Why Are Mathematicians So Bad at Arithmetic?

Thomas Mann once said, “A writer is someone for whom writing is more difficult than it is for other people.”

I believe the same applies to mathematicians doing arithmetic.

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It’s a running joke among mathematicians that they’re bad with numbers. This confuses outsiders, like hearing surgeons plead clumsiness, or poets claim illiteracy, or Rick Astley confess that actually he is going to give you up and let you down, maybe even run around and desert you.

Does it come from some false modesty? A skewed sense of humor?

No, some mathematicians insist: it’s really true, we’re bad at arithmetic.

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I’m choosing my words carefully: “mathematics” and “arithmetic” are not interchangeable. “Arithmetic” refers to calculations with numbers: 17.9 + 18.32, for example. “Mathematics,” meanwhile, is far broader: it tackles shape, structure, change, and all kinds of quantities.

The reality is that mathematicians aren’t professional arithmetic-doers, any more than musicians are professional players of scales.

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I’ve heard mathematicians lament that their ability with arithmetic peaked sometime in grade school. That sounds overblown, but they’re probably not wrong.

As early as high school, specific numbers start taking a back seat to patterns among numbers. You stop working with 7 and 9 and 22, and start working with an x or an n that can refer to all of them at once. As you move into more abstract realms, your arithmetic gets rusty.

And the more math you study, the more extreme this gets.

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When I began to teach 3D vectors two years ago, I realized I first had to teach it to myself, because I’d never actually learned it. My college courses skipped straight to “n-dimensional vectors.”

This is how mathematicians approach things: why discuss the 2D or 3D case when you can just climb the ladder of abstraction and cover all cases at once? Surely you can figure out the 3D specifics when you need them, right?

Well… maybe.

But going from abstract to concrete isn’t always as easy as you’d think.

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I’m not a professional mathematician, but I’m proud to say I have all the bad habits of one. To wit: my students are often surprised at my clumsiness with arithmetic. Just today I casually said “40,000” when I meant “400,000.”

This happens a lot.

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Now, are mathematicians actually that bad at arithmetic? Compared to engineers and accountants, perhaps. Compared to the average person on the street, of course not. The “bad at arithmetic” thing is probably overplayed. So why do mathematicians love to bring it up? Here’s one reply:

Imagine you’re an artist, and people are convinced that your job consists of rolling and unrolling canvasses. That’s it.

Week after week, people ask: How fast you can roll a canvas? What’s the biggest canvas you’ve ever unrolled? Can you come over and unroll my canvas for me this weekend? And so on, and so on.

You keep trying to clarify – to talk about the actual paint you put on the canvas – but they don’t really get it. They just laugh and say, “Oh, you artists!”

Wouldn’t you start pretending to be bad at canvas-rolling, just to change the subject?

That’s one answer. But I have to admit there’s another possibility:

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43 thoughts on “Why Are Mathematicians So Bad at Arithmetic?

  1. Sent this to my mum who likes to comment to others about how she was shocked I could become a mathematician without knowing my times tables. Love ya mum 🙂

  2. I am a professional mathematician. And I am not bad at arithmetic. At least far better than my students who want a numerical calculator for their exam in calculus. Even though I tell them it won’t be necessary nor helpful. But they use it on the level of calculating 3 times 4. Compared to this, I am a genius. If I concentrate I can multiply teo two digit numbers. I feel great.

    But okay, I have not divided two numbers on paper since 30 years. I too use a calculator for that, or an estimate. Have I lost basic skills? Do I need to worry?

  3. First: Serious question: does arithmetical calculation take place in a different part of the brain than mathematical reasoning? Because when I’m figuring out the mathematics of a problem (at my very, very low level – what’s the algebraic equation I’m trying to set up, for instance) and I have to do a minor calculation, it seems distracting to me – which is why I end up using a calculator so much for even trivial things. I’ve compared it to watching a non-English movie with subtitles – I really feel exhausted by switching back and forth between interpreting the action, tone, music and reading text. I suppose practice would help make switching easier.

    Second: a lot of talented writers can’t spell and mess up grammar. Poets, on the other hand, mess up grammar for a living.

    Third: When I took Mike Brown’s solar system astronomy mooc (yeah, giggle) to find out why he killed Pluto, he taught me something that changed my life: 5678. That is, 56=7*8. I’m still not sure about adiabatic processes and I need three pages of notes to keep the relationship between orbital period and radius straight, but that I remember. Except I keep screwing it up and coming up with 5*6=78. I’m hopelesss.

