Unusual Mathematical Ability:

Okay, so here is my completely random post of the week. I have a friend, now around 40, who has a rather astonishing ability to calculate numbers in his head. He can multiply three digit numbers by three digit numbers, divide three digit numbers by other numbers, and even do square roots, all in his head, in real time. I wouldn't believe it if he didn't prove it to me over the course of an evening: you can call out all sorts of calculations that most of us would take a minute to do on paper and he can answer back without even pausing to appear to think of the answer. As he tells it, he has had this ability ever since he was a little kid, and he can just "see" the numbers and how they multiply or divide.

Obviously being a human calculator isn't as cool as it was before, well, before the invention of the modern calculator. But I was wondering, has anyone ever heard of anything like this before? I know a lot of people who are extremely good at math, but I have never seen anything quite like this. I wondered if anyone else has.

And so endeth the completely random post of the week.

UPDATE: In response to the comment thread, my friend wrote in with a response:

Obviously being a human calculator isn't as cool as it was before, well, before the invention of the modern calculator. But I was wondering, has anyone ever heard of anything like this before? I know a lot of people who are extremely good at math, but I have never seen anything quite like this. I wondered if anyone else has.

And so endeth the completely random post of the week.

UPDATE: In response to the comment thread, my friend wrote in with a response:

When Orin sent me the link and I saw some of the posts, I wanted to respond both to try to eliminate some misperceptions but also because I thought people would find some of what I am saying interesting.

1) While it is true that computational abilities are not the same as mathematical abilities/logic/abstract reasoning, I am quite strong at all of these. Orin knows this, so when someone went after him saying that he does not know the difference, that was wrong.

2) I only estimate square roots, and I can usually go out to at least 4 or 5 decimal places, but I don't need to use tricks. In some cases, I can use some tricks I've developed to facilitate my computations, but they are tangential to the process and I can do any of the calculations without these, and for the most part, do.

3) While at least short-term memory is critical to the process of this ability, I do not memorize the answers. It would be impossible given the number of possible mathematical computations that can be asked/done. There are certain "sub"-calculations I sort of just "know" immediately, but that is just sort of a combination of being innate and from doing lots of math/computations over the years.

4) I find the division to be the most unusual part as I can go to as many decimal places as one wants as quickly as I can talk.

5) The limitation I have with the calculations is primarily with multiplication. I can at times do up to 4X4 digits (or 5 or 6 by 3 sometimes), but it depends on my focus/concentration. I am sure if I work on my memory, I can do more than that, not that that is so easy. With division, it is really just the 2 numbers I need to remember and as I say the string of decimal places, I don't need to recall everything I've said to that point. With addition/subtraction, it is essentially a "running tab," so again I don't need to remember a lot of numbers at any one time, unless I happen to fall behind. I can add 1, 2 or 3 digit numbers, sometimes more -- this is with saying them as quickly as you can put them in the calculator or computer and it can be many, many numbers.

The limitation with multiplication is the need to multiply the different pieces of the numbers (as one of the postings referred to) and remember those answers while also multiplying the other pieces that it needs to be added to.

6) While some of this can be taught in terms of the process, and while practice and improving one's memory helps, for the most part this is an innate skill. I have had this as long as I can remember. In fact, I was quicker with this when I was much younger and had fewer other things to think about and more time and interest in utilizing these skills.

7) Anyway, it's strange, but I've rarely talked about this (and certainly never written about it) before, certainly never close to in this level of detail or at this length. It is interesting to see what people think.

This type of ability is not so unheard of. Usually the people who have it, though, are autistic savants, like Rainman.

If your friend is otherwise normal, he is very rare.

But people who can write a symphony after composing it completely in their heads are the ones who stun me.

http://en.wikipedia.org/wiki/Category:Calculating_prodigies

and another here: http://en.wikipedia.org/wiki/Mental_calculator

Leonhard Euler and Carl Friedrich Gauss were noted for their powers of mental calculation. Also check out Zerah Colburn and Zacharias Dase.

