AcademyMeasurement and Vectors
Academy
Significant Figures
Level 1 - Physics topic page in Measurement and Vectors.
Principle
Significant figures report precision after the physical calculation is complete.
Notation
\(s\)
number of significant figures
none
\(d\)
decimal place
none
\(x_{\mathrm{raw}}\)
unrounded result
varies
\(x_{\mathrm{reported}}\)
rounded result
varies
Method
Rounding is a reporting step; guard digits preserve the calculation until the final value is chosen.
Guard digits
\[x_{\mathrm{raw}}\rightarrow \text{keep extra digits}\]
Products
\[\times,\div\Rightarrow s_{\mathrm{reported}}=\min(s_1,s_2,\ldots)\]
Sums
\[+,-\Rightarrow d_{\mathrm{reported}}=\text{least precise decimal place}\]
Round once
\[x_{\mathrm{raw}}\rightarrow x_{\mathrm{reported}}\]
Rules
Multiply divide
\[s_{\mathrm{reported}}=\min(s_1,s_2,\ldots)\]
Add subtract
\[d_{\mathrm{reported}}=\text{least precise decimal place}\]
Rounding error
\[|x_{\mathrm{reported}}-x_{\mathrm{raw}}|\le\frac{1}{2}\text{ place value}\]
Examples
Question
Report
\[2.36\times4.1\]
with appropriate significant figures.Answer
The inputs have 3 and 2 significant figures, so the product has 2:
\[2.36(4.1)=9.676\rightarrow9.7\]
Checks
- Exact counted numbers do not limit precision.
- Leading zeros are placeholders.
- Captive zeros are significant.
- Scientific notation makes precision explicit.