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## Training Your Mind With Mental Arithmetic

Many years ago, I was fascinated by a performance on TV by Arthur Benjamin, a man who describes his work as “mathematical”. To the delight of his audience and appropriate showmanship, he multiplies numbers of increasing size in his head, calculates the day of the week of any calendar date, and, for the grand finale, squares the 5-digit number. Calculations that result are too large for most pocket calculators. There is certainly no sleight of hand in Professor Benjamin’s performance; He’s more than happy to give some insight and hints about how his performance actually goes, and jokingly comments that he’s quite comfortable with it, as he doesn’t expect to see anyone else perform his show in the near future.

While it may not be your goal to be able to perform a task like Professor Benjamin’s, having some idea of how calculations and memorization tasks are performed are useful tools for improving and exercising your brain. You may not find yourself doing much mental multiplication in your day-to-day life, but some of the tips and tricks Professor Benjamin uses can help with those everyday memorizing tasks like keeping your pin number handy or remembering a telephone number. A friend who doesn’t have your address book handy – and, with a little practice, you can entertain at family reunions!

**Multiplication Tables**

Unfortunately, there is a little bad news. It’s true that you can’t even consider multiplying a number with more than one digit until you’ve mastered those pesky multiplication tables that some of us struggled with for ages in school. Force-feeding tables as facts, learning by rote, is one of the factors that makes mathematics an unpopular subject for many people. People who claim that they are unable to “get” math may be able to attribute their tables to a poor experience in learning. The secret to learning the tables, however, is to realize that there really aren’t that many different items 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 memorize, but in truth, there are very few. For starters, many facts appear twice; If you know what 6 times 7 is, you already know what 7 times 6 is. Multiplying by 0 and 1 has enough facts; Zero times zero; Anytime 1 is unchanged. Multiplying by 2 and 5 is easy to learn next; What remains after that is less than two dozen multiplication facts, and a simple way to remember them *Practice*. You can use extended memory tricks later in this article to remember these facts as well; But more about that later.

**Cross multiplication**

If single-digit numbers are within your capabilities, then multiplying two two-digit numbers really isn’t too far. In school, you may have been taught how to do this *Long multiplication*, which actually, indirectly, involves multiplying every possible pair of digits in question. With a bit of smarts, you can layout long multiplication sums in your head and quickly see the answer. The trick is to visualize all single-digit multiplications as correspondingly placed as two-digit answers. This is best illustrated by 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 next to each other, giving 3506. Now consider all the other options. Marks from questions; 7 times 2 (14) and 3 times 5 (15), and add those products to the middle number you already have. (There may be a carry on the left-most 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 brain when trying to perform such additions. We may not actually be able to remember all those intermediate calculations; In fact, we may even forget the question! On the Arthur Benjamin Show, it’s perhaps surprising to quickly switch to multiplying a number. *by himself *(*Classification*), because, well, there are fewer intermediate results to remember. There are half the numbers to memorize in the question, and the same goes for counting details. The same multiplication logic applies, though; For example, we consider 73 times 73 by first multiplying the digits in the first place, giving 4909, and then adding 7 times 3, which now appears twice, to the middle digit, giving 5329.

**hard stuff**

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 of the commonly used tricks involves modifying the calculation so that difficult multiplications are replaced by easier ones. For example, suppose you want to multiply 993 by 993. It’s a shame we don’t multiply by 1000, it would be easy. So why not add 7 to one of those 993 entries, and to be fair, we should subtract 7 from the other. 986 times 1000 is a very easy problem, and the answer is almost correct. With a little work, you can see how to write the correct answer without too much trouble.

However as the sums get larger, the more results we can remember, the smoother things will go. For example, we have already mentioned that sometimes we are asked to remember partial calculations and carry them to the end of the sum, or we may need to keep this question in our mind so that we do not forget it. Likewise, when it came to classifying two-digit numbers, only ninety of those answers were actually remembered. That sounds like a lot, but remember, there were only a hundred single-digit multiplication facts before. If we can find a smarter way to think about them and store them in our brains, we’ll save time and brain power later!

**Memorizing numbers**

The trick is to convert numbers (which we’re almost certain to find difficult to remember, being just strings of digits) into words (which are much easier to remember and perhaps prompt our brains to create pictures). For example we have a great ability to remember the words of a poem, or a song. There are systems that can do this. One of the simplest is to remember numbers by counting the letters in a word. (A ten-letter word can represent zero). For example, the phrase “Shall I have a drink, alcohol of course, after a heady chapter involving quantum mechanics?” Probably something you can eventually remember without much effort. Going back to numbers, you notice 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9 – that’s the first 15 digits of a mathematical constant. *pi*. With something like a credit card number within your reach, just form an appropriate phrase, and the mere act of thinking the phrase in the first place helps commit it to memory.

A more compact way to remember numbers is to replace numbers with letters. In the popular phonetic mnemonic system, numerals are represented by consonants, in fact, by consonant sounds. There are only ten different groups to memorize, and they are given convenient visual cues, for example, the sound of T (or the equivalent TH and D) represents 1, since the letter T has a downstroke. Given a number to remember, choose the sounds that match the numbers, and pad with the vowels to make words. This seems like a long and tedious way to remember numbers, but it works, especially if the word or phrase you come up with is completely ridiculous. Do you remember that the answer to 5329 came some time ago? Maybe it wasn’t a number you found particularly memorable. Using the phonetic method, conversion to consonants gives L, M, N, P. There are definitely some mental pictures you can think of to remember those letters. What about, for example, a little **LaMb** taking a **NaP**. It sounds outrageous, but it’s very simple to picture, and will stick in your mind, and when necessary, unwinding the picture back to the phonetics and then becoming a completely seamless process with some practice on the digits.

**what now**

You might want to check out Professor Benjamin’s demonstration of the work, and see if you can get some idea of where some of these techniques can be used. Listen to him using the phrase “cookie fission” to remember the number during his grade final calculation, especially for art. Anyway, I hope you enjoy the show, especially the audience’s apparent increase in wonder at his abilities as the show goes on, and, at the very least, the next time you feel the need to memorize a number, maybe you can. Try the phonetic mnemonic method. I believe, now, you can still remember the phrase to remember the answer to that square problem in this article!

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