Civil Thinking › Surveying Tutorials Module 2: Units, Significant Figures, and Field Notes

Significant Figures: Rules and Examples

Significant figures represent the precision of a measured value. They include all reliably known digits, plus one final digit that is estimated based on the instrument’s sensitivity. For example, a length measured as 114.8 mm using a ruler with 1 mm divisions includes three certain digits (114) and one estimated digit (0.8), totaling four significant figures.

Instrument Sensitivity and Significant Figures

In civil engineering, measurements are central to accuracy in design, construction, and surveying. The number of significant figures in a measurement depends on the precision of the instrument used. Instruments with finer least counts can detect smaller differences and thus record more significant figures.

For example, measuring the length of a beam:

Using a standard measuring tape (least count: 1 cm) may yield 12 cm → only 2 significant figures.
Using a ruler with millimeter divisions (least count: 1 mm) may give 12.0 cm → 3 significant figures.
Using a digital caliper (least count: 0.01 cm) may show 12.00 cm → 4 significant figures.

Though all three instruments measure the same object, the recorded values differ in significant figures due to the instrument’s sensitivity. A more precise tool allows civil engineers to capture and report measurements with greater confidence.

Key takeaway: The more precise the instrument, the more significant figures the measurement will have. Always choose an appropriate tool based on the required accuracy for structural, geotechnical, or surveying tasks.

Not all digits in a number contribute to its precision. Here’s how to tell which digits count:

Leading zeros—those appearing before the first non-zero digit—are not significant. For instance, in 0.056, the two zeros only serve to locate the decimal point. The significant figures are 5 and 6.
Trailing zeros—those after non-zero digits—are only significant if there’s a decimal point present. In 1500, the zeros may not be significant unless written as 1500. or in scientific notation like 1.500 × 10³.
Zeros between non-zero digits are always significant. For example, 305 has three significant figures: 3, 0, and 5.

Rules for Identifying Significant Figures

All non-zero digits (1–9) are always significant.
Any zeros between non-zero digits are significant.
Leading zeros (before the first non-zero digit) are not significant.
Trailing zeros are significant only when a decimal point is present.

Examples (Tabulated)

Table 1: Counting Significant Figures

Number	Evaluation	Significant Digits	No. of Significant Figures
0.0030420	Leading zeros (0.00) are not significant, 0 between 3 and 4 is significant, trailing zero after decimal is significant	3, 0, 4, 2, 0	5
3620	3, 6, 2 are significant, trailing 0 without decimal is not significant	3, 6, 2	3
3040	3 and 4 are significant, 0 between them is significant, trailing 0 without decimal is not significant	3, 0, 4	3
50050.	All digits including trailing 0 are significant, decimal makes last zero count	5, 0, 0, 5, 0	5
0.00301	Leading zeros not significant, 3, 0 (between), and 1 are significant	3, 0, 1	3
0.04030	Leading zeros not significant, digits 4, 0 (between), 3, and trailing 0 (after decimal) are significant	4, 0, 3, 0	4
430.045030	All digits including zeros are significant due to decimal point and placement between non-zero digits	4, 3, 0, 0, 4, 5, 0, 3, 0	9

Rounding in Multiplication and Division

When multiplying or dividing, your result must reflect the same number of significant figures as the value with the fewest significant figures. Use the following rules to round the result:

If the digit after the last significant digit is ≥ 5, round up.
If the digit after the last significant digit is ≤ 4, keep it unchanged.

Table 2: Multiplication and Division Examples

Operation	Raw Result	Least Sig. Figs.	Not-Rounded Result & Sig. Fig. Adjusted	Evaluation	Final Answer (Rounded-off & Sig. Fig. Adjusted)
3.4 × 4.62	15.708	2 (from 3.4)	15	Next digit is 7 → round up	16
340.3 × 231	78,579.3	3 (from 231)	78,500	Round to 3 sig figs → 78,600	78,600
5.678 × 0.12	0.68136	2 (from 0.12)	0.68	Next digit is 1 → stays	0.68
100.5 × 2.3	231.15	2 (from 2.3)	230	Round to 2 sig figs → 230	230
432.32 ÷ 32.10	13.4692	4 (from 32.10)	13.46	Next digit is 9 → round up	13.47
123.4 ÷ 5.6	22.0357	2 (from 5.6)	22	Next digit is 0 → stays	22
78.90 ÷ 0.456	173.0263	3 (from 0.456)	173	Next digit is 0 → stays	173

Steps for Rounding in Addition and Subtraction

Step 1: Line up the decimal points of all numbers.
Step 2: Perform the addition or subtraction.
Step 3: Look at all the numbers used. Find the one with the fewest digits after the decimal point. This is the least precise number.
Step 4: Round your final answer to the same number of decimal places as that least precise number.

This ensures that your result doesn’t appear more precise than the least accurate measurement.

Example 1:

      2.36
    +12.1
    -------
     14.46   (Raw total)

The number 12.1 has the least digits after decimal, so the final answer must be rounded to 14.5.

Example 2:

     45.67
   −  3.412
   --------
     42.258   (Raw result)

The number 45.67 has the least digits after decimal, so the final answer must be rounded to 42.26.

Using Scientific Notation

Scientific notation makes it easier to clearly show significant figures, especially with large or small numbers. It formats the number so only one digit appears before the decimal point, and all significant digits are preserved:

2.400 × 10³ → Four significant figures: 2, 4, 0, 0
2.40 × 10³ → Three significant figures: 2, 4, 0
2.4 × 10³ → Two significant figures: 2, 4

Module 2: Units, Significant Figures, and Field Notes