# Are all mathematicians human calculators?

I asked my dad why he did not major in math he said "because he is not good at math". I think I like math, and I think I'm ok at it, but I'm not gifted or anything like that, I just like math. I think I'd like to major in math, but I see all these documentaries about great mathematicians and they can all multiply and divide numbers off the top of their heads and I certainly cannot. I realize that we have calculators, but somehow I don't think I should go into math.

As a forum/site of math people, what do you think?

Sorry if this is kind of a random question with no definite answer and probably out of place. My tags are probably wrong too, sorry.

EDIT: Mostly, i think I'm worried that I don't have a high enough IQ to ever contribute anything to the field of mathematics. I think I'll only study the work of others and never have my own work. It seems like people who are successful in mathematics are people who are talented by nature, like they have a really high IQ (yes, IQ is just a number, but you know what I mean) and math comes easily to them. For those people, it seems like math is effortless, but I wouldn't know.

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"great mathematicians and they can all multiply and divide numbers off the top of their heads". This is simply not true, and is very far from what mathematics is about. – user27126 Nov 4 '13 at 4:16
Professional mathematics has very little to do with arithmetic... I've known several great mathematicians who were terrible at the latter. Of course, like with anything else, after years of practice you do usually develop a "familiarity" with numbers. – user7530 Nov 4 '13 at 4:16
In my experience the stereotype is actually that mathematicians are lousy at arithmetic. It’s greatly exaggerated — many mathematicians are very competent at routine computation — but very few are human calculators. – Brian M. Scott Nov 4 '13 at 4:18
There is no doubt David Hilbert is one of the most influential and universal mathematicians in 19th and early 20th century and yet he has Dyscalculia - the disability in learning and comprehending arithmetic. If you know the answer of what 7+5 is, you are better in arithmetic than Hilbert and bad arithmetic can't be used as an excuse to stop you from do math. – achille hui Nov 4 '13 at 4:34
@achillehui: Is there documented evidence of Hilbert's "dyscalculia"? (Also, he certainly doesn't have it now - he's dead.) – Nate Eldredge Nov 4 '13 at 17:04

Replace the captions in the below comic with "math fan" and "mathematician" and it will convey the spirit of the truth (link to original) :

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I momentarily thought this had a golden ratio easter egg. – anon Nov 4 '13 at 5:04
The author should be ashamed for missing the opportunity. – Malice Vidrine Nov 4 '13 at 5:09
While I imagine that Zach does not mind the use of his comic here, I recommend that when you probably violate copyright you at least do it in such a way that you give proper credit: smbc-comics.com/?id=1777 – Carsten S Nov 4 '13 at 11:17
I would like to add: smbc-comics.com/index.php?db=comics&id=2208#comic – geodude Nov 4 '13 at 13:26
Sincere apologies; I have it bookmarked by the image URL rather than the full page. It was not my intention to in any way obscure the origin of the work. :( – Malice Vidrine Nov 4 '13 at 13:52

We invented computers to avoid precisely that...

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Love it........ – Soner Gönül Nov 4 '13 at 8:50
And we will fight them in 10000 years. To much Dune... – Luc M Nov 4 '13 at 21:59
@LucM I prefer ODing on Asimov... "Nine times seven, thought Shuman with deep satisfaction, is sixty-three, and I don't need a computer to tell me so. The computer is in my own head. And it was amazing the feeling of power that gave him." – Dan Neely Nov 5 '13 at 1:16
A link for Asimov's 'The Feeling Of Power'. – Raymond Manzoni Dec 15 '13 at 21:23
I sometimes like doing math on paper, but I am thankful for the computer, as it lets me make mistakes much faster. ;) – J. M. Jul 12 at 11:14

The short answer is no.

Mathematics, at the more advanced levels, very quickly ceases to be computational as you described in your question. Your ability to carry out complicated calculations mentally, while neat and impressive, does not really matter once you're asked to prove theorems.

