Reference Library

The International System of Units (SI) defines seven base quantities. Every other physical unit is formed by multiplying and dividing these seven.

Every physical quantity has a dimension built from the base quantities. An equation can be correct only if every term shares the same dimension — a check that catches most algebraic errors without needing any numbers.

An order-of-magnitude estimate gives the power of ten nearest a quantity's true value. Estimating each factor in a calculation to one significant figure, then combining them, is often enough to judge whether a detailed result is plausible.

Significant figures indicate how precisely a quantity is known. In multiplication and division, a result carries as many significant figures as its least precise factor. In addition and subtraction, a result carries as many decimal places as its least precise term.

A vector has both magnitude and direction; a scalar has magnitude alone. Vectors combine by the same rules regardless of what they represent.

A vector is resolved into components along chosen axes, each multiplying a unit vector of length one.

The dot product returns a scalar and measures how much two vectors point the same way. The cross product returns a vector perpendicular to both, with magnitude equal to the area they span.

Symmetry in a problem should guide the choice of coordinates: Cartesian axes suit straight-line symmetry, polar or cylindrical coordinates suit circular symmetry, and spherical coordinates suit a single central point. A well-chosen system reduces the number of non-zero components, and often eliminates a coordinate entirely.

A derivative gives the instantaneous rate of change of one quantity with respect to another — the slope of its graph at a point.

An integral accumulates a quantity that varies continuously — the area under a graph, or the total change built up from a rate.

The gradient of a scalar field points in the direction of its steepest increase, with magnitude equal to the rate of increase in that direction. Many force and field laws are gradients of a potential.

Three forms recur throughout physics: exponential growth or decay (dy/dx = ky), oscillation (d²y/dx² = −ω²y), and relaxation towards equilibrium. Recognizing the form immediately gives the shape of the solution — exponential, sinusoidal, or exponential approach — without solving from scratch.

Measurement uncertainty has two sources. Random error scatters repeated measurements around the true value and shrinks with averaging. Systematic error shifts every measurement the same way and does not improve with repetition. Precision describes the scatter; accuracy describes closeness to the true value.

When a result is calculated from several measured quantities, their uncertainties combine to produce an uncertainty in the result.

Fitting a model to data finds the parameters that minimize the residuals — the differences between measured values and the model's predictions. Least squares, the most common criterion, minimizes the sum of squared residuals. Residuals scattered around zero with no remaining trend are evidence the model form is appropriate.

A measured value is reported with its uncertainty and units, rounded so both share the same last decimal place — 9.81 ± 0.02 m/s², not 9.81 ± 0.023 m/s². A number reported without its uncertainty implies unwarranted confidence in every digit shown.