Maths bridge › Significant figures, units and checking
Significant figures, units and checking
Rounding that matches the data, units carried through algebra, and the homogeneity check.
Sig figs follow the data
Quote answers to the same number of significant figures as the least precise input, and round only at the end: intermediate rounding walks the answer away from the truth one nibble at a time. Significant figures are not decimal places; 0.0025 has two of the former and four of the latter.
Units are a free error detector
Every equation in physics must be homogeneous: the units on both sides must match, because only like can equal like. Carrying units through the algebra costs seconds and catches rearrangement errors that nothing else will.
WORKED EXAMPLE
Checking a wave-speed formula
On a stretched string, v = √(T/μ), with tension in newtons and mass per unit length in kg m−1. Check the units.
A newton is kg m s−2. Dividing by kg m−1 gives kg m s−2 × m kg−1 = m2 s−2.
The square root of m2 s−2 is m s−1: a speed. The formula is homogeneous, and any misremembered version (T μ, or μ/T) would have failed the check.
YOUR TURN
Round like the data deserves
A resistor passes 0.62 A at a measured pd of 3.157 V. Compute R = V/I and decide how many significant figures the answer has earned before opening the working.
Show the working
R = 3.157/0.62 = 5.0919… Ω on the calculator.
The current carries only two significant figures, so the answer is 5.1 Ω. The voltmeter's four figures cannot rescue the ammeter's two.
Where physics leans on this: SI units and prefixes · Estimation and orders of magnitude. All eight skills: the maths bridge.