The confounding variable here is that being able to do quick mental arithmetic is usually indicative of mathematical familiarity or mathematical maturity - when you do a lot of math, you get exposed to tricks that make these computations faster, and you become better at them by practice. For example, I can divide $1024$ by $128$ in a split-second, because I know one is $2^{10}$ and the other is $2^7$, so the result is $2^{10-7} = 2^3 = 8$.

If you think you like math, just do a lot of math! Don't be afraid.

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+1 Don't be afraid – user7530 Nov 4 '13 at 4:25

I am thinking exactly the opposite. To me many mathematician are bad in doing explicit calculation. They are so concentrated on those abstract thinking and almost never do explicit calculations.

You might search for "Grothendieck prime 57", which is (I think) a canonical answer to your question.

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+1 for Grothendieck prime! – Alexander Shamov Nov 5 '13 at 22:28
Its funny because by getting it wrong, Grothendieck did us all a favour. – goblin May 31 '14 at 1:52

$$\text{arithmetic} : \text{mathematics} \quad::\quad \text{spelling} : \text{ literature}$$

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No.

Part of the curriculum for math majors is logic. Now, logic may not seem to math-like, but logic is the language of mathematics. Once you understand logic, you can move on to topics such as algebra (not that crap they teach you in grade school), set theory, combinatorics, graph theory, and all types of other good stuff. In learning this, you gain a fundamental understanding of numbers, how they work together, all the different things you can do with them, nice tricks to solve seemingly hard problems easily (this especially if your in to number theory), and whatnot. This doesn't account for your 'human calculator' description, but a solid understanding of numbers certainly helps.

Another thing is practice. The more you work with numbers, the more comfortable you get with them.

While it is true that there are some mathematicians who were naturally gifted (Ramanujan, Euler, Newton, and more), they are by no means normal. Normal mathematicians started just as you are: clueless, but willing to learn.

I don't know what grade level you are in, but I'm going to go ahead and assume your in high school. Make sure you know your algebra and trig very well before going into college. Nothing too crazy, but understand logs/exponentials, trig functions and their inverses, the unit circle, factoring of 2nd and 3rd degree polynomials, and graphing common functions (x, e^x, ln(x), x^2, x^3, etc etc). You will need a good grounding of this to get through calculus and into higher level math.

Sincerely, A current math major

edit: and now that I look at your recent history, I see that, no, you are not in high school. Or at least if you are, your learning induction and trying to prove the cauchy-schwarz inequality. Whelp, oh well, I already wrote this out :P

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Actually, you're right, I am in high school. The logic/calculus stuff was a result of trying to go through Spivak's Calc – Kat Nov 4 '13 at 4:54
ahh, ok. Then be assured, your ahead of the game. I didn't learn any calculus or logic till college, and I'm turning out fine...I think... – Nick Nov 4 '13 at 4:58
Thanks :) Though, I don't really get logic or calc tbh, but I plan to figure it out eventually. – Kat Nov 4 '13 at 5:04
Logic helps everything to me.Except I am stuck at trigonometry :/ – ministic2001 May 12 at 9:18

Although some of the great mathematicians definitely were also phenomenal calculators, I do not think that this is necessary for being a mathematician.

In general, I would say that mathematicians and pseudomathematicians may be divided into three categories (of course, dividing people into categories is always problematic, so what follows is not meant to hold without exceptions):

• People that remember 1000 digits of $\pi$, multiply ten-digit numbers in several seconds, and similar stuff. In fact, this has usually more to do with autism than with mathematical talent. Although there may be some exceptions, these people usually are not real mathematicians, since their focus is very far away from understanding things. Our local radio once hosted a boy that holds a national record in the number of digits of $\pi$ remembered. However, he seemed to know almost nothing about why the number $\pi$ is useful.
• Great calculators. Some mathematicians in history, e.g. Leonhard Euler, were exceptionally skillful in performing algebraic and numerical calculations. This kind of people should not be confused with those mentioned in the first point. In fact, before computers have been invented, mastering the craft of performing calculations was a must. It still may be useful today, but computers make things little bit easier also for people that are not so good at these things.
• Horrible calculators. Although some skill in performing calculations is undoubtedly useful, it does not appear to be the most important thing. In fact, there are also some really horrible calculators amongst professional mathematicians. Mathematics, as well as the rest of science, philosophy, and art, is mainly a creative discipline. Most of really good mathematicians certainly have a sort-of artistic soul. Creativity is what matters the most.

