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It is mentally more difficult to divide 2 numbers than it is to multiply them. If you ask me what is 3 * 27, i will immediately tell you 'a little less than 90'.

However, I really have no idea what is 3 / 27. It is much harder to do this operation, I cannot even get an approximate estimate value, it just seems really difficult, and the logic on how to proceed is not at all obvious.

Multiplication and division are both fundamental mathematical operations that we learn in elementary school, and theoretically they are on the same footing, and yet division takes about 5 times more mental effort than multiplication.

I know that the question is mathematical, but it seems to me that it has to do with the mind / psychology, evolution or phenomenology. I doubt mathematicians can answer this question.

Similar question: Why is Division harder than Multiplication?

What is the reason?

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  • 4
    But is it really a philosophical question? The feeling of mental effort is subjective. It's a bit like asking - why is it harder to run than to walk, no?
    – Frank
    Feb 19, 2023 at 4:36
  • 7
    I dunno, honestly. 3/27 is "a little more than 0.1", a result I came up with about as fast as "3*27=a bit less than 90". An exact result would indeed be more difficult, because it mobilizes decimal numbers which are less instinctive than integers to manipulate in one's head. Also on the specific case of 3/27 the exact result has an infinite number of digits, which will definitely never happen with multiplication.
    – armand
    Feb 19, 2023 at 5:48
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    Division is not "fundamental", it is introduced as the inverse of multiplication. More to the point, kids learn multiplication table, but not division table. The mental conversion takes extra effort. Also, in your example the result of division is a fraction rather than an integer, which takes even more effort to process. On computers typical division algorithms also run longer than multiplication algorithms, although computers implement operations differently than human neural network.
    – Conifold
    Feb 19, 2023 at 6:06
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    The example given in the question confuses the issue a bit, because we can see "3x27 is a little less than 90" just as fast as we can see "3/27 is a little more than 1/10" just by rounding 27 to 30. In fact, we can immediately recognise that 3x27 is 81 and 3/27 is 0.11111... just by knowing our multiplication tables, without actually doing any calculation.
    – Stef
    Feb 19, 2023 at 17:16
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    @AgentSmith That doesn't seem true to me at all (the implicit claim in the question doesn't seem true to me, either). For example, if you ask me 374*187, I might take a while even to give an estimate, but I will give you an exact answer to 374/187 pretty much instantly. Feb 19, 2023 at 20:05

9 Answers 9

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We can argue that multiplication is a forward reasoning process, from causes to effects, while division is a backward reasoning process, from effects to causes. Backward reasoning is generally more difficult.

Multiplication is forward: If I multiply 3 by 27, what will I get? This is analogous to: If I set a marble on a slope, where will it roll?

Division is backward: 81/3 is really the question "What did I need to multiply 3 by in order to get 81?" This is analogous to looking at a marble a little way from a slope and trying to infer what point it started rolling from, a harder problem.

(Here I am accepting the premise of the problem for the sake of argument, and I think there's some truth to it.)

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    Why do you think the second analogy is a harder case than the first? (Maybe poor analogy?)
    – minseong
    Feb 19, 2023 at 23:20
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    I think what you call forward reasoning is also called deduction, and backward reasoning is called induction. So you could also say it's analogous to calculus, where differentiation (by hand) is easier than integration (by hand).
    – Flounderer
    Feb 20, 2023 at 2:37
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    From a pure mathematical point of view the distinction you make does not exist. Which number do I have to divide by 3 in order to get 27? Is a backwards formulation of multiplication and If I divide 81 into 3 equal pieces, how large are the pieces is a forward formulation for division.
    – quarague
    Feb 20, 2023 at 7:30
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    @quarague and if we were on math.se, I would have given a completely different answer.
    – usul
    Feb 20, 2023 at 14:15
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    I'm not convinced by this. Can you give a more concrete basis for relating the operators to backwards & forwards in time? Is a marble really rolling down a slope like multiplication? Perhaps you mean something like, vector-sums are like multiplication, & don't have a formal inverse (because they aren't a division algebra)
    – CriglCragl
    Feb 20, 2023 at 23:32
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The answers on the question on Math.SE you cite provide an insight that what you call more difficult might in fact be more perceived difficulty:

The usual algorithms for multiplication and division, that is those that are commonly taught to school kids, are such that multiplication is computationally simpler to carry out. On average multiplication requires fewer steps in the algorithm. For instance multiplying two three-digit numbers is quite easy but dividing even single digit numbers (e.g., 1/7) may require lots of iterations compared to input size. Another cause of the perceived difficulty of multiplication vs. division is that the multiplication table for the first ten numbers is commonly drilled to death, and is commutative, so one only needs to memorize about half of the entries. The division table for the same numbers is usually not drilled at all (not even shown) and is not symmetric so memorizing it will require more work.

and another answer:

If you use logarithm tables then dividing is no harder than multiplying, because subtraction is no harder than adding. [...] That's how it was done before electronic calculators.

