Decimal Places and Significant Figures
These answer different questions. Decimal places count digits after the point and preserve absolute precision — useful for money, where a cent is a cent regardless of the amount. Significant figures count from the first non-zero digit and preserve relative precision — useful in science, where being accurate to 1% matters equally at any scale.
0.00123456 to 3 decimal places is 0.001. To 3 significant figures it is 0.00123. The first discards nearly all the information; the second keeps it.
Rounding Chart
Rounding 3.14159265:
| Rounded to | Decimal places | Significant figures |
|---|---|---|
| 0 | 3 | — |
| 1 | 3.1 | 3 |
| 2 | 3.14 | 3.1 |
| 3 | 3.142 | 3.14 |
| 4 | 3.1416 | 3.142 |
| 6 | 3.141593 | 3.14159 |
The Methods Disagree at Exactly 0.5
| Method | 2.5 becomes | 3.5 becomes | Used in |
|---|---|---|---|
| Round half up | 3 | 4 | Schools, everyday arithmetic |
| Round half to even | 2 | 4 | Statistics, finance, IEEE 754 |
| Round half down | 2 | 3 | Rare |
| Truncate | 2 | 3 | Integer division in programming |
Round half to even — banker's rounding — exists because always rounding 0.5 upward introduces a systematic upward bias. Over thousands of transactions that bias accumulates into real money, which is why financial systems and most statistical software default to it.
Significant Figure Rules
- All non-zero digits are significant.
- Zeros between non-zero digits are significant: 1002 has four.
- Leading zeros are never significant: 0.0012 has two.
- Trailing zeros after a decimal point are significant: 1.200 has four.
- Trailing zeros in a whole number are ambiguous: 1200 may have two, three or four. Scientific notation removes the ambiguity — 1.2 × 10³ is unmistakably two.
Rounding Errors Compound
Rounding intermediate results and then continuing to calculate propagates error through every subsequent step. The rule in numerical work is to carry full precision throughout and round only the final answer. The 1991 Patriot missile failure at Dhahran traced to a small timing rounding error accumulating over 100 hours of operation.
Frequently Asked Questions
Why does 0.1 + 0.2 not equal 0.3 on a computer?
Because binary floating point cannot represent 0.1 or 0.2 exactly, any more than decimal can represent 1/3. The result is 0.30000000000000004. Financial software uses decimal or integer cent arithmetic to avoid this.
How many significant figures should I report?
No more than the least precise measurement in the calculation. Multiplying a value known to 3 significant figures by one known to 5 gives a result justified to 3.
Does rounding twice give the same answer?
Not always. Rounding 2.44 to one decimal gives 2.4, then to a whole number gives 2. Rounding 2.44 directly to a whole number also gives 2 — but 2.46 rounds to 2.5 and then to 3, while direct rounding gives 2. Always round once, from the original.