Short Answer
Rounding a number in scientific notation to a given number of significant figures is a two-part operation: round the coefficient to the required significant digits, then normalize if the rounding carries into a new power of ten. The exponent is not a significant figure; it only sets the order of magnitude. This reference page follows ISO 80000-1, ASTM E29, and NIST SP 811 conventions, with notes on common classroom half-up practice. Use our significant figures calculator to verify results after reading the rules.
Rule Statement
Given a × 10n, where 1 ≤ |a| < 10, to round to k significant figures:
- Identify the first k significant digits of the coefficient a. Leading zeros are never significant; trailing zeros in the coefficient are significant.
- Inspect the (k+1)th digit and any following digits.
- If the following part is less than half a unit in the kth place, leave the kth digit unchanged. If greater than half, increase the kth digit by 1. If exactly half, apply the tie convention: ISO/ASTM/NIST use round-half-to-even; many school texts use round-half-up.
- If rounding changes the coefficient to 10.0… or larger, renormalize: write it as 1.00… × 10n+1 and keep exactly k significant figures.
- Do not change the exponent unless renormalization occurs. Do not use the exponent to determine significant figures.
Related rules: rounding rules, sig figs in scientific notation, banker’s rounding.
Visual Digit Map: Where Significant Figures Live in Scientific Notation
Consider 6.02214076 × 1023. The coefficient is 6.02214076; the exponent is 23. For five significant figures, the digit map is:
| Position | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Coefficient digit | 6 | 0 | 2 | 2 | 1 | 4 | 0 | 7 |
| Role for 5 sig figs | keep | keep | keep | keep | keep | decide | drop | drop |
The sixth digit is 4, so 6.02214076 × 1023 rounds to 6.0221 × 1023 at five significant figures. The exponent 23 remains unchanged.
Worked Examples
Example 1: No carry, exponent unchanged
Round 3.14159 × 105 to 3 significant figures.
- First three significant digits: 3, 1, 4.
- Next digit is 1, which is less than 5.
- Leave the third digit unchanged.
Result: 3.14 × 105.
Example 2: Tie with half-even
Round 1.2500 × 102 to 2 significant figures under ISO/ASTM/NIST.
- First two significant digits: 1, 2.
- Next digit is 5, and all following digits are zero, so it is an exact tie.
- The retained digit 2 is even, so leave it unchanged.
Result: 1.2 × 102. Under half-up, the result would be 1.3 × 102.
Example 3: Carry and renormalization
Round 9.999 × 102 to 3 significant figures.
- First three significant digits: 9, 9, 9.
- Next digit is 9, so round up.
- 9.99 + 0.01 = 10.00, which is not normalized.
- Rewrite as 1.00 × 103.
Result: 1.00 × 103, not 10.0 × 102 and not 1.00 × 102.
Counter-Examples: Common Rounding Traps
- Dropping significant zeros. 2.50 × 103 has three significant figures. Rounding to two gives 2.5 × 103; the zero was significant and must be dropped only when reducing sig figs.
- Counting the exponent. In 4.56 × 107, the digits 4, 5, and 6 are significant. The 7 is an exponent, not a significant digit.
- Forgetting renormalization. 9.999 × 102 to 3 sig figs is 1.00 × 103. Writing 10.0 × 102 is not proper scientific notation.
- Double rounding. Rounding 1.2345 first to 1.23 and then to 1.2 can differ from rounding directly to two significant figures. Round once to the target precision.
- Ignoring tie convention. 1.25 × 102 to two sig figs is 1.2 × 102 under half-even, but 1.3 × 102 under half-up. State the convention.
Convention Comparison Table
| Convention | Rule at exact half | 1.25 × 102 to 2 sig figs | Typical use |
|---|---|---|---|
| Half-even (banker’s) | Round to nearest even retained digit | 1.2 × 102 | ISO 80000-1, ASTM E29, NIST SP 811, GUM |
| Half-up | Always increase retained digit by 1 | 1.3 × 102 | Many school curricula and informal calculations |
| Truncation | Drop extra digits without rounding | 1.2 × 102 | Conservative limits, some software defaults |
| Half-away-from-zero | Increase magnitude at half | 1.3 × 102 | Some spreadsheet functions |
For metrology and regulated testing, use the tie rule specified by the governing standard or contract. See ASTM E29 and ISO 80000.
Standards Citation
Authoritative references include:
- ISO 80000-1:2009, Quantities and units — Part 1: General, Clause 7.3.3: rounding to a required number of significant figures; exact halves are rounded to the nearest even digit.
- ASTM E29-13a, Standard Practice for Using Significant Digits in Test Data to Determine Conformance with Specifications, Section 6: significant-digit rounding for conformance; ties round to the nearest even digit.
- NIST SP 811, Guide for the Use of the International System of Units (SI), Section 7.9: rounding numerical values and retaining significant figures.
- JCGM 100:2008 (GUM), Evaluation of measurement data — Guide to the expression of uncertainty in measurement, Clause 7.2.6: report numerical values with an appropriate number of digits and avoid excessive precision.
Precision is not the same as accuracy. Rounding controls how many digits are reported; it does not improve the underlying measurement. See accuracy vs precision and measurement uncertainty.
Common Mistakes
- Rounding the exponent. The exponent is an integer order of magnitude; it is not rounded to significant figures.
- Leaving the coefficient unnormalized after carry. Always ensure 1 ≤ |coefficient| < 10.
- Misreading trailing zeros. In scientific notation, trailing zeros in the coefficient are significant: 3.20 × 104 has three sig figs.
- Applying half-up when the standard requires half-even. This can cause conformance failures in ASTM E29 testing.
- Rounding intermediate results. Keep extra digits during calculation and round only the final reported value.
Practice Problems
- Round 8.7654 × 10-4 to 3 significant figures using half-even. Answer: 8.77 × 10-4 (next digit 5, following 4 makes it greater than half).
- Round 1.005 × 103 to 3 significant figures using half-even. Answer: 1.00 × 103 (exact tie, retained 0 is even).
- Round 9.9999 × 106 to 4 significant figures. Answer: 1.000 × 107.
- Round 4.5600 × 102 to 2 significant figures. Answer: 4.6 × 102.
Quick Reference Table
| Task | Action |
|---|---|
| Identify sig figs | Count all digits in the coefficient; ignore exponent and leading zeros. |
| Round to k sig figs | Keep k digits; inspect the (k+1)th digit and beyond. |
| Exact half | Use half-even for ISO/ASTM/NIST; use half-up only if specified. |
| Coefficient reaches 10 | Renormalize to 1.00… × 10n+1. |
| Report | Keep exactly k significant figures; do not add or drop significant zeros. |
FAQ
Does the exponent count as a significant figure?
No. In a × 10^n, significant figures are counted only in the coefficient a. The exponent n gives the order of magnitude.
What happens if rounding makes the coefficient 10?
Renormalize. For example, 9.99 × 10^2 rounded to two significant figures becomes 1.0 × 10^3.
Should I use half-up or half-even?
For scientific, engineering, and metrology work, follow ISO 80000-1, ASTM E29, and NIST SP 811: use half-even for exact ties unless a contract or instructor specifies half-up.
How do I round a negative number in scientific notation?
Round the absolute value of the coefficient using the same significant-figure rules, then restore the negative sign. The exponent is unchanged unless renormalization occurs.
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