See, that strikes me as an extension of something that many people could do to a lesser extent. For the math feats, though, I have no idea where they even come from. I don't think I'd know 7*8 is 56 except that I memorized my math tables. How does one visualize a three digit number? What are you visualizing, hundreds of little dots dancing around? I have to think it requires some kind of trick...

Additionally, if you write down any group of single digit numbers, he can quickly glance at them and tell you exactly what they add up to.

Some people are just wired differently.

At least in the early seasons.

Sort of a random gift they gave her, as she was a reporter without any mathematic responsibilities whatsoever.

I miss that show.

As a Fuller Brush Man in Silver Spring and Georgetown, I sold lots of *** to the young wives of young lawyers, especially to those whose husbands had put "NO SOLICITORS" signs on the door.

mathematicalability, as opposed to "calculating" ability. As I understand it, Gauss, Euler, and von Neumann are exceptions: most of the people with exceptional calculating talents are not unusually gifted at doing high-level abstract mathematics. (And, for what it's worth, I know a few Ph.D. mathematicians who are abysmal at doing sums in their heads.)I've actually met two fellow students/friends in a GR course and another in a topology course I've taken who had impressive computational abilities. Oddly enough they weren't that great at the basic linear algebra needed for the courses. However, they were great guys and one also had an almost encyclopedic knowledge of basketball statistics dating back to the days of mikan. That last quality was the most awesome to me, even though i could only unfortunately test it w/whatever little (comparatively) bball knowledge i had--asking for charles shackleford and tim leglers stats were a breeze for him.Pfft. If they can't at least calculate the eigenvalues of a 3x3 matrix to 2 digits precision in their heads, I am unimpressed.

From there you break it into the difference of squares, ie, 500 * 300 = (400 - 100) * (400 + 100), which means you now only need to calculate 400 squared, 100 squared, and 2 * 100.

Assuming you know the squares (tough, but I imagine one could do it?), you won't need paper.

Basically you need the square of the average, square of half the difference, and the difference (half the difference times 2).

I spoke with a neurologist once, with regard to Alzheimer's, and he said that the current theory is that memory is somehow carried in the brain as a manner of holograph. That would explain why memories are not, as in a computer, either there or deleted ... they begin clear and then details fade over time. I wonder if this is a simlar phenomenon, the ability to do the seemingly impossible simply because the mind/brain has the ability to see interelations not obvious from the outset.

At Interior Dept, one of the attorneys had been blind from birth, and handled Alaska matters. He'd had to cultivate his memory, essentially creating a map (which he had never seen) of a massive state, and a title search capability for it, in his mind. It was stunning.

I once had to go to him for info on some tiny parcel of no great significance. He sat there and rattled off everything of interest about it. Its location, how it had come into gov't possession, the date, all modifications to the title since, its uses over the last fifty years, etc., etc. I'm not talking Grand Canyon National Park here, just some tiny parcel that had once been a military outpost and had been given to Interior sometime after WWII.

Surely You're Joking, Mr. Feynman, the physicist frustrates the hell out of an abacist with mental math and a stroke of luck. If you haven't yet read his autobiography, put it on the short list.When I was in law school I phoned PG&E with a complaint about my bill, ran the math in my head when the PG&E guy on the line explained how my bill was computed, and showed him their error in about as much time as it took to say it when he finished. He got his supervisor on the line, explained what had happened and the supervisor asked if I was using one of them new-fangled hand calculators. I said I was doing it in my head and he offered me a job, but gave up when I replied that I was in law school and clerking for SEC Enforcement.

When I was a drinker, it was a fun party trick where I convinced others that I wasn't drunk because I could still multiply 121 x 79 = 9,559 in my head after several drinks; I impressed a client or two by being able to do quick price-quantity calculations while they were still fiddling with their calculator; and occasionally I freak out a salesperson by telling them what the final price with discount and taxes will be before they finish entering it into a computer. And it helped me get a 780-equivalent or higher on every math section of every standardized test I've taken in my life.

But in a world where the deli counterperson is stymied because I give her $12.13 for a $6.88 purchase, it's not a particularly useful skill; my girlfriend had been dating me for several months before I had occasion to mention to her that I could do it, and lots of other people who know me relatively well don't know that I can do this.