I would say that there are two different levels of mathematics: the craft of mathematics and the art of mathematics. Mastering the craft is certainly useful, but you do not have to be phenomenal in it, in order to be good at mathematics. The art of mathematics is much more important and I doubt that any important mathematician in history was solely a craftsman.

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There are three kinds of people in the world: those who can count, and those who can't. – Michael Shaw Nov 4 '13 at 19:19
+1 for the distinction art/craft of mathematics. I really like that. – Daniel Robert-Nicoud Dec 15 '13 at 21:47
I am an autist who loves math but the only thing first thing when I learn a topic,is to find the reason behind it. – ministic2001 May 12 at 9:19

I certainly didn't study mathematics to develop abilities to make trivial calculations at high speeds, albeit I was always good at that sort of thing in elementary school.

Math is a lot more than just calculations. In fact, it has almost nothing to do with calculation.

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Numerical analysis and statistics excepted, perhaps – user7530 Nov 4 '13 at 4:23
But no one sane does numerical analysis by hand or —even less— mentally! – Mariano Suárez-Alvarez Nov 4 '13 at 4:45
@Mariano, Gauss did numerical analysis by hand, but of course he did not have computers, and we can't really say now if he's sane. :) On my part: I could do numerics by hand, but why should I bother? – J. M. May 3 at 3:36
No one sane does numerical anaysis today. Gauss did not have any options... – Mariano Suárez-Alvarez May 3 at 3:57

Not only is great prowess at mental arithmetic not necessary (as everyone has sufficiently pointed out), but knowledge of mathematics can help you avoid doing a lot of calculation. You probably know the famous (and apparently even true) story about Gauss in elementary school: Instead of adding the numbers from 1 to 100, he figured out that their sum is equal to 50 * (1 + 100). Bypassing boring calculations is a great reason to develop mathematical insight, in my opinion...

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In considering this i think the phrase "human calculators" is limiting. A calculator is programmed at doing one thing: rendering a numeric value/answer based only on numeric input. While a human being is obviously much more. You see where i'm headed with this. So the question itself is actually limited to a simple "yes" or "no". Of course my answer is NO.

Mathematicians are much more that "human calculators".

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No, because of intelligence and creativity, something we have a great deal of trouble implementing into computers, and it may even be impossible to get computers to mimic. The best computers today can solve maybe 1% of the questions on this site, while lousy calculators we are superior problem solvers.

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Think about this: when you were 17, was the mathematics you were doing in school very hard arithmetic problems? For me it was geometry, algebra, trigonometry, calculus: things that were of a different kind, not harder-of-the-same-kind.

If higher mathematics were just multiplying bigger numbers faster, it wouldn't be very interesting.

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The number of famous mathematicians in history who were also great mental calculators are few and far between. Euler, who became blind in his latter years, is said to have been able to perform calculations in his head. One story is that Euler passed away while he was mentally calculating the orbit of the moon. Emilie du Chatelet is also another famous example. It has been determined from his extant notes that Newton was not a mental calculator, though he did enjoy working out numerical calculations to something like 50 decimal places. Gauss has also been named as a calculating prodigy. Doron Zeilberger, though, has recently brought to light that Gauss' famous boyhood deduction of summing consecutive integers up to 100 was actually determined with inductive examples (as shown in his Tagebuch, i.e. diary) and not by other more direct means (as had previously been promulgated). The reader should take caution that we are talking about famous people in history with many admirers and followers who very likely would not fall short of exaggerating their accomplishments. In the 20th century, we have Alexander Aitken and John von Neumann. There are several anecdotes about von Neumann, who was able to solve the two trains and a fly problem the long way by summing geometric series in his head, according to Eugene Wigner. Feynman was perhaps not a calculating prodigy in the same league as a real mental calculator, but he did explain some of his tricks and techniques with regards to mental calculations.

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## protected by Michael Greinecker♦Nov 4 '13 at 9:36

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