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Results
| Time | Input | Op | n | Mode | Notation | Preserve | Int Zeros | Counted | Result |
|---|
Most recent computation details
Example Data
| Input | n | Mode | Notation | Int Zeros Significant | Rounded Output | Sig Fig Count |
|---|---|---|---|---|---|---|
| 0.00340 | 2 | HALF_UP | Plain | No | 0.0034 | 3 |
| 1200 | 3 | HALF_UP | Plain | No | 1200 | 2 |
| 1.230e3 | 4 | HALF_UP | Scientific | Yes | 1.230E+3 | 4 |
| -45.600 | 3 | HALF_EVEN | Plain | Yes | -45.6 | 5 |
| 7.8900E-4 | 2 | DOWN | Engineering | Yes | 0.8E-3 | 5 |
Significant Figures Quick Reference
| Number | Interpretation | Significant Figures |
|---|---|---|
| 0.00340 | Leading zeros not significant; trailing decimal zero is significant | 3 |
| 1200 | No decimal shown; trailing zeros ambiguous by default not significant | 2 |
| 1200. | Decimal point indicates trailing zeros are significant | 4 |
| 1.200×103 | Mantissa digits determine significant figures | 4 |
| -45.600 | Decimal trailing zeros are significant | 5 |
| 0.000 | No non-zero digits present | 0 |
| 7.8900E−4 | Mantissa 7.8900 has five significant digits | 5 |
Rounding Modes Side‑by‑Side
| Input | n | Half‑up | Half‑even | Down (toward zero) | Away from zero |
|---|---|---|---|---|---|
| 2.445 | 3 | 2.45 | 2.44 | 2.44 | 2.45 |
| 7.05 | 2 | 7.1 | 7.0 | 7.0 | 7.1 |
| 1.245×103 | 3 | 1.25×103 | 1.24×103 | 1.24×103 | 1.25×103 |
Rows chosen to show “. . .5” tie cases. Half‑even minimizes systematic bias in large datasets.
Operations Rule Cheat Sheet
| Operation | Rounding Rule | Example (unrounded) | Final (rounded) |
|---|---|---|---|
| Multiplication | Fewest significant figures among factors | 4.56 × 1.4 = 6.384 | 6.4 (2 s.f.) |
| Division | Fewest significant figures among operands | 12.0 ÷ 3.00 = 4.000… | 4.00 (3 s.f.) |
| Addition | Fewest decimal places among terms | 12.11 + 18.0 + 1.013 = 31.123 | 31.1 (1 decimal) |
| Subtraction | Fewest decimal places among terms | 1200 − 3.45 = 1196.55 | 1197 (0 decimals) |
| log10(x) | Mantissa decimals = s.f. in x | log10(3.40) = 0.5314789… | 0.531 (3 decimals) |
| 10y | Result s.f. = decimals in y | 10−2.30 = 0.0050119… | 5.0×10−3 (2 s.f.) |
Formulas and Rules Used
• Ignore leading zeros. Zeros between non-zero digits are significant.
• Trailing zeros are significant only when a decimal point is present, or when specified.
• Scientific notation: significant figures equal the digits in the mantissa.
Rounding to n significant figures
Let x ≠ 0. Define k = ⌊log10(|x|)⌋ and s = 10n−1−k.
Round y = round(s·|x|) using the selected mode, then output sign(y)·y/s.
Supported modes: Half-up, Half-even, Down (toward zero), Away from zero.
Formatting
• Plain: decimals = max(0, n−1−⌊log10(|x|)⌋).
• Scientific: one digit before the decimal, exponent base 10.
• Engineering: exponent is a multiple of 3. Mantissa has n digits total.
How to Use This Calculator
- Enter a number. You may include scientific notation, like 7.89e−3.
- Choose whether to count only or round to a specified number of digits.
- Select a rounding mode, output notation, and whether to keep trailing zeros.
- Toggle the integer trailing zeros option if your integers include measured zeros.
- Click Compute. Review the results table and step-by-step explanation.
- Export your session Results as CSV or PDF for documentation or reports.
Tip: When working with values like "1200", trailing zeros can be ambiguous. Use scientific notation or enable the integer trailing zeros option to declare intent.
When Are Calculations Rounded Off Based on Significant Figures?
General principle. Carry full precision through intermediate steps (or at least one guard digit beyond the final requirement) and round once at the end, unless you must report intermediate results. If you round an intermediate value, keep one extra digit and round again only at the final step.
Multiplication and division. Round the final answer to match the factor with the fewest significant figures. Example: 4.56 (3 s.f.) × 1.4 (2 s.f.) = 6.384 → 6.4 (2 s.f.).
Addition and subtraction. Round the final answer to the fewest decimal places among the terms. Example: 12.11 + 18.0 + 1.013 = 31.123 → 31.1 (one decimal place).
Mixed operations. Evaluate in logical order. Apply the decimal-place rule to addition/subtraction parts and the significant-figure rule to multiplication/division parts. Round only once at the end. If an intermediate must be shown, keep one extra digit.
Exact numbers and defined constants. Counted quantities (e.g., 3 beakers) and defined conversion factors (e.g., 1 in = 2.54 cm exactly) have infinite significant figures and do not limit rounding.
Functions (logs, powers). For log10(x), the number of digits after the decimal (mantissa) should equal the significant figures in x. For 10y, the result should have as many significant figures as the number of digits after the decimal in y.
Trailing zeros and notation. Use scientific notation to make intended significant digits explicit (e.g., 1.20×103 has three significant figures) or enable the “integer trailing zeros significant” option when appropriate.
Rounding ties. Default is half-up here. Half-even (“bankers”) is available to reduce bias in repeated rounding.
Quick examples.
- 0.00340 (3 s.f.) ÷ 2.0 (2 s.f.) = 0.00170 → 0.0017 (2 s.f.).
- 1200 + 3.45 = 1203.45 → 1203 (no decimals because 1200 limits decimal places).
- (2.50 × 3.1) + 0.004 = 7.75 + 0.004 = 7.754 → 7.8 (final 2 s.f.; add then round once).
Notes and Limitations
- Zero values are shown with the requested number of implied significant digits.
- This tool uses double-precision arithmetic; extreme magnitudes may introduce tiny rounding artifacts.
- For exact formatting control, prefer scientific notation with preserved trailing zeros.