One of the odd mental abilities that we all have until about five years of age and some persons retain into adulthood is eidetic visual memory. All children and some adults can just glance at, e.g., a stack of library books and retain the complete image for an extended period. You can ask a person with this ability to give you the title, author, and binder color of the fifth book in, on the sixth shelf down, in the third stack and he can do this just by consulting his visual memory, the same way as we would go over and examine the book if we were physically in front of the stack.

Newsradiowas the best television program that has ever aired. Just brilliant. Constantly brilliant.grownin the retelling.Take 341x 115, for example. While it appears impossible to do in your head, if you break it down into three parts: 341 X 100 = 34100 which is pretty easy

341 X 10 = 3410 which is also pretty easy

341 X 5 = 1705 (which happens to be easy, because it is half of the above number).

Remember the results and add them together.

I can't do it. I can't the first result while calculating the second. But I can do so with shorter, easier numbers. 74 X 7? That's 490 (70x7) + 28 (7X 4) or 518. I could imagine someone with better memory and who is better at math being able to do the 3 digit numbers in the same way that I can do two digit numbers (kind of like most of us can remember aces and kings while playing cards, and some folks can remember almost the whole deck while playing-just deeper, more recallable short-term memory).

Also, practice helps. I was better at math when I was an engineering student twenty years ago than I am now. You get better memorizing cards when you play alot, etc.

Steve

This comment made me think. Isn't this ability analogous to spelling? Some of the brightest people I know (one of them a Presidential appointee (to a nonpartisan body in a slot reserved for a Dem)) are atrocious spellers, despite being able to work wonders with the English language.

Arithmetic, like spelling, is merely a tool. One can be a wizard with a hammer and nail, but that doesn't make one an architect.

Yes. There's a young high school student - Jay Greenberg, age 14 - who can do that; he's just issued a CD on Sony with his 5th Symphony and a Quintet for Strings - London Symphony Orchestra and the Julliard String Quartet, two groups that don't go in for amateur stuff, so I guess (I haven't had a chance to play it yet) that it's pretty good stuff.

The story goes that he was in elementary school at the time, so he was probably older than four. But not much older.

Truth be told a lot of people who spent substantial amounts of time working with numbers used to get really good at basic math in the old days. My father (born in '35) was an engineer for most of his life and he could do a startling amount of math in his head.

if I recall it focused mostly on chess grandmasters (multiple simultaneous games and blindfolded).

However, my answers are usually incorrect.

Does that matter?

Multiplying 3-digit numbers is more time-consuming, but if you want, for example, to multiply 242x373, it is much easier if you realize that you are multiplying (200+42)x(300+73), so the total answer is 200x300, which is easy, plus 42x300, which is fairly easy, plus 73x200, which is fairly easy, plus 42x73, which is a bit harder. The main trick is too keep all those things in your head. In short, you have reduced the problem to simple algebra.

Sometimes, it helps just to have a lot of practice. I've had one or two occasions when I startled the heck out of someone because they needed to know 45 squared, and I knew off the top of my head that it was 2025--and of course, because I did that calculation a lot when I was in junior high (I can't remember why). It is also possible to interpolate with surprising accuracy, if you do it regularly. What's the square root of 17? This came up in a chemistry class at USC, and because I knew that it wasn't much larger than the square root of 16, I guessed at 4.1, which startled a number of the other students. (It is actually 4.1231.)

...computers. (Before the advent of machines that "computed," "computers" used to be the title for people who "computed.")

I've had one or two occasions when I startled the heck out of someone because they needed to know 45 squared, and I knew off the top of my head that it was 2025--and of course, because I did that calculation a lot when I was in junior high (I can't remember why).Another interesting quirk of memory is the way that some item persists while the things around it fade, such as remembering the result of a particular calculation but not why you did it. Is it the nature of the thing remembered and the thing forgotten that accounts for that, or is the subconscious mind keeping some things and discarding others based on some unknown criteria?

In Engineering Physics in college, one day in the second semester, when the teacher asked a question, there was silence for a few moments, then I told everyone I was too tired to think, so people started digging for their calculators (which they had stopped bothering to get out, since I would have the answer before they could punch it in). Yes, that included cube roots out to several decimal places.

