Rounded Value
3.14
Decimals
2
Mode
Half Up
Rounded (half up)
3.14
3.14159265 → 3.14
10
0
100
0
1000
0
Difference
-0.0016
% Error
0.0507%
round(x, N) = Math.round(x × 10^N) / 10^NShift the decimal point N places right, round to the nearest integer, then shift back. This is the standard half-up rounding method.
Where:
x= The number to roundN= Number of decimal places to keeperror = |rounded - original| / |original| × 100%The percentage error introduced by rounding, calculated as the absolute difference divided by the original value.
Where:
rounded= The rounded resultoriginal= The original number before roundingInputs
Result
The 3rd decimal digit is 1 (less than 5), so the 2nd decimal stays at 4. Result: 3.14.
Inputs
Result
At exactly 0.5, different modes disagree. Half-up gives 3, but banker's rounding gives 2 (rounds to nearest even). This demonstrates why mode choice matters.
Inputs
Result
Rounding to 2 decimals: the 3rd decimal is 3 (< 5), so result is 9876.54. Nearest 10: 9880. Nearest 100: 9900. Nearest 1000: 10000.
The six main rounding modes are: half-up (standard, 0.5 rounds up), half-down (0.5 rounds toward zero), half-even or banker's rounding (0.5 rounds to nearest even), ceiling (always rounds up), floor (always rounds down), and truncation (removes decimal digits).
| Mode | 2.5 | 3.5 | -2.5 | Best For |
|---|---|---|---|---|
| Half-up | 3 | 4 | -3 | General use |
| Half-down | 2 | 3 | -2 | Conservative estimates |
| Half-even | 2 | 4 | -2 | Financial/statistical |
| Ceiling | 3 | 4 | -2 | Worst-case estimates |
| Floor | 2 | 3 | -3 | Best-case estimates |
| Truncate | 2 | 3 | -2 | Removing decimals |
Banker's rounding (half-even) rounds 0.5 to the nearest even number. This eliminates systematic bias that occurs with standard rounding, where 0.5 always rounds up. Over many transactions, banker's rounding produces more accurate totals.
| Value | Standard (Half-Up) | Banker's (Half-Even) | Bias |
|---|---|---|---|
| 0.5 | 1 | 0 | Half-up biases up |
| 1.5 | 2 | 2 | Same |
| 2.5 | 3 | 2 | Half-up biases up |
| 3.5 | 4 | 4 | Same |
| Sum of 4 | 10 | 8 | Banker's is unbiased |
To round to N decimal places, look at the digit in position N+1 after the decimal point. If it is 5 or greater (with standard rounding), increase the Nth digit by 1. If less than 5, keep the Nth digit. For example, 3.14159 to 2 decimal places: look at the 3rd decimal (1), which is less than 5, so the result is 3.14.
| Number | 0 Places | 1 Place | 2 Places | 3 Places |
|---|---|---|---|---|
| 3.14159 | 3 | 3.1 | 3.14 | 3.142 |
| 2.71828 | 3 | 2.7 | 2.72 | 2.718 |
| 99.995 | 100 | 100.0 | 100.00 | 99.995 |
| 0.5678 | 1 | 0.6 | 0.57 | 0.568 |
Floor always rounds toward negative infinity (down for positives, more negative for negatives). Ceiling always rounds toward positive infinity. Truncation removes decimal digits, effectively rounding toward zero. They differ only for negative numbers and non-integers.
| Value | Floor | Ceiling | Truncate |
|---|---|---|---|
| 2.7 | 2 | 3 | 2 |
| -2.7 | -3 | -2 | -2 |
| 2.0 | 2 | 2 | 2 |
| -0.5 | -1 | 0 | 0 |
Use rounding to decimal places when the number of digits after the decimal matters (e.g., currency to 2 decimal places). Use significant figures when measurement precision matters and the magnitude varies (e.g., scientific data). Rounding to 2 decimals treats 1.23 and 1234.56 the same way; sig figs preserve relative precision.
| Context | Method | Example | Result |
|---|---|---|---|
| Price | 2 decimal places | $19.995 | $20.00 |
| Lab measurement | 3 sig figs | 0.05678 g | 0.0568 g |
| Construction | Nearest 1/8 inch | 3.127 in | 3.125 in |
| Population | Nearest 1000 | 1,234,567 | 1,235,000 |
Rounding simplifies numbers while preserving their approximate value for practical use.
Different rounding modes serve different purposes: standard for everyday use, banker's for finance, and floor/ceiling for bounds estimation.
Understanding which rounding mode to apply prevents systematic errors in calculations and reporting.
Last Updated: Mar 9, 2026
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