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Rounding Calculator
Decimal Places · Significant Figures · Nearest Value · 6 Rounding Modes
Round any number to decimal places, significant figures, or the nearest integer/ten/hundred/thousand. View all 6 rounding modes simultaneously — including banker's rounding, ceiling, floor, and truncate.
Quick Examples
All 6 Rounding Modes Comparison
| Mode | Result | Description |
|---|
How This Result Was Calculated
Rounding Modes Explained
Half Up
Round Half Up (Standard)
The most common rounding method. When the digit after the rounding position is exactly 5, the result is rounded toward positive infinity. Example: 2.5 → 3, −2.5 → −2.
Half Down
Round Half Down
When exactly halfway, rounds toward negative infinity. This is the mirror of Half Up. Example: 2.5 → 2, −2.5 → −3.
Banker's
Round Half to Even (Banker's Rounding)
When exactly halfway, rounds to the nearest even number. Eliminates statistical bias over many rounding operations. Used by default in Python 3, IEEE 754, and most financial systems. Example: 2.5 → 2 (even), 3.5 → 4 (even).
Ceiling
Ceiling (Always Round Up)
Always rounds toward positive infinity, regardless of the fractional part. The smallest integer (or value at precision) not less than the input. Example: 2.1 → 3, −2.9 → −2.
Floor
Floor (Always Round Down)
Always rounds toward negative infinity, regardless of the fractional part. The largest integer (or value at precision) not greater than the input. Example: 2.9 → 2, −2.1 → −3.
Truncate
Truncate (Round Toward Zero)
Drops the fractional part without rounding. Rounds positive numbers down and negative numbers up (toward zero). Equivalent to integer division in most languages. Example: 2.9 → 2, −2.9 → −2.
What Is Rounding?
Rounding is the process of replacing a number with an approximation that is simpler and easier to work with, while keeping the value close to the original. Every day, people round numbers when paying for goods, reporting measurements, and reading statistics. A price of $3.9967 becomes $4.00 at the register; a temperature of 98.6°F is a rounded version of 37°C; a poll result of 52.47% gets reported as "52%."
The fundamental rule of rounding is to look at the digit immediately after the target position. If that digit is 5 or greater, the last kept digit increases by one (rounding up). If it is less than 5, the last kept digit stays the same (rounding down). This is round half up, the method taught in most schools. However, six distinct rounding modes exist, each suited to different applications.
Decimal Places vs. Significant Figures
Two different concepts control where rounding happens:
Decimal Places
Decimal places count the digits after the decimal point. Rounding 3.14159 to 2 decimal places gives 3.14 — the result always has exactly 2 digits after the point. This is the most intuitive approach for everyday money calculations (2 decimal places = cents) and for displaying results in tables.
Significant Figures
Significant figures (sig figs) count meaningful digits starting from the first non-zero digit. The number 0.00456789 has 6 significant figures: 4, 5, 6, 7, 8, 9. Rounded to 3 sig figs it becomes 0.00457. The leading zeros are not significant — they are just placeholders. Scientists and engineers use significant figures because they convey the precision of a measurement rather than its magnitude.
Nearest Value
Rounding to the nearest ten, hundred, thousand, or other unit is common in statistics and reporting. The population of a city reported as "2.4 million" has been rounded to the nearest hundred thousand. This approach is also used when rounding money to the nearest dollar or the nearest cent.
The 6 Rounding Modes
| Mode | 2.5 | 3.5 | −2.5 | 2.1 | 2.9 |
|---|---|---|---|---|---|
| Half Up | 3 | 4 | −2 | 2 | 3 |
| Half Down | 2 | 3 | −3 | 2 | 3 |
| Banker's (Half Even) | 2 | 4 | −2 | 2 | 3 |
| Ceiling | 3 | 4 | −2 | 3 | 3 |
| Floor | 2 | 3 | −3 | 2 | 2 |
| Truncate | 2 | 3 | −2 | 2 | 2 |
Why Rounding Matters
Finance and Accounting
Banks, stock exchanges, and tax agencies specify exact rounding rules because systematic bias in rounding can transfer large sums of money over millions of transactions. The US Internal Revenue Service mandates rounding to the nearest dollar on tax returns. Many European central banks use banker's rounding to prevent cumulative rounding errors in currency conversion. The infamous "salami slicing" computer fraud exploited the fact that half-cent fractions were always rounded down, allowing tiny thefts from each transaction to accumulate into large sums.
