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All the mathematicians whose biographies are given in our archive exhibited extraordinary mental powers. In this article we look at a few mathematicians who have shown extraordinary powers of memory and calculating. We also look at a number of people who had no mathematical skills, usually no education, yet were able to display feats of mental arithmetical skills which astounded their contemporaries and today still astound us.

First we mention John Wallis whose calculating powers are described in [2]:-

[Wallis] occupied himself in finding (mentally) the integral part of the square root of3 × 10^{40}; and several hours afterwards wrote down the result from memory. This fact having attracted notice, two months later he was challenged to extract the square root of a number of53digits; this he performed mentally, and a month later he dictated the answer which he had not meantime committed to writing.

This, although quite remarkable, is rather typical of the feats we shall describe in this article. It is the combination of exceptional memory and calculating ability which seems to combine in many of those we consider. However, in one respect Wallis is very different from others we describe in that he was 53 years old when he performed the above feats. Most of the others we describe were at the height of their powers when young children, often around 10 years of age.

Having looked at a mathematician from an early period, let us next consider von Neumann whose feats of memory are described by Herman Goldstine in [8]:-

As far as I could tell, von Neumann was able on once reading a book or article to quote it back verbatim; moreover he could do it years later without hesitation. He could also translate it at no diminution in speed from its original language into English. On one occasion I tested his ability by asking him to tell me how the 'Tale of Two Cities' started. Whereupon, without pause, he immediately began to recite the first chapter and continued until asked to stop after about ten or fifteen minutes.

Von Neumann's ability to do mental arithmetic is the source of a large number of stories which no doubt have grown the more impressive with the telling. It is hard to decide between fact and fiction. However, it is clear that multiplying two eight digit numbers in his head was a task which he could accomplish with little effort. Again it would appear that von Neumann's 'almost perfect' memory played a large part in his ability to calculate.

Other mathematicians who have exhibited great powers in mental arithmetic include Ampere, Hamilton and Gauss. Only one mathematician has ever described in detail how he was able to perform incredible feats of memory and calculating. This is A C Aitken, see [9] , [1] and [7], and we shall consider his methods later in this article. First, however, let us describe the feats of a number of calculating prodigies who had no mathematical training.

Zerah Colburn's powers in calculating are described in [2], [12] and [13]. He was born in Cabut, Vermont, U.S.A. in 1804 and he visited Europe in 1812, when only eight years old to demonstrate his skills:-

He could instantly give the product of two numbers each of four digits but hesitated if both numbers exceeded10,000. Among questions asked him at this time were to raise8to the16th power; in a few seconds he gave the answer281,474,976,710,656which is correct. ... he worked less quickly when asked to raise numbers of two digits like37or59to high powers. ... Asked for the factors of247,483he replied941and263; asked for the factors of171,395he gave5, 7, 59and83, asked for the factors of36,083he said there were none. He, however, found it difficult to answer questions about numbers higher than1,000,000.

Colburn is interesting for a number of reasons. Firstly he influenced Hamilton to take up mathematics, secondly that he exhibited a common characteristic of most uneducated calculating prodigies, namely that his abilities diminished when he underwent education. This may be due to the simple fact that such calculating abilities require continual practice for many hours each day and education occupies too much time to allow this to continue. Colburn is also interesting in that he was able to give an idea of how he carried out the calculations, the main method being by factorising the numbers concerned:-

Asked for the square of4,395he hesitated but on the question being repeated he gave the correct answer, namely19,316,025. Questioned as to the cause of his hesitation, he said he did not like to multiply four figures by four figures, but he said 'I found out another way; I multiplied293by293and then multiplied this product twice by15'. On another occasion when asked for the product of21,734by543he immediately replied11,801,562; and being questioned explained that he had arrived at this by multiplying65,202by181.

George Parker Bidder was born in 1806 at Moreton Hampstead in Devonshire, England. He was not one to lose his skills when educated and wrote an interesting account of his powers in [3]. Again it is worth noting that other members of his family had exceptional powers of memory and calculating. One of his brothers knew the Bible by heart, another brother, who was an actuary, had the misfortune of having all his books destroyed in a fire. This was not the problem it might have been to an ordinary person since he was able, in the period of six months, to rewrite them from memory. One of Bidder's sons was able to multiply two numbers of 15 digits but he was slow, and less accurate, compared with his father. Bidder explained how the sound of numbers was much more important to him than their visual representation, something that Aitken was also to emphasise. Bidder wrote, see [3]:-

... if I endeavour to get any number of figures that are represented on paper fixed in my memory, it takes me a much longer time and a very great deal more exertion than when they are expressed or enumerated verbally. ... if required to find the product of two numbers each of nine digits which were read to me, I should not require this to be done more than once; but if they were represented in the usual way, and put into my hands, it would probably take me four times to peruse them before it would be in my power to repeat them, and after all they would not be impressed so vividly on my imagination.

