Several years ago, I was captivated by a performance on TV by Arthur Benjamin, a man who describes his act as "mathemagic".
To the delight of his audience and with suitable showmanship, he multiplies numbers of ever-increasing size in his head, calculates the day of the week of any given calendar date, and, for a grand finale, squares a 5-digit number, a calculation whose result is too large for most pocket calculators.
There is certainly no sleight of hand involved in Professor Benjamin's performance; he is quite happy in fact to give some insight and clues into exactly how his performance is done, and jokingly remarks that he is quite comfortable with that, as he does not expect to see anyone else performing his show in the immediate future.
While being able to deliver a performance like Professor Benjamin's might not be your goal, having some idea how the calculations and the 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 daily life, but some of the tips and tricks that Professor Benjamin uses can be of help for those daily memorization tasks such as keeping your PIN number to hand or recalling a telephone number of a friend without having your address book handy - and, with a bit of practice, perhaps you can entertain at a family reunion! Multiplication tables Unfortunately, there is a little bad news.
It is true that you cannot even consider multiplication of numbers with more than one digit until you have mastered those pesky multiplication tables that some of us struggled over for so long at school.
Being force-fed the tables as facts, to be learned by rote, is for many people one of the factors that makes mathematics an unpopular subject.
People who claim that they are unable to "get" math may be able to attribute that to the bad experience when first learning their tables.
The secret to learning the tables, however, is to realize there are not in fact all that many different items to remember.
Think about it for a moment - you need to practice multiplying two single digit numbers, 0 through 9.
In theory there are 100 multiplication facts to remember, but truthfully, there are considerably less.
For a start, many of the facts appear twice; if you know what 6 times 7 is, then you already know what is 7 times 6.
Multiplying by 0 and 1 are simple enough facts; anything times zero is zero; anything times 1 is unchanged.
Multiplying by 2 and 5 are the next easiest to learn; what remains after that is less than a couple of dozen multiplication facts, and the simplest way to remember those is practice.
It is possible you could use the memory tricks detailed 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 is actually not all that far away.
At school, you may have been taught how to do long multiplication, which actually, indirectly, involves multiplying every possible pair of digits in the question.
With a little bit of smarts, you can layout the long multiplication sum in your head and quickly see the answer.
The trick is to visualize all of the single-digit multiplications as two-digit answers laid out accordingly.
It is best illustrated by an example.
Suppose for instance 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 think of all the other selections of digits from the question; 7 times 2 (14) and 3 times 5 (15), and add those products to the middle digits of what you have already.
(There may be a carry to the left-most digit).
In this case, 3796 is indeed the answer.
With a little practice, you can multiply two-digit numbers quite readily, but often, something happens to our minds when trying to perform such sums.
We may not in fact be able to remember all those intermediate calculations; in fact, we may even forget the question! It is perhaps unsurprising then, in the Arthur Benjamin show, he soon switches to multiplying a number by itself (squaring), because, well, there are fewer intermediate results to remember.
The question has half as many numbers to remember, and the same goes for the calculation details.
The same multiplication logic applies, though; for example, we consider 73 times 73 first by multiplying the digits in place, giving 4909, and then the 7 times 3, which now appears twice, gets added to the middle digits, giving 5329.
Tougher stuff There are some more sophisticated techniques used to square three- and four-digit numbers which the interested reader may wish to research.
As a hint, one of the tricks commonly used involves modifying the calculation so difficult multiplications are replaced by easier ones.
For example, suppose you wished to multiply 993 by 993.
It is a shame that we were not multiplying by 1000, that would be easy.
So why not add 7 to one of those 993 entries, and to be fair, perhaps 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 may see a method for writing down the correct answer without too much trouble.
As the sums get bigger though, the more results we can remember, the smoother things will go.
For instance, we have already mentioned that at times we may be asked to remember partial calculations and carry them through to the end of the sum, or we may simply have to store the question away in our minds so we do not forget it.
Likewise, when it came to squaring the two-digit numbers, in actual fact there are only ninety of those answers to remember.
