Why is Division harder than Multiplication?

Both conceptually and computationally it feels easier to see that:

$6 \cdot 3.7 = 22.2$

than it is to see that

$22.2 \div 6 = 3.7$.

Thoughts about the roots of this asymmetry?

An analogous question might be asked of anti-differentiation and differentiation...

• On the last point, see math.stackexchange.com/questions/20578/…. – lhf Oct 16 '14 at 10:13
• Informally, both division and anti-differentiation require some degree of guessing. – lhf Oct 16 '14 at 10:16

3 Answers

If you use logarithm tables then dividing is no harder than multiplying, because subtraction is no harder than adding.

To find $A \div B$ :

• look up $a = \log A$
• look up $b = \log B$
• find $c = a - b$
• look up $C = 10^c$

Then you have $C = A \div B$. That's how it was done before electronic calculators.

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.

The story with derivation vs. anti-derivation is quite different. It is easy to compute the derivative of even mildly complication elementary functions because the rules of differentiation of sums, products, compositions, and the fundamental functions are quite straightforward. For integration though, there are very few general rules and they only apply to very particular forms of the integrand. Moreover, the integral of an elementary function need not be elementary at all so we really can't expect anything systematic. I'm not sure what you'd accept as the reason for this other than it is a fact of life.

The situation becomes somewhat more comparable to that of multiplication vs. division when considering derivative and anti-derivative of analytic functions. Given a power representation of an analytic function one can differentiate and integrate term by term and so the problem of obtaining the integral or derivative as a power series becomes automatic, with differentiation requiring multiplying the coefficients by integers, and integration requiring division.

I'm not sure division really is harder; I think it's just that the common algorithm for division is harder. DanielV's answer points out that if you calculate via a table of logarithms, multiplication and division are equally easy. (Or equally difficult, I suppose.) The so-called “russian peasant algorithm” for multiplication has a corresponding algorithm for division that is almost identical, neither easier nor harder. To calculate $23\times 57$ with this algorithm, we write two columns of numbers, starting with $1$ and $23$, and each line contains numbers that are twice the previous line:

$$\begin{array}{crr} & 1 & 23 \\ & 2 & 46 \\ & 4 & 92 \\ & 8 & 184 \\ & 16 & 368 \\ & 32 & 736 \\ \end{array}$$

Then we mark the rows with stars so that the left-hand numbers add up to $57$. We do this by subtracting the left-hand numbers successively from 57, starting at the bottom, until the total reaches 0:

$$\begin{array}{crr} * & 1 & 23 & 1\\ & 2 & 46 \\ & 4 & 92 & \\ * & 8 & 184 & 9\\ * & 16 & 368 & 25\\ * & 32 & 736 & 57\\ \end{array}$$

(Here $57-32 = 25; 25-16 = 9; 9-8=1,$ and $1-1=0$.)

Finally we add the numbers from the middle column in the starred rows, obtaining $23\times 57 = 23 + 184 + 368 + 736 = 1311$.

To perform division is almost the same. To divide $1370$ by $29$ we write two columns as before:

$$\begin{array}{crr} & 1 & 29 \\ & 2 & 58 \\ & 4 & 116 \\ & 8 & 232 \\ & 16 & 464 \\ & 32 & 928 \\ & 64 & 1856 \\ \end{array}$$

Then we subtract the right-hand numbers from $1370$, starting from the bottom, marking the rows where subtraction is possible:

$$\begin{array}{crrr} *& 1 & 29 & 7\\ * & 2 & 58& 36\\ * & 4 & 116 & 94\\ * & 8 & 232 & 210 \\ & 16 & 464 \\ * & 32 & 928 & 442\\ & 64 & 1856 & 1370\\ \end{array}$$

Then we add the left-hand numbers in the starred rows, obtaining the quotient $1+2+4+8+32 = 47$; the remainder, $7$, is in upper-right-hand corner.