Yes, it requires more calculations for both human and machines, but to call that lot more mental effort... I wouldn't go that far, specially when making approximations and not exact results. In fact, at O-notation complexity level both are constant (O(1)) on a computer.

As a bonus, a related, more interesting question can be found in MathOverflow.SE (linked in the Math.SE post mentioned): Why is differentiating mechanics and integration art?. This, I would indeed call a lot more mental effort, with no mental shortcuts.

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Division is nothing more than the multiplication of one number by the inverse of another number:

5/6 = 5 * 1/6

Computing the inverse of a number is the extra step that is not required in the multiplication of two numbers and that's why it's more complicated. By inverse I mean that a number multiplied by its inverse = 1

In order for division to be as "easy" as multiplication, all you have to do is memorize the inverse of all the numbers that you memorized in your multiplication tables. But requiring your number to be in decimal (base 10) format makes the calculations intensive.

Finally I will argue that any division that produces a whole number will be calculated nearly as fast by a person as multiplication

Quick!!! 20/5 81/9 100/10. See what I mean.

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  • Are you saying that when you have to calculate 5/6, you first calculate 1/6 = 0.1666..., and then you multiply that number by 5?
    – Stef
    Feb 19, 2023 at 17:22
  • 1
    Yes. The short hand division techniques you learned in school obscure it. They take advantage of your memorization of the multiplication tables so 27/3 = 9 is really from memory and you don't have to convert to decimal and then multiply.
    – user64314
    Feb 19, 2023 at 17:27
  • I didn't learn any shorthand division techniques in school. I only learned long division. To calculate 5/6 I either rewrite it as 1 - 1/6, or I do long-division, finding 5 = 0 + 5, 50 = 48+2, and 20 = 18+2, concluding that 5/6 = 0.8333... I would not even have thought about multiplying 5 and 0.1666..., it sounds way too bothersome!
    – Stef
    Feb 19, 2023 at 17:32
  • By this logic, one could argue that multiplication is just division of one number by the inverse of another. So multiplication is division with an extra step.
    – Opifex
    Feb 19, 2023 at 19:12
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The first thing to consider is training.

You can immediately tell that 3 x 27 is approximately 90, probably because you notice 27 is not so far from 30 and you know what 3 x 30 is.

Being trained as a physicist and having to divide things fairly often, I can immediately tell you tat 3/27 is not so far from 0.1, because I know that 3/30 = 0.1, and thanks to my experience I could recognize it[1].

However I think there is more than that.

I would say it hangs on is that multiplication and division are fundamentally different when you consider how they interact with ensemble of numbers. When you multiply two integers, you get an integer. When you divide an integer by another, you can get a rational number. Division is powerful enough that it can produce numbers of a new kind.

Now if you work in the space of rational numbers, and accept the true representation of number to be of the form a/b, then the two operations are as simple.

a/b x c/d = (a x c)/(b x d) [2 integer multiplications]

(a/b) / (c/d) = (a x d)/(b x c) [2 integer multiplications]

Of course you then need more work if you want to convert them from or to decimal numbers. A work that is much simpler to do for multiplication[2]. But this is not the division fault, we just prefer our base 10 representation to rational numbers[3].

To conclude, I think that division is more complicated on integers, because integers are fundamentally not suited to represent the result of a division. And since human brains and computer alike prefer integers over other number formats[4], division is harder for us.

Another related question is why we consider 0.1111 more informative than 3/27 to begin with, they are after all representing the same number. Strictly speaking there is no information gained by performing the computation.


[1] A poor estimate still, a more experience person would have recognized 3/27 = 3/3^3 = 1/9 = 0.1111 immediately.

[2] Because in decimal representation, numbers look like a/10^n, so that multiplication is a/10^n x b/10^m = (a x b)/10^(n + m) which is immediately a decimal number. While the division is (a/10^n) / (b/10^m) = 10^(n - m) x a/b and we end up having the same problem as with integers, that a/b can not be represented as a decimal number in general.

[3] In fact with training you start to appreciate other format, and for many scientists the answer to 3/27 really is 1/9.

[4] A decimal number is really just an integer with a dot.

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In the case of addition and multiplication commutative property is satisfied. It is easier to understand a problem that can be considered in a given order and vice versa. Since understanding is rather difficult, a lot more mental effort is needed in subtraction and division.

See, what happens when it satisfies commutative property.

When the numbers are the same we need not have more mental effort. (except in the case of zero)

Eg:

a) 2/2

b) 1.23456789/1.23456789

These two are interchangeable. Therefore they are commutative. And division becomes easier.

So we don't need to take the inverse of the divisor. That is, we need not take division as multiplication by the inverse (In the first eg: 2/2 as 2*1/2) and argue that more mental effort is needed for division because of this type of complexity.