The last non-mathematician whose feats we describe is Dase. He is of particular interest since his talents were investigated by Gauss, Encke and other mathematicians. As examples of Dase's calculating ability here is an example: 79532853 × 93758479 = 7456879327810587, time taken 54 seconds. He multiplied two 20 digit numbers in 6 minutes; two 40 digit numbers in 40 minutes; two 100 digit numbers in 8 hours 45 minutes. Gauss commented that he thought that someone skilled in calculation could have done the 100 digit example in about half that time with pencil and paper.

Although Dase had no mathematical education he offered to use his powers to help mathematics. He was shown how to use the formula

π/

_{4}= tan^{-1}(^{1}/_{2}) + tan^{-1}(^{1}/_{5}) + tan^{-1}(^{1}/_{8})

and, using this, he calculated π correctly to 200 places over a period of about two months. In his spare time, between 1844 and 1847, he calculated the natural logarithms of the first 1005000 numbers to 7 decimal places.

In the 19^{th} century there seems to have been quite a popular entertainment watching extraordinarily gifted children perform calculations on stage. Of course since there was a market for such performances children who seemed gifted in this way were encouraged to practise hard so that they could earn money. Colburn, Bidder and Dase all gave stage performances.

Another type of lightning calculator was Trueman Henry Stafford from Royalton, Vermont in the United States. Stafford chose not to perform on stage although, as a ten year old, he had calculating abilities to rival any of the others. Had he performed on stage he might well have brought the crowds in according to the description given by H W Adams, see [12], [12]:-

Multiply in your head365,365,365,365,365,365and365,365,365,365,365,365. He flew around the room like a top, pulled his pantaloons over the tops of his boots, bit his hands, rolled his eyes in their sockets, sometimes smiling and talking, and then seeming to be in agony, until, in not more than one minute, said he133,491,850,208,566,925,016,658,299,941,583,225!

Stafford took a degree at Princeton and became a professional astronomer. His ability to perform mental calculations slowly departed as he grew older.

In many ways the most interesting of all these people was A C Aitken. The reason is that

did not exhibit his talent at a young age like most of those described above. Rather he developed the skill as he explained, see [7]:-

Only at the age of15did I feel I might develop a real power and for some years about that time, without telling anyone, I practised mental calculation from memory like a Brahmin Yogi, a little extra here, a little extra there, until gradually what had been difficult at first became easier and easier...

Aitken became an excellent professional mathematician holding the chair of mathematics in Edinburgh in Scotland. He put his memory to good use in many ways in his role as a mathematician. To give just one example, see [6]:-

When he examined a new number of a mathematical journal he had only to scan it page by page, turning the pages over at a rate which the ordinary reader would record only half dozen lines or so. Subsequent discussion made it clear that he registered all the material. And, as he said, he never forgot what he had once seen.

Aitken's life is described in [7], his mental abilities are recorded and analysed by himself in [1] and by an Edinburgh psychologist in [9]. The most fascinating aspect of Aitken's ability to do mental calculation, to my [EFR] mind, is the way in which he was able to combine his calculating skill and extraordinary feats of memory with a deep understanding of the methods of numerical mathematics.

In [9] Hunter describes Aitken reciting the first 1000 digits of π to him:-

Sitting relaxed and still, he speaks the first500digits without error or hesitation. He then pauses, almost literally for breath. The total time taken is150sec. The rhythm and tempo of speech is obvious; about five digits per second separated by a pause of about^{1}/_{2}sec. The temporal regularity is almost mechanical; to illustrate, each successive block of fifty digits is spoken in exactly15sec.

Then Aitken recited correctly the second 500 digits of π. Here however Hunter reports that Aitken hesitated and sometimes corrected himself. When asked why he found the second 500 much harder than the first 500 Aitken had an interesting answer. Partly he said it was due to tiredness since recall requires a great effort. More interesting however was his other reason:-

Before the days of computing machines there was a kind of competition (human, I mean) in seeing how far they could calculate π. In1873, Shanks carried this to707decimals; but it was not until1948that it was discovered that the last180of these were wrong. Now, in1927I had memorised those707digits for an informal demonstration to a students society, and naturally I was rather chagrined, in1948, to find that I had memorised something erroneous. When π was calculated to1000and indeed more decimals, I re-memorised it. But I had to suppress my earlier memory of those erroneous digits,180of them...

So Aitken's problem was that he could not forget the incorrect 180 digits!

You can see the 1000 digits of the decimal expansion of π as Aitken recited them.

What about his ability to calculate? Firstly when presented with a number, Aitken instantly saw a variety of different ways of expressing it. For example

1961 = 37 × 53 = 44

^{2}+ 5^{2}= 40^{2}+ 19^{2}

occurred to him instantly. In fact a number instantly appeared to him in a form particularly suitable for the required task. For example if asked for the decimal expansion of ^{1}/_{851} he would think of 851 as 23 × 37, if asked for the square root of 851 then he thought of it as 29^{2} + 10, if asked for the decimal expansion of ^{17}/_{851} then he would think of it as almost 0.02.

