Uncertainties
Uncertainties should be expressed to one significant figure, and the measurement value should be given to the same number of decimal places. For example, 2.058 \pm 0.027\m should instead be given as 2.06 \pm 0.03\m.
Notation
When writing an absolute uncertainty, the unit comes after the uncertainty:
When writing a percentage uncertainty, the unit comes after the value:
Addition and subtraction
When adding or subtracting two measurements, always add the absolute uncertainties.
Multiplication and division
When multiplying or dividing two measurements, always add the percentage uncertainties.
When squaring a measurement, double the uncertainty. When taking the square root, halve the uncertainty.
Relationships
When a = k \times b, we say that a is directly proportional to b, or:
The value k is the proportionality constant. A graph mapping a to f(b) will have a gradient of k and will pass through (0, 0) (with f(x) representing the type of relationship, such as x^2 or \frac{1}{x}).
- If f(x) = x, then a is directly proportional to b. The graph is a straight line.
- If f(x) = \frac{1}{x}, then a is inversely proportional to b. The graph is a rectangular hyperbola.
- If f(x) = x^2, then a is proportional to b squared. The graph is a parabola.
Power relationships
A power relation between the variables a and b with a constant power n takes the form:
This can be graphed using a logarithmic scale on both axes to get a linear graph. If we take logarithms of each side and expand, we get:
If we compare this to the line equation y = mx + c, we can see that the gradient is given by the power n, and the y-intercept by \log k.