But the brain, just like the body, atrophies without exercise.

He also couldn't get a date to save his life; C'est la vie.

while Jingolaw relates that:

In my experience arithmetic is mainly useful to mathematicians when they need to settle a restaurant bill. This is not to say the mathematics of arithmetic is disinteresting (especially to number theorists like myself), rather my point is that studying arithmetic has little to do with the ability to perform long division quickly.

It is interesting that everyone seems to be able to tell the difference between the ability to run fast and the study of physiology and yet even educated members of the public cannot tell the difference between arithmetic and mathematics.

I think that a lot of this is just practice. I was born in 1952, which means that there were no calculators available at prices affordable by students until I was, say, twenty-one. Doing the calculations yourself, I think, helps a lot in acquiring the ability to do them in your head.

Von Neumann is given the following puzzle: Two motorcycles are 100 miles from each other and travelling toward each other at 50 mph each. A bee passes the first at 100 mph at this point in time. It proceeds forward until it hits the second motorcycle, whereupon it bounces off and travels back to the first motorcycle at 100 mph and so on, bouncing off the motorcycles, until the two motorcycles hit each other, squashing the unfortunate bee. From the point in time of passing the first motorcycle, how far did the bee travel?

Von Neumann answered immediately: 100 miles.

The questioner then commented "You know a lot of people take forever to do this since they calculate the infinite sum" to which Von Neumann replied "Is there another way?"

(There is: It takes one hour for the motorcycles to hit each other, so the bee is travelling at 100 mph for one hour. You don't need to add up how far it travels each segment.)

I suspect that this is because the two are used synonomously by 99.9% of the population. Essentially, they meaen the same thing... EXCEPT to the small percentage of highly trained experts.

In other words, they only have different meaning in the jargon of professionals.

Example: the public refers to the "Armed Forces" as the "army". Members of branches of the armed forces other than the Army (capital A) take offense to this.

Same thing here, only with a smaller number of people who care.

Moreover, my understanding is that anyone of decent intellect who has trained extensively with an abacus (still used in many regions of Asia) will develop this sort of ability eventually. It is not a particularly unique or interesting skill. And, as others have noted, is not a sign of any sort of creative genius. I have seen students with remarkable calculation ability who struggle with simple word problems. ("John has a pizza and 4 friends to divide it with..." etc.)

My friend (we'll call him "Jeff," as that is his name), does, indeed, have a phenomenal ability to perform even complex arithmetic on the fly, but is also quite gifted in mathematics in general. As a freshman in college he was asked to be a Teaching Assistant for senior-level mathematics classes, and by sophomore year he was co-teaching integral and differential calculus at the request of the Chairman of the Mathematics Department. He has learned, studied, and taught mathematics subjects that I cannot recall or describe because I don't even pretend to understand them.

So, while I meant to relate Jeff's prowess at party-trick-arithmetic to that of Prof. Kerr's friend, I also meant exactly that Jeff does, indeed, have a gift for mathematics.

Of course, as Jeff recognizes, it was his early ability to perform rapid calculations that made the study of more complex mathematics somewhat easier. I'm curious to know if any of the people with the party-trick-arithmetic ability described in these comments nonetheless have otherwise limited mathematic abilities.

You must also recognize, Lior, that in general conversation, "mathematics" is used as shorthand for numeric calculation or "arithmetic." The blurred distinction is reinforced when number theorists are prone to say "the mathematics of arithmetic" in one paragraph, and "the difference between mathematics and arithmetic" in another.

I am not a mathematician, but in my experience with higher level math, this is not true. And that was precisely Lior's point. Arithmetic has about as much to do with math, as running has to do with the study of human physiology. East Asia, with all of its amazing calculating phenoms (and test scores to prove it), has relatively few successful mathematicians.

When you are trying to write a proof of some novel theorem, calculating ability has exactly zero value.

"You can ask a person with this ability to give you the title, author, and binder color of the fifth book in, on the sixth shelf down, in the third stack and he can do this just by consulting his visual memory, the same way as we would go over and examine the book if we were physically in front of the stack."

