case most experts use another short cut. Start, for example, with a meal check for $4.00. 15 is exactly 1/10 of a hundred plus ½ of that tenth. The 15% tip on $4.00 then is 40¢ (1/10) plus 20¢ (1/12 the tenth), or 60¢.

This method is called "breakdown."
Could you use the method we first mentioned, proportionate change, to find the tip? Yes, though multiplication works in a different fashion from division. In division you double both numbers to keep the relationship the same. In multiplication, to keep the relationship the same you must cut one of them in half if you double the other. ½ x 2 is 1—so the problem will have the same answer.
 
To use proportionate change on this tip, you would double the 15 to 30, and cut the $4.00 in half to get $2.00. 30% of $2.00 is (3 x 2) 60¢ again.

Which is the better short cut? Neither. Each one fits certain combinations of numbers better than the other does, and it is helpful to know both so you can select the easier of the two for any one case.

This is the central fact about short cuts. There are literally hundreds of short cuts, from the quick method of squaring a number ending in 5 (when did you last have to square a number ending in 5?) to multiplying by 11 in one operation (a little more useful, but still pretty specialized). Of the many available, only four types of short cut are really applicable to enough problems to be worth the trouble of learning, unless for the reward of knowing a number oddity to impress people.

These four short cuts have certain similarities and certain differences. Learn them well, learn how to recognize which is most valuable in any one case, and you can add to your already advanced handling of basic numbers the extra advantage of frequent "overleaps" in lightning calculation.

Dig out your pad again, or better yet get a fresh one, and prepare to enter another fascinating aspect of numbers.

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