Rounding Calculator
Round decimal numbers or integers to specific decimal places or significant figures. Compare standard math, Bankers rounding, floor, and ceiling rules instantly.
123.46
The deciding digit is 6. Since it is 5 or greater, the target digit is rounded UP.
Alternative Rules Comparison
| Rounding Rule | Resulting Value | Difference |
|---|---|---|
| Round Half Up (Math) ★ | 123.46 | +0.003300 |
| Banker's (Half to Even) | 123.46 | +0.003300 |
| Round Up (Ceiling) | 123.46 | +0.003300 |
| Round Down (Floor) | 123.45 | -0.006700 |
Step-by-Step Mathematical Derivation
The Mathematics of Numerical Rounding
Rounding is the process of replacing a number with a simpler value that has a shorter representation. Rounding is widely used in scientific analysis, bookkeeping, taxation, and shopping.
Rounding Modes
This tool supports two primary rounding targets:
- Decimal Places: Rounding to the nearest integer, tenth (0.1), hundredth (0.01), or thousands (1000). This preserves absolute physical units.
- Significant Figures (Sig Figs): Rounding to a specified number of digits starting from the first non-zero left digit. This preserves relative precision, which is essential in physics and chemistry.
Important Rounding Rules
Different industries use different rounding rules to handle halfway cases (ending in 0.5):
- Round Half Up (Standard Math): If the fractional part is 0.5 or greater, it rounds up. For example, 1.5 to 2 and -1.5 to -2.
- Banker's Rounding (Round Half to Even): If the fractional part is exactly 0.5, it rounds to the nearest even number. Thus, 2.5 to 2, and 3.5 to 4. This is used in financial auditing to avoid cumulative statistical upward bias.
- Round Up (Ceiling): Rounds the number up toward positive infinity (2.1 to 3, -2.9 to -2).
- Round Down (Floor): Rounds the number down toward negative infinity (2.9 to 2, -2.1 to -3).
Frequently Asked Questions About Rounding
What is Banker's Rounding?
Banker's Rounding, also known as Round Half to Even, is an algorithm for rounding numbers. When it encounters a halfway case (like 2.5 or 3.5), it rounds to the nearest even number. Thus, 2.5 rounds to 2, and 3.5 rounds to 4. This avoids upward rounding bias over large datasets.
How do you count significant figures?
Significant figures are counted starting from the first non-zero digit on the left. All subsequent digits, including zeros at the end of decimal numbers, are considered significant. Leading zeros are not significant.
How does decimal place rounding work?
Decimal rounding works by identifying the target decimal place (like hundredths). You look at the digit to its immediate right (the deciding digit). If it is 5 or greater, round up by adding 1 to the target digit. If it is less than 5, keep the target digit the same and discard remaining numbers.
Does this support Banker's rounding?
Yes. Apply ceiling, floor, half-up, or Banker's rounding (half-to-even) with step-by-step breakdowns.