I could that in college, and sometime beyond - I'd forgotten all about it.

I don't know when I lost the ability (maybe I haven't, and just haven't needed to do it lately), but on thinking about it it may be related to having more

things. Hmmmmmm.I can still tell you if my desk doesn't look

exactlythe same way as it did when I last left it, and what's changed. AND I can reach into my (relatively large) purse and find anything I want with out looking, though I wouldn't be able to list all the contents if you asked me to.On the numbers side, when I was in calculus class in college (boy, was

thata disaster!), the only thing I could do easily was look at a progression of numbers on the blackboard and tell you what the next one in the series should be. I never knew how I knew; I just did. I'm pretty sure there's a formula for calculating the next number, but I could never memorize it and certainly couldn't do it in my head (and this was in pre-calculator days). The teacher was positive I had to be cheating (since I was so bad at everything else in the class), but she couldn't figure out how.She was a lousy teacher - glad something I did drove her nuts. :-D

By the way, a similar skill that comes naturally to some savants, but is also learnable by others through simple tecnhiques and practice, is memorization of long strings of numbers--sixty digits was the max for the technique I learned.

He claimed that people would walk past his dorm room, hollar out two such numbers, and he'd give them the answer quickly.

Likewise with taxes--he would run taxes in his head at the cash register.

I don't think he was phenomenal (up to 3 or 4 digit numbers) in that respect, though the young man was very knowledgeable about number theory and other pieces of higher math.

Jimbino said:

"I haven't yet met a lawyer who can tell me why "buy high, sell low" is just as good a strategy as "buy low, sell high," for example."

okaaay -- you stumped this lawyer. Why is buying for more than you sell as good a strategy as buying for less than you sell? I mean, unless you have an incredibly crooked tax accountant.

His story "Gulf" covers some of this.

Tractenburg is the name of an author of a system for teaching kids to be fast calculators. There are various systems for developing memory. Suzuki showed that any random group of schoolkids could be developed into skilled violinists.

So the ability is some mix of aptitude and training, and most of us don't get the training, because we go to government schools which are focused on the least common denominator.

Orin, very likely your friend is an autistic savant. He may not even know it. I cannot do this with mathematics, but I can do all of the same with law. My abilities in this regard appear to scare people who oppose me at hearings, as I can "see" the exact words of statutes and cases to the most detailed footnote.

"And there's a person I know who can recite the precise numerical sections of ECPA corresponding to any bit of text in the statute -- not as impressive or useful as an ability to do math, and probably the sad sign of a bright mind wasted."

"You can ask a person with this ability to give you the title, author, and binder color of the fifth book in, on the sixth shelf down, in the third stack and he can do this just by consulting his visual memory, the same way as we would go over and examine the book if we were physically in front of the stack."

I can do both of these, and more.

"Scientific American in August had an article on the 'expert mind' if I recall it focused mostly on chess grandmasters (multiple simultaneous games and blindfolded)."

I could repeatedly beat IBM's mainframe computer predecessor to "Big Blue" at 3-dimensional tic-tac-toe by age 7, my favorite pastime when my father used to take me to IBM where he worked.

"I have the ability to do this.

However, my answers are usually incorrect.

Does that matter?"

An autistic savant's answers are almost always correct, so for this unique ability, it does matter.

Probably why there is such an irrational fear against ever giving me my bar admission, because I have this talent others do not have, that makes me "different," and many people have "never seen that before." I am a firm believer people like your friend should be applauded and admired, given every opportunity to be a success for all of us.

"Robert Heinlein and John Campbell suspected these skills could be taught. Lightening calculations and visual recall."

No, this is something that an autistic savant just does at anytime, anyplace, "on the fly," with no need to "practice." It is not something that can be taught to create in a *normal* person that abilities possed by an autistic savant.

"Some people are just wired differently." Exactly, it is called autism, savant style.

The unfortunate problem, is most autistic savants are hated by everyone, never get the opportunities achived by so many others, and this is a prejudice that is very difficult to overcome.

In 50 years, through a J.D./M.B.A. and a US National Equestrian Championship, I have not been able to find the answer -- 99.9% of the people upon first meeting me treat me as a retard.