Science and Engineering
In science, the number of significant figures in a result should reflect the precision of the least precise measurement used to obtain it. If a length is measured to 3 sig figs and a width to 4 sig figs, the area should be reported to 3 sig figs — reporting more would imply a precision the measurement does not possess. Engineers routinely round intermediate results to avoid propagating floating-point errors through complex calculations.
Computing and Software
IEEE 754, the international standard for floating-point arithmetic in CPUs, uses round half to even (banker's rounding) as its default mode. Python 3's built-in round() function also uses banker's rounding. Java, C, and JavaScript use round half up. These differences can cause subtle bugs when porting numerical code between languages — always check the rounding mode when precision matters.
How to Round to Significant Figures
To round a number to n significant figures:
- Find the first non-zero digit — this is significant figure 1.
- Count n digits from there; that is your rounding position.
- Apply the standard rounding rule at that position.
- Replace all digits to the right with zeros (for integers) or drop them (for decimals).
Example: Round 0.004567 to 2 sig figs. The first significant digit is 4 (position −3). Two sig figs takes us to the 5 (position −4). Look at the next digit (6 ≥ 5), so round up: 0.0046.
Rounding Errors and Accumulation
A single rounding operation introduces an error of at most half a unit in the last place (ULP). But when rounding is applied repeatedly across thousands of calculations, these tiny errors can accumulate. Banker's rounding minimises this by ensuring that half-way cases round up about half the time and down about half the time (to the nearest even digit), producing a statistically unbiased result over large datasets.
Frequently Asked Questions
What is the difference between rounding to decimal places and significant figures?
Decimal places fix the number of digits after the decimal point, so 3.14159 rounded to 2 dp is 3.14. Significant figures count all meaningful digits from the first non-zero digit, so 0.00456789 rounded to 3 sig figs is 0.00457 — still 3 significant digits (4, 5, 7).
What is banker's rounding (round half to even)?
Banker's rounding rounds a number that is exactly halfway to the nearest even integer. So 2.5 → 2 (even) and 3.5 → 4 (even). This eliminates the upward statistical bias of always rounding 0.5 up, and is the default in IEEE 754, Python 3, and many financial systems.
What is the difference between ceiling and floor rounding?
Ceiling always rounds toward positive infinity (smallest value ≥ input). Floor always rounds toward negative infinity (largest value ≤ input). For positive numbers, ceiling rounds up and floor rounds down. For negative numbers, ceiling rounds toward zero and floor rounds away from zero. Example: ceil(−2.3) = −2, floor(−2.3) = −3.
What is truncation and how does it differ from floor?
Truncation drops the fractional part without rounding — it always rounds toward zero. For positive numbers, truncate equals floor. For negative numbers, truncate equals ceiling. truncate(2.9) = 2 and truncate(−2.9) = −2, whereas floor(−2.9) = −3.
How do you round to the nearest ten, hundred, or thousand?
To round to the nearest N, compute round(x ÷ N) × N. For example, 1234.5 to the nearest hundred: round(1234.5 / 100) × 100 = round(12.345) × 100 = 12 × 100 = 1200.
Why does rounding matter in finance and science?
In finance, rounding errors in currency conversion and tax calculations can compound across millions of transactions. Banker's rounding prevents systematic upward bias. In science, the number of significant figures communicates measurement precision — too many figures imply false accuracy; too few lose information.
What are the 6 main rounding modes?
The 6 main modes are: Round Half Up (most common — 0.5 goes toward +∞); Round Half Down (0.5 goes toward −∞); Round Half to Even / Banker's (0.5 goes to nearest even, eliminates bias); Ceiling (always toward +∞); Floor (always toward −∞); Truncate (drop fractional part, always toward zero).