We know that fractions (or decimals) are more difficult to comprehend (and to express in written form) than whole numbers. So,another significant reason for complexity is,

Among the four fundamental operations in maths, division is the only operation that produces fractions using whole numbers. And more effort is needed to express fractions (or decimals).

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  • you are saying that subtraction and division are not commutative? Feb 19, 2023 at 14:00
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    Without making changes in the numbers, just by interchanging the numbers we will not get the same answer. So subtraction and division are not commutative. Feb 19, 2023 at 14:12
  • 1
    This is an important property, I think, indeed. When I see 3 * 27 or 27 * 3 I will evaluate both as roughly 30 + 30 + 30. On the other hand, while 27 / 3 is easy (I know my multiplication tables) I can't use its result for 3 / 27 immediately: I can reduce it to 1/ 9, but I still then need to evaluate that fraction. Feb 20, 2023 at 11:40
  • 1
    @DennisKozevnikoff Subtraction and division are not commutative: 1 - 2 and 2 - 1 are different, and the same holds for 1 / 2 vs 2 / 1
    – j4nd3r53n
    Feb 20, 2023 at 15:49
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As I understand it, and as far as I can find, only one animal has been able to learn to do division, Alex the African Grey parrot. Whereas some capuchins can do multiplication, also apes, parrots, and other animals.

In Abstract Algebra, to be a Division Algebra is quite a strict criteria on a number system, and is limited to real and imaginary numbers over 1, 2, 4, or 8 dimensions. This has been linked to fundamental physics, in M-Theory.

Whole number multiplication will only include whole number answers, and finite rational numbers. Whereas finding the number that multiplied by itself gives two, involves an irrational number that we can only approximate in calculations.

Operators seem very intuitive to us, but a good case can be made that their fundamental mechanism is one of sorting: 'Richard Feynman Computer Science lecture, hardware software and heuristics'. In this view of code as more fundamental than math, we can understand that division is just a more intensive process on average than multiplication, and so bound to be more cognitively taxing. See details of the algorithms involved relating to computational intensity here: 'Why is division more expensive than multiplication?' on StackOverflow

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I think the real question is: Why are multiplication and addition so easy? Most mathematical operations on two numbers are not straightforward - just take exponentiation or logarithms.

If you start with the natural numbers: 1, 2, 3, ..., then addition and multiplication are simple to understand: if you have a bag of 5 apples and another with 7 apples, how many do you have? You just count the apples in the first bag, then continue with the second: that's addition, and multiplication comes from counting the total number of apples in a number of same-sized bags. Those operation are easy to understand, and all you need is to learn the addition- or multiplication tables.

Division is more awkward: you have your apples and want to divide them up into piles of the same size, and very often there are some left over. And of course, as others have pointed out, division is not commutative, so you can't rearrange the expression. There is a trick to making easier, sometimes, for whole numbers: Prime factors. Most whole numbers are a product of multiple prime numbers - the prime factors that occur both above and below in the fraction can be eliminated, thus simplifying the expression, for example:

27 / 75 = 3 * 3 * 3 / 3 * 5 * 5 = 3 * 3 / 5 * 5 = 9 / 25 ~ 10 / 25 = 2 / 5 = 4 / 10 = 0.4

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Division of small numbers doesn’t usually produce simple results. 2x3 = 6. Simple. 2/3 = 0.66666. Not simple.

The method that we learn at school is that we look at the first two digits or so and guess what the first digit of the result is. And then we multiply the divisor by that digit and subtract it from the dividend. And hope we didn’t guess too high. And if we guessed too low then we have to correct it. It is complicated. Multiplication, you just write down a few rows of numbers.

For computers, the difference is huge. Your computer can multiply all digits of a product simultaneously. But for division, it does it like humans. It guesses two bits or four bits of the result, then adjusts the dividend, and repeats. That usually takes about 10 times longer.

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As other answers point out, division with small numbers often leads to "not nice" results...but mostly what happens is...

A self-reinforcing lack of practice and familiarity. You practice addition from the time you're an infant. You practice multiplication a lot in other areas of math, why? Because students are more comfortable with multiplication and the teacher doesn't want to deal with remedial division practice. Now the students are even more comfortable with multiplication and division lags even further behind.

But the areas that you do practice probably feel perfectly comfortable and natural. If I ask what's 187/25, you might hesitate. If I ask you how many quarters (25 cent pieces) in $1.87, you can probably see right away that there's 7 quarters (plus 12 cents left over). 270/27 = ? Easy. 400/33 = ? Probably also pretty simple. But something like 100/12.5 is a little harder because you don't use it as much.

Some of these involve "tricks", not direct division--but that's the same as multiplication. You can see this with multiplication by 6, 7, and 8...if you think multiplying 7*8 is harder than 7*9 because 7*9 has a nice trick you can use. Or for that matter 7*19 (multiply by 20 and then subtract 7). You're just a lot more comfortable with the multiplication tricks because you use them all the time.

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