Disclaimer: I'm a beginner with summation and sigma notation.

Background: Since division is the inverse of multiplication, and multiplication is repeated addition, it seems (at first glance) possible to rewrite $\frac a b = c$ in sigma notation--i.e., summation notation.

Assumptions: I've read that the upper bound must be a whole number, so when a, b, and c are whole numbers, this seems correct: a = $\sum_{i=1}^b c$

Problem: But that fails when b is $\frac m n$ since b isn't a whole number.

Question: Is it not always, or is it ever, possible to represent division in terms of sigma notation?


1 Answer 1


Careful there, cowboy! You need to be consistent in your assumptions throughout your reasoning.

Division is always the inverse of multiplication, but multiplication is not always repeated addition. Take, for example, $ \frac{3}{2} \times \frac{5}{4} $. How do you add $\frac{3}{2}$ to itself $\frac{5}{4}$ times (or, for that matter, add $ \frac{5}{4} $ to itself $\frac{3}{2}$ times)?

Multiplication is defined as repeated addition for integers: numbers that have vanishing fractional part. For those sorts of numbers, I'm sure you can see that $$ a = \sum_{k=1}^b{c} $$ indeed.

Defining multiplication elsewhere requires new techniques. For fractions $\frac{a}{b}$, $\frac{c}{d} $, we reuse our previous definition of multiplication of integers to define $$ \frac{a}{b}\times\frac{c}{d} = \frac{a\times c}{b\times d} $$

Defining multiplication for the real numbers (numbers with decimal expansions, even infinitely long ones) gets a bit tricky, because you need some sort of process capable of handling things with infinite precision. The trick is to define $ r\times s $ to be (assuming $r,s\geq0$) the smallest real number $ k $ characterized by the following property: given any pair of fractions $ p,q $ in the ranges $ 0\leq p\leq r$ and $ 0\leq q\leq s$, we have $pq<k$.

It's not immediately obvious how to turn these last two examples into some sort of sigma notation, although it can be done. (In fact, I think the way to do it most consonant with the rest of my answer involves reinventing the Riemann integral from calculus!)

But there's a bigger point here. You'll notice I haven't mentioned division at all. In order to write division in sigma form, you have to transform it into multiplication first, so you might as well ask the question "When and how can multiplication be written in sigma form?" As I hope I've shown, the best way to approach that is to think about how you define multiplication on whatever sort of numbers you are working with.

  • $\begingroup$ Interesting! Since $\frac 3 2 \times \frac 5 4$ = $\frac 1 2$ $\times$ $\frac 1 4$ $\times (3 \times 5)$ = $$\sum_{i=1}^{15} \frac 1 8 $$ so could we generalize to this? $$\sum_{i=1}^{ac} \frac 1 {bd} $$ But that breaks if irrational numbers are involved--e.g., a=$\pi$. For the real numbers, I think you were referring to Cauchy sequences (which is still beyond my level of math) that I read about in the Wikipedia page on Multiplication. Thank you! $\endgroup$ Mar 12, 2017 at 17:32
  • $\begingroup$ Exactly right! That's actually the generalization I was thinking of when I mentioned that you could do it in a non-obvious way. $\endgroup$ Mar 12, 2017 at 18:50
  • $\begingroup$ Actually, when I wrote my definition for the reals, I was thinking of them as Dedekind cuts, not Cauchy sequences; my set-theoretic intuition has been shaped by Rudin's Principles in that way. The definition for Cauchy sequences is really simple (you just multiply corresponding terms of the sequences), so maybe I should have gone for that. $\endgroup$ Mar 12, 2017 at 18:53

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