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Training Your Mind With Mental Arithmetic
Several years ago, I was captivated by a performance on television by Arthur Benjamin, a man who describes his act as “mathematical.” To the delight of his audience and with proper spectacle, he multiplies numbers of ever-increasing size in his head, calculates the day of the week for a given calendar date, and for a grand finale, squares a number of 5 digits, a calculation whose result is too large for most pocket calculators. There is certainly no sleight of hand involved in Professor Benjamin’s performance; in fact, he’s more than happy to give insight and hints as to exactly how his act is done, jokingly commenting that he’s very comfortable with it, as he doesn’t expect to see anyone else performing his show in the immediate future.
While being able to deliver a performance like Professor Benjamin’s may not be your goal, having an idea of how calculations and feats of memorization are done are useful tools to improve and exercise your mind. You may not necessarily find yourself doing a lot of mental multiplication in your day-to-day life, but some of the tips and tricks Professor Benjamin uses can be helpful for those daily memorization tasks, such as keeping your PIN number handy or remember a phone number from a friend without having your address book handy, and with a little practice, you might be able to entertain yourself at a family gathering!
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. Feeding the tables as facts, which are learned by rote, is for many people one of the factors that makes mathematics an unpopular subject. People who claim they can’t “get” math can put it down to bad experience when first learning their tables. The secret to learning the tables, however, is to realize that there really aren’t that 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 the truth is, there are far fewer. To begin with, many of the facts appear twice; if you know what 6 times 7 is, you already know what 7 times 6 is. Multiplying by 0 and 1 are simple enough facts; anything for zero is zero; nothing for 1 changes. Multiply by 2 and 5 are the next easiest to learn; what’s left after that is less than a couple of dozen multiplication facts, and the easiest way to remember them 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 capabilities, multiplying two two-digit numbers is actually not that far off. In school, you may have been taught how to do this long multiplication, which actually, indirectly, involves multiplying all possible pairs of digits in the question. With a little intelligence, you can work out 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 are multiplying 73 by 52. First consider 7 times 5 (35) and 3 times 2 (06) as two-digit numbers, and place them side by side, giving 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 in the leftmost digit). In this case, 3796 is actually the answer.
With a little practice, you can multiply two-digit numbers quite easily, but something often goes wrong in our heads when we try to perform these additions. In fact, we may not be able to remember all those intermediate calculations; in fact, we can even forget the question! Perhaps unsurprisingly, in Arthur Benjamin’s show, he soon turns to multiplying a number by itself (square), because, well, there are fewer intermediate results to remember. The question has half numbers to remember, and the same goes for the details of the calculation. The same logic of multiplication applies, however; for example, we first consider 73 times 73 by multiplying the digits in place, giving 4909, and then 7 times 3, which now appears twice, is added to the middle digits, giving 5329.
There are some more sophisticated techniques used to square three- and four-digit numbers that the interested reader may want to investigate. As a hint, one commonly used trick is to modify the calculation so that difficult multiplications are replaced by easier ones. For example, let’s say you want to multiply 993 by 993. It’s a shame we weren’t multiplying 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 can 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 go. For example, we have already mentioned that sometimes we are asked to remember partial calculations and carry them through to the end of the addition, or we simply need to store the question in our minds so that we do not forget it. Similarly, when it comes to squaring two-digit numbers, there are actually only ninety of these answers to remember. That sounds like a lot, but remember that there used to be only a hundred single-digit multiplication facts. 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!
Memorization of numbers
The trick is to convert numbers (which will almost certainly be difficult for us to remember, being just a string of digits) into words (which are much easier to remember and perhaps inspire our minds to create images). We have a much greater ability to remember poetry or song lyrics, for example. There are systems to do this. One of the simplest is to remember the digits by counting the letters in 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 of quantum mechanics?” it’s perhaps something you could eventually memorize without too much effort. Converting back to digits, you remembered 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, which are the first 15 digits of the mathematical constant Pine tree. Something like a credit card number is within your reach, you just have to come up with a suitable phrase and simply thinking of the phrase in the first place will help you remember it.
An even more compact way to remember numbers is to replace the digits with letters. In the popular phonetic mnemonic system, digits are represented by consonants, in fact, by the sound of the consonants. There are only ten different groups to remember and they are given convenient visual cues, for example the sound of T (or equivalently TH and D) represents 1, as the letter T is underlined. Given the number to remember, choose the sounds that correspond to the digits and fill them in with vowels to form words. It seems like a long and winding road to remembering a number, but it works, especially if the word or phrase you come across is completely ridiculous. Remember answer 5329 a while back? Maybe it wasn’t the kind of number you found particularly memorable. Using the phonetic method, a conversion to consonants gives L, M, N, P. There are probably some mental images you could think of to remember these letters. What about, for example, a little LaMb taking a NaP. It sounds outrageous, but this is much easier to imagine and will stick in your mind, and when necessary, switching back to the image to phonetics and then to digits can become a completely seamless process with a little practice.
You might want to check out Professor Benjamin’s performance of his act and see if you can get an exact idea of where some of these techniques can be used. Listen in particular to Art using the phrase “cracker fission” to remember a number during the calculation of his final grade. In any case, I hope you enjoy the show, especially the audience’s obvious increase in amazement at his skill as the show progresses, and at the very least, the next time you find yourself needing to memorize a number, perhaps you could try the phonetic mnemonic method. I think, right now, you can still remember the phrase to remember the answer to this previous squares problem in this article!
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