units - Why are significant figure rules in Multiplication/Division different than in Addition/Subtraction?

I've never understood specifically why this is.

Here's what I mean.

---

In Addition/Subtraction, what matters are the digits after the decimal point. So for example:

1.689 + 4.3 =

  1.629
+ 4.3XX

---------
  5.929
---------
  5.9

This makes sense to me. I filled in uncertain values with X, and it makes sense why I can't use the 0.029 in the answer - because I added it to an uncertain value.

---

However, I don't understand the rules when it comes to Multiplication/Division. The same little trick with X's doesn't help me here.

I know that what matters in Multiplication/Division are the significant figures. So for example:

12.3 * 4.6 =

  12.3
*  4.6
-------
   738
  492X
-------
  56.58
-------
  57

The answer is 57 according to significant figure rules of Multiplication/Division, but I just can't make sense of those rules like the way I did with Addition/Subtraction.

Does anyone have an intuitive explanation for the significant figure rules of Multiplication/Division?

Answer

The number of significant figures is a representation of the uncertainty of a number. $123.4$ has an uncertainty of $0.1$ since the first uncertain digit is usually included. So, your multiplication of $$12.3 \times 4.6$$ is better represented as $$(12.3 \pm 0.1) (4.6 \pm 0.1).$$ I'm going to use @barrycarter's trick of using scientific notation to better represent the values $$(1.23 \pm 0.01) \cdot 10^1 \times (4.6 \pm 0.1) \cdot 10^0.$$ $$(1.23 \pm 0.01) (4.6 \pm 0.1) \times 10^1.$$ From this we can see that the number with more significant digits has less relative uncertainty. Relative uncertainty is approximately the measured uncertainty (also called absolute uncertainty) divided by the value.

Multiplying out: $$((1.23)(4.6) + (1.23)(\pm 0.1) + (4.6)(\pm 0.01) + (\pm 0.01)(\pm 0.1)) \times 10^1.$$ $$(5.658 + (\pm 0.123) + (\pm 0.046) + (\pm 0.001)) \times 10^1.$$ The number of digits we can write down is determined by the total uncertainty, which here is dominated by $\pm 0.123$. So, our result is $$(5.658 \pm 0.123) \times 10^1.$$ $$(5.7 \pm 0.1) \times 10^1.$$ $$57 \pm 1$$ which results in $57$.

To summarize, in multiplication, the largest relative uncertainty dominates the uncertainty of the final answer. This translates into the the number with fewer significant digits determining the significant digits of the final answer. This contrasts with addition, where the largest absolute uncertainty determines the uncertainty of the answer.