    • Your second point is actually about stereotypes. Some writers are really bad with spelling, but most of them are very talented with grammar (it’s what make them talented writers). Which doesn’t mean that they can’t make mistakes from time to time, since language isn’t an exact science anyway.

      And saying that poetry is just messed up grammar looks like overgeneralization to me. Poets don’t “mess up” grammar. They play with grammar. Just like we can play with physics or maths to do fun or interesting things. You aren’t messing up the rules of physics when you find a way to levitate with a bit of liquid nitrogen and a magnet. Of course there are some poets who intentionally bend our language rules, but what makes a poet isn’t the hapakes or the neologisms. It’s their control over language.

      But more generally, yes, practive makes things easier. That’s how you learn new things, in fact – and that’s also how you forget things, by a lack of practice.

      • It’s not grammar. It’s poetics and rhetorics. Tropes and Schemes. Not unlike substitutions and commutations and transforms in math.


        His tale was strange and wonderful.

        This tale of his was strange and wonderful.

        This strange and wonderful tale was his.

        This tale, strange and wonderful, was his.

        This, his tale was, strange and wonderful.

        His tale was interesting, if by “interesting” you mean both strange and wonderful.

        Sworn testimony, a bald account, a blatant lie, a tall tale; whatever it was he said, it was wonderfully strange.

        Strange was his tale, and full of wonders.

        The tale was his; his tale was strange; this tale was wonderful.


        The emphasis shifts to suit the writer’s purpose — and key bits can be isolated to different locations in the expression.
        The relationships can be distributed or associated or yoked or implied. In any case where the grammar and syntax have been used correctly the “diagram” of the sentence will be topologically identical.

        Students who are good at “math” and bad at “writing” have often been poorly instructed — in my experience.

    • To your first question:
      I’m not a neuroscientist, but I am a mathematician however. There’s this recent study, where 15 mathematicians and 15 non-mathematicians had fMRI’s done while solving both low-level and high-level math problems (high-level as in, undergrad-ish level it seems):

      http://www.pnas.org/content/113/18/4909

      They conclude, among other things, that the same part of the brain were used when the mathematicians solved high-level problems and when both groups solved low-level problems. So it seems like you’re not really using a different part of the brain.

      Nevertheless, I’m also one of the mathematicians struggling with arithmetic, but I suppose it’s only a matter of habit. I’m equally bad, or worse, at linear algebra, as I’m never really using it.

    • I think that with arithmetic, it’s the way that you learned how to add/subtract/multiply that makes you see your process. Not the brain region. My husband and I calculate numbers differently, and I always want to know how he gets to his results because sometimes he beats me. So speed can also be due to your strategy.

  4. I’m not a mathematician, but as a linguist I got used to perceive rhythm patterns. With time, I progressively lost the habit to deal with scales (since scales aren’t really important in rhythm patterns). I could totally have said “Just today I casually said “40,000” when I meant “400,000.””.

    I guess it’s similar with mathematicians. They focus on what’s important in their respective fields of interest.

  5. My exercise group partner at university, when I was studying mathematics, used to repeat a quote from one of his earlier professors, he was from Yugoslavia, “mathematics is the only subject that has no need of numbers”.

    • That’s an excellent quote. Indeed, one can do mathematics without even thinking about numbers, but be it in physics, chemistry, economy, sociology, biology, medicine even… you’ll always need numbers one way or another. Be it some moles of this, or some amount of money, or people, or amount of medicine to give a patient… The only exception to this is, indeed, maths, but I suppose maybe one could also extend it to the very pure breeds of physics.

      The spirit of the quote overrules these technicalities though.

  6. Many years ago, I was having a discussion with a computer scientist about his research. He said that he carried out research on pseudo-random number generators. I asked him if he could recommend a good pseudo-random generator. His reply was that he knew little about these generators in practice. And he genuinely didn’t seem to be interested in practice.

  7. A few years ago I attended what was essentially a one man show of a mathematician showing off his arithmetic skills. He was fully capable of multiplying five digit numbers in his head. (Among other mathematical parlor tricks.) I really wish I could remember the man’s name, but unfortunately names are another thing mathematicians are often bad at.

    He went out of his way to emphasize that he was not a savant, and he talked a little about how he worked with a psychologist to work out strategies that improved his mid-calculation memory and worked out a set of tricks to simplify the mental calculation. In a way, he made a Math problem out of mental calculation, and the result was a person with very good arithmetic skills.

    If you want to talk about the average arithmetic ability of mathematicians, you could make the case that this man raises the average all by himself.