Numbers filled Aitken's world. He said:-

If I go for a walk and if a motor car passes and it has the registration number731, I cannot but observe it is17times43. ... When I see a bus conductor with a number on his lapel, I square it ... this isn't deliberate, I just can't help it. ... [Given a number] is it a prime of the form4n+1, and so expressible as the sum of two squares in one way only? Is it the numerator of a Bernoullian number, or one occurring in some continued fraction? And so on. Sometimes a number has almost no properties at all, like811, and sometimes a number, like41, is deeply involved in many theorems that you know.

Aitken showed an ability divide, and thus to calculate decimal expansions of rationals, which other calculators could not do. It was always a rule at the calculating exhibitions of those described above that, when the audience was asked to suggest sums, division sums were banned. Aitken however was able to use certain tricks. On asked to compute the decimal expansion of ^{1}/_{697} he explained his method. He saw immediately that 697 = 17 × 41. Then:-

I mentally worked out a41st and divided it by17at the same time. I did two things. That's severe by the way. A double process is very severe. You see you've got to run one decimal and divide it at the same time by something else. You've got to alternate back and forward I've got to be aware that a41st is point, 0, 2, 4, 3, 9,and be dividing by17along.

However many divisions were carried out by Aitken using clever tricks. He explained how many decimal expansions could be carried out by short division. He said, see [9]:-

One can divide by a number like59, or79, or109, or599, and so on, by short division. Take for example,^{1}/_{59}, which is nearly^{1}/_{60}. Set the division out thus

6 ) 1.0 1 6 9 4 9 1 5 2... ------------------------ 0.0 1 6 9 4 9 1 5 2 5...

Here we have the decimal for^{1}/_{59}, obtained by dividing1by60; as we obtain each digit we merely enter it in the dividend, one place later, and continue with the division. As another example, consider^{5}/_{23}. Write it as^{15}/_{69}. Then proceed

7 ) 15.2 1 7 3 9 1 3 0... ------------------------ 0.2 1 7 3 9 1 3 0 4...

In fact^{5}/_{23}= 0.2173913043478260869565, a recurring decimal with a period of22digits. One could equally well have written it as^{65}/_{299}, then carrying out division by3, two digits at a time, and entering in the dividend two places further along. There are other possibilities: For example, the mental calculator is, or should be, very familiar with the factorisation of numbers; he should know not merely that23time13is299, but that23times87is2001. For example^{5}/_{23}is equal to^{435}/_{2001}; and if we note that435is the same as434.999999999..., we have another method, in which, as we obtain the digits, we subtract them from the dividend, so many places later. Thus in the present case

2 ) 434 782 608 695 652 ... ------------------------- 217 391 304 347 ...

For example,217from999gives782, which we then divide by2, obtaining391; this, subtracted from999, gives608; and so on. My aim has been to demonstrate, in these various rather simple examples, some part of the repertoire, the armoury of resource upon which a mental calculator may draw, and in regard to the choice of which he must make instantaneous decisions, and keep to them.

When asked to explain exactly what he does when asked to multiply 123 by 456 Aitken replied:-

I see at once that123times45is5535and that123times6is738; I hardly have to think. Then5535plus738gives56088. Even at the moment of registering56088, I have checked it by dividing it by8, so7011, and this by9gives779. I recognise779as41by19. And41by3is123, while19by24is456. A check, you see; and it passes by in about1sec.

Why cannot we all then, having learned Aitken's tricks, perform rapid calculations as he could. Aitken, when asked what he believed made him more able to calculate than the average person, put it down to his ability to easily remember numbers. He said:-

I can put aside in storage for a future occasion a result that has already been obtained. I know I can bring it out correctly. .. [this] is what distinguishes the calculator from the man in the street. The man in the street forgets the stages in between.

Some feats of memory are certainly visual in nature. The person concerned can 'see' the numbers of objects which have been committed to memory and in some sense read them off as if they had been written on paper. Aitken said that his memory could work that way but it slowed him down.

Seeing would put me off. If I were forced to visualise, I would be much slower.

However, when asked to recite the digits of π backwards Aitken had, interestingly, no other option but to bring the numbers into a visual form and read them off backwards from his visual image. His 'much slower' comment may be true but the speed was still impressive. The last 50 digits of the expansion required 18 seconds for him to recite them forwards and 34 seconds to recite them backwards.

We end with two final illustrations of Aitken's memory. While serving in the Otago Company at Armentières during World War I, the platoon book was destroyed. Aitken recited the names and numbers of all the members of his platoon. The second illustration comes from the testing of Aitken's powers by the Psychology Department in Edinburgh. In the 1930's Aitken had been tested by the Department. One test involved 25 random words which were selected from a dictionary. These words were read to Aitken and he was able to repeat them. When Hunter interviewed Aitken in 1961 he had before him a record of the 1930's test and he asked Aitken if he remembered being asked to recite a random sequence of words. Indeed Aitken did and was still able to recite them correctly nearly 30 years later.

**References (14 books/articles)**

**Article by:** *J J O'Connor* and *E F Robertson*

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