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Training Your Mind With Mental Arithmetic
Several years ago, I was captivated by a television performance by Arthur Benjamin, a man who calls his act “mathemagic.” To the delight of his audience and with appropriate staging, he multiplies larger and larger numbers in his head, calculates the day of the week of a given calendar date and, for a grand finale, raises to square a 5-digit number, a calculation whose result is too large for most pocket calculators. There’s certainly no sleight of hand involved in Professor Benjamin’s performance; he’s actually quite happy to give insight and hints as to exactly how his performance is done, and jokingly remarks that he’s quite comfortable with it, as he doesn’t expect to see anyone another to play his show in the immediate future.
While being able to deliver a performance like Professor Benjamin’s might not be your goal, getting a sense of how calculations and memorization feats are performed are useful tools for improving and exercising your mind. You might not necessarily find yourself doing a lot of mental multiplication in your daily life, but some of the tips and tricks Professor Benjamin uses can be useful for everyday memorization tasks such as keeping your PIN handy. or call back a phone number. a friend without having your address book handy – and, with a little practice, maybe you can entertain yourself at a family reunion!
Unfortunately, there is a bit of bad news. It’s true that you can’t even consider multiplying numbers with more than one digit until you’ve mastered those pesky multiplication tables that some of us struggled with for so long in school. Being force-fed with tables as facts, to be memorized, is for many people one of the factors that makes math an unpopular subject. People who claim they are unable to “understand” math may attribute this to the bad experience when learning their tables. The secret to learning the tables, however, is to realize that there are actually not many different elements to remember. Think about it for a moment – you need to practice multiplying two single-digit numbers, from 0 to 9. In theory, there are 100 multiplication facts to remember, but in truth, there are far fewer. To begin with, many facts appear twice; if you know what 6 times 7 is, then you already know what 7 times 6 is. Multiplying by 0 and 1 are pretty simple facts; anything multiplied by zero equals zero; any times 1 is unchanged. Multiplying by 2 and 5 are the next easiest to learn; what’s left after that is less than a few dozen multiplication facts, and the easiest way to remember that is practice. You may be able to use the memory tricks detailed later in this article to remember these facts as well; but more on that later.
If single-digit numbers are within your abilities, then multiplying two double-digit numbers is actually not that far off. At school, you may have been taught to do long multiplication, which is actually, indirectly, multiplying all possible pairs of digits in the question. With a little cleverness, you can format the long multiplication sum in your head and quickly see the answer. The trick is to visualize all single-digit multiplications as two-digit answers arranged accordingly. It is best illustrated with an example. For example, suppose we multiply 73 by 52. First consider 7 times 5 (35) and 3 times 2 (06) as two-digit numbers, and place them next to each other, which gives 3506. Now think about all the other digit selections in the question; 7 times 2 (14) and 3 times 5 (15), and add these products to the middle digits of what you already have. (There may be a carry over to the leftmost digit). In this case, 3796 is indeed the answer.
With a little practice, you can easily multiply two-digit numbers, but often something happens in our minds when we try to perform such sums. We cannot in fact remember all these intermediate calculations; in fact, we can even forget the question! Perhaps unsurprisingly then, on Arthur Benjamin’s show, he soon moves on to multiplying a number by himself (squared), because, well, there are fewer intermediate results to remember. The question has half the numbers to remember, and so do the details of the calculation. The same multiplication logic applies though; for example, we first consider 73 times 73 by multiplying the digits in place resulting in 4909, then the 7 times 3, which now appears twice, is added to the middle digits resulting in 5329.
There are more sophisticated techniques used to square three and four digit numbers that the interested reader may wish to research. As a guideline, one of the commonly used tricks is to modify the calculation to replace difficult multiplications with easier ones. For example, suppose you want to multiply 993 by 993. Too bad we don’t multiply by 1000, that would be easy. So why not add 7 to one of those 993 entries, and to be fair, maybe we should subtract 7 from the other. 986 times 1000 is a much easier problem, and the answer is almost correct. With a little work, you might see a method to write the correct answer without too much trouble.
However, as the sums increase, the more results we remember, the smoother things will run. For example, we have already mentioned that sometimes we may be asked to remember partial calculations and continue them until the end of the sum, or we may simply have to store the question in our mind so that we do not forget it. Likewise, when it comes to squaring two-digit numbers, there are really only ninety of these answers to remember. That sounds like a lot, but remember there were only a hundred single-digit multiplications before. If we can find a smarter way to think about them and store them in our minds, we’ll save time and brain power later!
The trick is to convert numbers (which we’re almost certain to find hard to remember, being just a string of numbers) into words (which are much easier to remember and perhaps inspire our minds to create pictures) . We have a much greater ability to remember poetry or song lyrics, for example. There are several systems for doing this. One of the easiest is to memorize numbers by counting the letters of a word. (A ten-letter word could represent zero). For example, the sentence “Can I have a drink, alcoholic of course, after the heady chapters on quantum mechanics?” is perhaps something you could possibly memorize without much effort. Converting back to digits, you remember 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9 – that’s the first 15 digits of the mathematical constant ft. Something like a credit card number is within reach, just make up an appropriate phrase, and just thinking of the phrase in the first place will help memorize it.
An even more compact way to remember numbers is to replace numbers with letters. In the popular phonetic mnemonic system, numbers are represented by consonants, in fact, by the sound of consonants. There are only ten different groups to remember, and they are given handy visual cues, for example, the sound of T (or equivalently TH and D) represents 1, since the letter T has a downstroke. Given the number to remember, choose the sounds that correspond to the numbers and complete them with vowels to form words. Remembering a number seems like a long and winding road, but it works, especially if the word or phrase you come up with is completely ridiculous. Remember 5329 replied a short time ago? Maybe it wasn’t the kind of number you found particularly memorable. Using the phonetic method, a conversion to consonants results in L, M, N, P. Surely there are mental images you could think of to remember these letters. How about, for example, a little Lamb take a Nap. It sounds outrageous, but it’s a lot easier to imagine, and it will stick in your mind, and if necessary, unrolling the image into phonetics and then numbers can become a perfectly smooth process with a little practice.
And then ?
You might want to check out Professor Benjamin’s performance of his act, and see if you can get an idea of exactly where some of these techniques could be used. Listen in particular to Art who uses the phrase “cookie fission” to remember a number when calculating his final grade. Either way, I hope you enjoy the show, especially the obvious increase in audience surprise at his ability as the show progresses, and, at the very least, the next time you need to memorize a number, maybe you could try the phonetic mnemonic method. I believe that right now you can still remember the phrase to remember the answer to that quadrature problem earlier in this article!
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