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 of those and store them in our mind, we will save time and brain power later! Memorizing numbers The trick is to convert numbers (which we are almost certain to find difficult tor remember, being just a string of digits) into words (which are much easier to recall and perhaps inspire our minds to create pictures).
We havce a far greater ability to recall poetry, or song lyrics, for example.
There are may systems of doing this.
One of the simplest is to remember digits by the count of letters in a word.
(A ten-letter word could represent zero).
For example, the phrase "May I have a drink, alcoholic of course, after the heady chapters involving quantum mechanics?" is perhaps something that you could eventually memorize without too much effort.
Converting back to digits, you have remembered 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9 - that is the first 15 digits of the mathematical constant pi.
Something like a credit card number is well within your reach, simply devise a suitable phrase, and the mere act of thinking up the phrase in the first place will help commit it to memory.
An even more compact way to remember numbers is to replace digits with letters.
In the popular phonetic mnemonic system, digits are represented by consonants, in fact, by the sound of 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, since the letter T has one downstroke.
Given the number to remember, pick out the sounds that correspond to the digits, and pad them out with vowels to make words.
It seems a long and torturous path to remember a number, but it works, particularly if the word or phrase you come up with is thoroughly ridiculous.
Remember that 5329 answer a little while back? Perhaps it was not the sort of number you found particularly memorable.
Using the phonetic method, a conversion to consonants gives L, M, N, P.
There are surely some mental pictures you could think of to remember those letters.
What about, for instance, a little LaMb taking a NaP.
It sounds outrageous, but that is far simpler to picture, and will stick in your mind, and when necessary, unwinding the picture back to phonetics and then to digits can become a thoroughly smooth process with some practice.
What next? You might wish to check out Professor Benjamin's performance of his act, and see if you can get some idea exactly where some of these techniques might be used.
Listen out in particular for Art using the phrase "cookie fission" to remember a number during his grade finale calculation.
In any case, I hope you enjoy the show, particularly the obvious increase in the audience's surprise at his ability as the show goes on, and, at the very least, the next time you find yourself needing to memorize a number, perhaps you could give the phonetic mnemonic method a try.
I believe, right now, you can still remember the phrase to remember the answer to that squaring problem earlier in this article!
To the delight of his audience and with suitable showmanship, he multiplies numbers of ever-increasing size in his head, calculates the day of the week of any given calendar date, and, for a grand finale, squares a 5-digit number, a calculation whose result is too large for most pocket calculators.
There is certainly no sleight of hand involved in Professor Benjamin's performance; he is quite happy in fact to give some insight and clues into exactly how his performance is done, and jokingly remarks that he is quite comfortable with that, as he does not expect to see anyone else performing his show in the immediate future.
While being able to deliver a performance like Professor Benjamin's might not be your goal, having some idea how the calculations and the 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 daily life, but some of the tips and tricks that Professor Benjamin uses can be of help for those daily memorization tasks such as keeping your PIN number to hand or recalling a telephone number of a friend without having your address book handy - and, with a bit of practice, perhaps you can entertain at a family reunion! Multiplication tables Unfortunately, there is a little bad news.
It is true that you cannot even consider multiplication of numbers with more than one digit until you have mastered those pesky multiplication tables that some of us struggled over for so long at school.
Being force-fed the tables as facts, to be learned by rote, is for many people one of the factors that makes mathematics an unpopular subject.
People who claim that they are unable to "get" math may be able to attribute that to the bad experience when first learning their tables.
The secret to learning the tables, however, is to realize there are not in fact all that many different items to remember.
Think about it for a moment - you need to practice multiplying two single digit numbers, 0 through 9.
In theory there are 100 multiplication facts to remember, but truthfully, there are considerably less.
For a start, many of the facts appear twice; if you know what 6 times 7 is, then you already know what is 7 times 6.
Multiplying by 0 and 1 are simple enough facts; anything times zero is zero; anything times 1 is unchanged.
Multiplying by 2 and 5 are the next easiest to learn; what remains after that is less than a couple of dozen multiplication facts, and the simplest way to remember those is practice.
It is possible you could use the memory tricks detailed 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 is actually not all that far away.
At school, you may have been taught how to do long multiplication, which actually, indirectly, involves multiplying every possible pair of digits in the question.