      • I think so. That talking about Mathemagic does sound familiar. Thank you so much. I have been wondering about that for some time now.

    • Here is Art Benjamin’s TED Talk where he gets two of the 3-digit squares wrong.

      I can also square 3- and 4-digit numbers in my head fairly quickly. 5-digits, I can do, but most would take 30+ seconds. I don’t know about Benjamin’s techniques, but in my case it’s a blending of being naturally great at both arithmetic and math, while having a good, but not great short term memory. I use my math skills to reconfigure arithmetic problems to compensate for unexceptional memory.

      For example, someone wrote 65×67 in the comments. To calculate that virtually instantly, I would (mentally) rewrite it as:

      (66-1)(66+1) =
      66^2 – 1 =
      (50 + 16)^2 – 1 =
      (50^2 + 2 * 50 * 16 + 16^2) – 1 =
      (2500 + 1600 + 256) – 1 =
      (4100 + 256) – 1 =
      4356 – 1 =
      4355

      None of these steps is difficult at all. I compensate for my lack of memory by avoiding multi-digit adding and subtracting as much as possible, especially avoiding carries. Most of the math skills part came into devloping, not employing, the techniques I use to do mental arithmetic.

      To me, what’s most impressive about Art Benjamin is ability to listen to a problem given orally and interpret it so quickly so he can give an answer.

      There is definitely a huge correlation between arithmetic and math skills. Some of that is due to a those being bad at arithmetic never having the opportunity or inclination to discover they are good at math. However, much of it is just pure problem solving skills and ability to attack problems in numerous ways.

  8. I love the slide about 8×7=56

    Just this Monday, after a 4-week complete break over the Christmas hols, I was preparing a lesson on solving fractional algebraic equations. Imagine my horror when after a whole page of workings to produce the wrong answer, I saw right at the beginning that I had calculated 9×6 as 56.

    I have been mentally beating myself up ever since! My multiplication tables are about as second nature to me as breathing (which seems contrary to the article). At least I don’t feel quite so bad now!

    • I am a ‘hobbyist’ mathematician,(meaning when I get home from boring school I crack out the pencil and paper and examine what new question to work on.) and while working on simple proofs, I calculated 6*4 to be, 26… I made some basic algebra errors to, like those that an even divided by even CAN be odd, leading to after a long proof the 36 is the only triangular even odd number. Checking through it, the face palms were practically endless. In class, I am usually quizzed, but they do questions on basic arithmetic, and nearly everytime, I am the slowest. I can’t imagine if I worked on a Fermat Last Theorem like proof and did something like that…

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  10. So true. 🙂 About 20 years ago, while at Northwestern, I had lunch every Thursday with some legendary mathematicians, but the arguments about calculating tips got so bad that I ended up getting a tip chart. But even that didn’t help, because true mathematicians don’t trust anything but their own calculations.

    Mathematicians relationship to calculation is different from their relationship to concrete problems. Most mathematicians use concrete objects to experiment on, and figure out their general results based on these experiments. Mathematics is not about calculation.

  11. Now, contrary to the subject, many mathematicians are very good at calculation. But they don’t like to be thought of a calculators, and go out of their way to dispel this idea that they crunch numbers all day.

    Regarding your analogy to painters being judged by their ability to roll and unroll canvas, I think a better one would be an artist who cannot draw a straight line. But what artist uses straight lines? (Peit Mondrian, shut up, you are distracting me!) Variations of weight and texture are more dynamic, And plenty of artists don’t use line at all!

  12. This cracked me up. Such a brilliant post. So mathematicians do end up using calculators like the regular folks when it comes to ‘numbers’. Hmmm… very interesting. Do you enjoy teaching mathematics?
    What do you do when you have to deal with numbers?

  13. Lol, I love it. As someone with ph.d. In astrophysics I can confirm this. During my undergrad and grad classes all of my math professors stopped solving every problem when it reached arithmetic. One of them referred to it as a “farmers’ market math”

  14. This is too true for me. Always getting embaressed in front of my non-maths learning friends for being the idiot who just can’t do simple subtraction while at tescos with budget money

  15. I’ve been trying to explain this to my wife for sometime but from the opposite perspective. She keeps saying ‘you’re so good at math’ and I keep trying to explain that I’m just quick with mental arithmetic / big round number thinking. When it comes to actual MATH though I got lost somewhere around Euclid.

  16. Arithmetic bears the same relation to mathematics that penmanship does to literature. Penmanship involves producing by hand the elements that are used to produce literature, but bring good at penmanship isn’t really what we means when we say someone is good at “writing”.

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