# Why don't we use absolute error while calculating the product of two uncertain quantities?

I've found a rule that says, "When two quantities are multiplied, the error in the result is the sum of the relative error in the multipliers."

Here, why can't we use absolute error? And why do we've to add the relative errors? Why not multiply them?

Please give me an intuition to understand the multiplication of two uncertain quantities.

• You use the term error, and then in your last sentence mention uncertainty. An error is different from an uncertainty, as the latter is usually taken to be a variance/standard deviation and propagates differently than an approximation error. Which of the two do you mean? Jul 31 at 10:38
• Related: Is propagation of uncertainties linear? and links therein. Jul 31 at 11:56

It basically comes from calculus (or more generally just the mathematics of change).

If you have a quantity that is a product $$z=x\cdot y$$, then the change in this value based on the change of $$x$$ and $$y$$ is$$^*$$ $$\Delta z=x\Delta y+y\Delta x$$. So then it is straightforward that

$$\frac{\Delta z}{z}=\frac{x\Delta y+y\Delta x}{xy}=\frac{\Delta x}{x}+\frac{\Delta y}{y}$$

The reason you don't use absolute uncertainty or multiply the relative uncertainties is the same reason why $$(a+b)^2\neq a^2+b^2$$. It's just not the result you get when you do the math.

$$^*$$We are neglecting the term $$\Delta x\cdot\Delta y$$ in $$\Delta z$$, since ideally $$\Delta x and $$\Delta y to the extent that $$\Delta x\cdot\Delta y\ll xy$$ such that $$\Delta x\Delta y/xy$$ is much less than both $$\Delta x/x$$ and $$\Delta y/y$$.

• I think you should add that it's somewhat similar to the Leibnitz product rule as well Aug 1 at 7:46

You get the best intuition if you just take two easy numbers with a possible error and multiply it. Choose 100±1 and 200±4 the relative errors are 1/100 or 1% and 4/200=2%.
Now multiply and you get for the positive error 101*204=20604=20000+604 or an error of about 3%. Multiplying the absolute error would give you 1*4 instead of 604, multiplying the relative errors would give you 2/10000 or 0.02%.
Try it with any two other numbers.