With a little bit of smarts, you can layout the long multiplication sum in your head and quickly see the answer.
The trick is to visualize all of the single-digit multiplications as two-digit answers laid out accordingly.
It is best illustrated by an example.
Suppose for instance 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 think of all the other selections of digits from the question; 7 times 2 (14) and 3 times 5 (15), and add those products to the middle digits of what you have already.
(There may be a carry to the left-most digit).
In this case, 3796 is indeed the answer.
With a little practice, you can multiply two-digit numbers quite readily, but often, something happens to our minds when trying to perform such sums.
We may not in fact be able to remember all those intermediate calculations; in fact, we may even forget the question! It is perhaps unsurprising then, in the Arthur Benjamin show, he soon switches to multiplying a number by itself (squaring), because, well, there are fewer intermediate results to remember.
The question has half as many numbers to remember, and the same goes for the calculation details.
The same multiplication logic applies, though; for example, we consider 73 times 73 first by multiplying the digits in place, giving 4909, and then the 7 times 3, which now appears twice, gets added to the middle digits, giving 5329.
Tougher stuff There are some more sophisticated techniques used to square three- and four-digit numbers which the interested reader may wish to research.
As a hint, one of the tricks commonly used involves modifying the calculation so difficult multiplications are replaced by easier ones.
For example, suppose you wished to multiply 993 by 993.
It is a shame that we were not multiplying by 1000, that would be easy.
So why not add 7 to one of those 993 entries, and to be fair, perhaps 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 may see a method for writing down the correct answer without too much trouble.
As the sums get bigger though, the more results we can remember, the smoother things will go.
For instance, we have already mentioned that at times we may be asked to remember partial calculations and carry them through to the end of the sum, or we may simply have to store the question away in our minds so we do not forget it.
Likewise, when it came to squaring the two-digit numbers, in actual fact there are only ninety of those answers to remember.
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 of those and store them in our mind, we will save time and brain power later! Memorizing numbers The trick is to convert numbers (which we are almost certain to find difficult tor remember, being just a string of digits) into words (which are much easier to recall and perhaps inspire our minds to create pictures).
We havce a far greater ability to recall poetry, or song lyrics, for example.
There are may systems of doing this.
One of the simplest is to remember digits by the count of letters in a word.
(A ten-letter word could represent zero).
For example, the phrase "May I have a drink, alcoholic of course, after the heady chapters involving quantum mechanics?" is perhaps something that you could eventually memorize without too much effort.
Converting back to digits, you have remembered 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9 - that is the first 15 digits of the mathematical constant pi.
Something like a credit card number is well within your reach, simply devise a suitable phrase, and the mere act of thinking up the phrase in the first place will help commit it to memory.
An even more compact way to remember numbers is to replace digits with letters.
In the popular phonetic mnemonic system, digits are represented by consonants, in fact, by the sound of 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, since the letter T has one downstroke.
Given the number to remember, pick out the sounds that correspond to the digits, and pad them out with vowels to make words.
It seems a long and torturous path to remember a number, but it works, particularly if the word or phrase you come up with is thoroughly ridiculous.
Remember that 5329 answer a little while back? Perhaps it was not the sort of number you found particularly memorable.
Using the phonetic method, a conversion to consonants gives L, M, N, P.
There are surely some mental pictures you could think of to remember those letters.
What about, for instance, a little LaMb taking a NaP.
It sounds outrageous, but that is far simpler to picture, and will stick in your mind, and when necessary, unwinding the picture back to phonetics and then to digits can become a thoroughly smooth process with some practice.
What next? You might wish to check out Professor Benjamin's performance of his act, and see if you can get some idea exactly where some of these techniques might be used.
Listen out in particular for Art using the phrase "cookie fission" to remember a number during his grade finale calculation.
In any case, I hope you enjoy the show, particularly the obvious increase in the audience's surprise at his ability as the show goes on, and, at the very least, the next time you find yourself needing to memorize a number, perhaps you could give the phonetic mnemonic method a try.
I believe, right now, you can still remember the phrase to remember the answer to that squaring problem earlier in this article!
Source...