S1.3 Mathematics

Manipulate before substituting

Rearrange equations symbolically first, keep brackets around measured quantities, and substitute values only after isolating the required variable.

Read proportionality

If $y\propto x^n$, changing $x$ by a factor $k$ changes $y$ by $k^n$. Use this to check whether a numerical answer has the right scale.

Choose the tool

Use geometry and trigonometry for components, logarithms for exponential relationships, rates for per-time quantities and estimation for order-of-magnitude checks.

Common trap

Do not round intermediate values unnecessarily or treat a proportionality as equality without the constant.

Classify quantities

Scalars have magnitude only; vectors have magnitude and direction. Mass, power and potential difference are scalar examples, while momentum, velocity and force are vectors.

Resolve components

Choose axes, draw the vector and resolve it into perpendicular components. Recombine components with signs and geometry, not by adding magnitudes blindly.

Read a free-body diagram

Draw only forces acting on the chosen object, label directions and calculate the resultant. Reversing a force or scaling another changes the vector sum, not just its length.

Common trap

Do not add vectors as if they were scalars, and do not include a force exerted by the object on its surroundings in the object’s own free-body diagram.

Convert prefixes first

Rewrite all quantities in compatible SI units before comparing or substituting. Keep a prefix table in mind: k=10^3, m=10^-3, micro=10^-6, n=10^-9 and so on.

Round with evidence

Keep guard digits during calculations, then round the final value to the precision justified by the data or uncertainty. An uncertainty is usually reported to one significant figure unless the context requires otherwise.

Check the unit

Use dimensional analysis to test an equation and confirm that the final unit matches the requested physical quantity.

Common trap

Do not compare raw prefixes as if they were the same unit, and do not report an uncertainty with more precision than the measurement supports.

Use absolute uncertainty rules

For sums or differences, add absolute uncertainties. For products, quotients and powers, add fractional uncertainties with the appropriate power multiplier.

Convert at the end

Find the fractional uncertainty first, then multiply by the measured value for absolute uncertainty or by 100 for percentage uncertainty. Keep the physical quantity and uncertainty in matching units.

Watch derived differences

When a result is formed by subtracting two measured values, the absolute uncertainties can be large compared with the small difference, producing a larger percentage uncertainty.

Common trap

Do not add percentage uncertainties for addition, and do not report an uncertainty with a different unit from the quantity.

Plot honestly

Label axes with quantity and unit, choose a scale that uses the graph area, and plot points with uncertainty bars. A best-fit line should represent the trend rather than join every point.

Use the gradient

Read two well-separated points on the best-fit line, not necessarily two measured points. Include the gradient’s units and use the intercept when the model predicts an offset.

Test a relationship

Transform variables so the predicted relationship becomes linear. A line through the origin supports direct proportionality; a non-zero intercept can indicate an offset or systematic error.

Read special features

Use areas for accumulated quantities, maxima or minima for turning points, and error bars to judge whether differences are significant.

Common trap

Do not draw an exact line through every point or calculate a gradient from adjacent noisy points when the task asks for a best-fit gradient.