Revision summary

State a result's value, its unit where applicable, and justified precision. Explain dimensionless ratios, and include direction for a vector. Use the linked explanations when you need to reconstruct a method.

Base units and prefixes

Preserve case, including milli m and mega M. Square an area conversion factor and cube a volume conversion factor. A quantity symbol and a unit symbol have different roles.

Volume V, or v under a stated convention, has SI unit m³. Use volume for density, the selected contact face for average pressure, and total outside area for a complete coating. Count all six block faces, both cylinder ends when included, and the whole spherical surface; do not substitute a sphere's projected circle for its surface area.

Temperature symbols θ or T follow the stated convention. Thermodynamic temperature uses K; Celsius temperature uses °C. T/K = θ/°C + 273.15, and a temperature interval has equal numerical values in K and °C. Do not use ratios of Celsius readings as thermodynamic-temperature ratios.

Force

N = kg m s^-2

Energy

J = N m = kg m² s^-2

Power

W = J/s = kg m² s^-3

Charge

C = A s

Potential difference

V = J/C = kg m² s^-3 A^-1

Pressure

Pa = N/m² = kg m^-1 s^-2. The fixed unit definition is 1 atm = 101325 Pa; actual atmospheric pressure varies.

Homogeneity: compare every added term and both sides using base units. A unit check can reject an equation, but cannot prove it or determine a dimensionless numerical factor. For an estimate, state plausible assumptions and avoid false precision.

Errors and measurement choices

• Accuracy: closeness to a reference. Precision: agreement between repeats.
• Random variation: produces scatter; suitable repetition and a mean can reduce its effect on the estimate.
• Systematic effects: persist in the mean. Correct a stable zero offset by subtracting its signed value; check calibration for a scale-factor error.
• Resolution: smallest indicated change, not automatically the total measurement uncertainty.
• Choose enough range and suitable uncertainty, then control alignment, zero, loading and timing as appropriate. Match each improvement to the actual limitation.

The arithmetic mean is <x> = x = (Σx_i)/n: add the specified readings and divide by their count. The mean has the same unit as the readings.

Derived uncertainty

Absolute, fractional, percentage

Δx has the unit of x and is nonnegative. Fractional = Δx/|x|; percentage = fractional × 100%.

Sum or difference

Add the relevant absolute uncertainty contributions conservatively. A common fixed offset can cancel in a difference.

Product or quotient

For small relative uncertainties, add fractional or percentage contributions.

Power xⁿ

Multiply its fractional contribution by |n|. An exact constant adds no measured uncertainty.

Numerical limits

Choose input combinations from the expression's behaviour. For positive L/t, maximum uses high L and low t; minimum uses low L and high t.

Retain guard digits, then report an uncertainty and value with compatible precision. Large relative uncertainties or a denominator range containing zero require more care. These uncertainty estimates are not automatically statistical confidence intervals.

≪ means much less than; ≫ means much greater than. A small-uncertainty approximation needs a suitable scale comparison such as Δx/|x| ≪ 1. Lowercase δx may denote a small signed change, while this section uses Δx for nonnegative uncertainty; follow the stated convention.

Vectors and components

• Scalars include mass, temperature, energy and speed. Vectors include displacement, velocity, acceleration and force. A minus sign alone does not define a vector.
• Addition: preserve arrow lengths and directions, place them head to tail, and draw the resultant from the first tail to the final tip.
• Subtraction: A - B = A + (-B). Reverse B, then add; do not merely subtract magnitudes.
• Resolution: adjacent component uses cosine, opposite component uses sine. Identify the reference angle and choose signs from the axes.
• Reconstruction: magnitude = √(x² + y²); determine direction with the correct quadrant.
• Velocity change: Δv = v_final - v_initial. Its magnitude need not equal the change in speed.

Components are an equivalent representation of the original vector. Do not count them as extra forces alongside it.

A triangle's interior angles sum to 180°. Similar triangles have equal corresponding angles and proportional corresponding sides, so scaling a vector at a fixed angle scales both of its perpendicular components by the same factor.