Physics › Maths bridge › Significant figures, units and checking
Significant figures, units and checking
Rounding to a precision the data justify, units carried through algebra, and the homogeneity check.
Rounding a result
Keep extra digits during working and round the final result to a precision justified by the measurement uncertainties. Rounding part-way through moves the answer away from the value the data support.
Quoting the result to the fewest significant figures in the data is a useful school-level rule for multiplication and division when uncertainties are not given; for addition and subtraction, decimal-place precision matters. Exact constants do not limit the result.
Significant figures are not decimal places: 0.0025 has two significant figures but four decimal places.
Checking an equation by its units
Every equation in physics must be homogeneous: the units on both sides match. Carrying units through the algebra catches many rearrangement errors.
Homogeneity is necessary but not sufficient. A homogeneous equation may still carry the wrong numerical factor or the wrong sign, so passing the check does not prove an equation correct; failing it proves one wrong.
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 immediately.
GUIDED PRACTICE
Rounding a calculated resistance
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 result should be quoted to before opening the working.
Show the working
R = 3.157/0.62 = 5.0919… Ω on the calculator.
No uncertainties are stated, so the school-level rule for a division applies: the current carries only two significant figures, so quote 5.1 Ω. The extra figures on the pd cannot improve the precision the current allows.
Where physics leans on this: SI base and derived units · Estimation and orders of magnitude. All eight skills: the maths bridge.