Significant figures, counting, rounding and calculation rules
× and ÷: min(significant figures)
Significant figures are a property of the writing, not of the number: "0.50", "250" and "250." carry 2, 2 and 3 figures while a calculator only sees floats. The tool therefore reads the value as written, zeros included, counts its figures and implicit uncertainty, rounds a raw value to n figures, and applies the course rule to a calculation while naming which one: multiplication and division keep the minimum significant figures, addition and subtraction stop at the least precise decimal place. Two different rules, never one.
3
- Reading
- "250." carries 3 significant figure(s)
- Scientific notation
- 2.50e+2
This writing asserts 3 significant figure(s), that is a value known to ± 0.5.
Significant figures are a property of the writing, not of the number: "0.50", "250" and "250." carry 2, 2 and 3 figures, while a calculator only sees floats. That is why this tool asks for the value as written, zeros included.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulas× and ÷: min(significant figures)
× and ÷: min(significant figures)
+ and −: least precise decimal place
"250" = 250 ± 5
"250." = 250 ± 0.5
The significant-figures rule governs multiplication and division; the decimal-places rule governs addition and subtraction. Mixing them up is the most frequent mistake of the chapter. Every writing also carries an implicit uncertainty: half of the last asserted decimal step.
- Significant figure
- · every written digit from the first non-zero one, trailing zeros included when a decimal point is present.
- Leading zeros
- · never significant: "0.0034" carries two figures, not five.
- Trailing zeros of an integer
- · ambiguous. The convention used here does not count them; "250." or "2.50e2" assert them.
- Implicit uncertainty
- · half the step of the last asserted decimal: "12.3" means 12.3 ± 0.05.
Validity domainSignificant-figure rules are a convenient approximation of true uncertainty propagation: they give the right order of magnitude, not an exact interval, and the uncertainty-propagation tool does better when uncertainties are known.
Significant-figure rules are a convenient approximation of true uncertainty propagation: they give the right order of magnitude, not an exact interval, and the uncertainty-propagation tool does better when uncertainties are known. They apply to the final result: in a multi-step calculation, keep one or two extra figures in the intermediates and round only at the end. EXACT numbers, counts and defining factors such as the 2 in 2πr, carry infinitely many significant figures and never limit the result; the tool treats both of its operands as measurements.
Reading the resultThe count is read on the written string, never on the value: that is why the entry is free-form and keeps your zeros.
The count is read on the written string, never on the value: that is why the entry is free-form and keeps your zeros. Subtracting two close values destroys figures, and that is a real phenomenon: the difference becomes smaller than the uncertainty of the measurements, and the decimal-places rule records it where a figure count would miss it.