How to calculate uncertainties?

Question

A question requires me to make a calculation involving variables with uncertainties, giving my answer with its result uncertainty.

How do I do that?

Answer 1

Firstly, you should understand fractional uncertainty $= \frac{\Delta x}x$ and percentage uncertainty $= \frac{\Delta x}x \times 100$

Also be careful where you are given a length with its uncertainty and required to calculate, for example, density of a cube. The volume of the cube is the length$^3$, and so the power rule applies (see below).

Now for some examples:

1. Addition/Subtraction.
   $x = a \pm \Delta a$
   $y = b \pm \Delta b$
   $x + y = (a + b) \pm (\Delta a + \Delta b)$
   $x - y = (a-b) \pm (\Delta a + \Delta b)$
2. Multiplication & Power rule.
   $x = abc^2$
   $\Delta x = \pm \left[ \frac{\Delta a}a + \frac{\Delta b}b + 2\frac{\Delta c}c \right]\cdot abc$
3. Division & Power rule.
   $x = \frac{ab}{c^2}$
   $\Delta x = \pm \left[ \frac{\Delta a}a + \frac{\Delta b}b + 2\frac{\Delta c}c \right]\cdot \frac{ab}{c^2}$
4. $\log$ / $\ln$
   $T = x \pm \Delta x$
   $\log T = \log x \pm \Delta \log T$
   $\Delta \log T = \pm \left[ \log (x + \Delta x) - \log x \right]$

Leave the result uncertainty to the same number of significant figures as the given uncertainty.

Measures you should know:
\begin{align} 10^{12} & = \text{Tera (T)} \\ 10^9 & = \text{Giga (G)} \\ 10^6 & = \text{Mega (M)} \\ 10^3 & = \text{kilo (k)} \\ 10^{-1} & = \text{deci (d)} \\ 10^{-2} & = \text{centi (c)} \\ 10^{-3} & = \text{milli (m)} \\ 10^{-6} & = \text{micro ($\mu$)} \\ 10^{-9} & = \text{nano (n)} \\ 10^{-12} & = \text{pico (p)} \end{align}