Round any number to decimal places, significant figures, nearest 10/100/1000, or a custom interval. Includes standard, banker's, ceiling, floor, and truncation modes.
Rounding is the process of reducing the precision of a number while keeping its value close to the original. It is one of the most frequently performed operations in everyday mathematics, from calculating tax to reporting scientific measurements. Our free rounding calculator supports five distinct rounding methods — standard rounding, banker's rounding, ceiling, floor, and truncation — and handles decimal places, significant figures, nearest multiples, and custom intervals.
Standard rounding (also called "round half up") is the most familiar method taught in schools. The rule is straightforward:
| Original Number | Rounded to 1 dp | Rule Applied |
|---|---|---|
| 3.14 | 3.1 | 4 < 5, round down |
| 3.15 | 3.2 | 5 ≥ 5, round up |
| 3.145 | 3.1 | 4 < 5, round down (to 1 dp) |
| 7.95 | 8.0 | 5 ≥ 5, round up (carry propagates) |
| −2.5 | −2 | Half rounds away from zero (round half up) |
These are two different ways of specifying precision, and students frequently confuse them.
Decimal places (dp) count digits after the decimal point. The number 3.14159 rounded to 2 dp is 3.14. The number 0.001234 rounded to 2 dp is 0.00 — this is almost always unhelpful for very small numbers.
Significant figures (sf) count from the first non-zero digit. 3.14159 to 3 sf is 3.14. 0.001234 to 3 sf is 0.00123 — much more useful for small numbers. 1,234,567 to 3 sf is 1,230,000.
| Number | 2 dp | 2 sf | 3 sf |
|---|---|---|---|
| 3.14159 | 3.14 | 3.1 | 3.14 |
| 0.004567 | 0.00 | 0.0046 | 0.00457 |
| 12345.6 | 12345.60 | 12000 | 12300 |
| 0.1005 | 0.10 | 0.10 | 0.101 |
Standard rounding introduces a systematic bias: every ".5" case rounds up, which means in large datasets the sum always rounds up more than down. This is a well-known problem in financial calculations.
Banker's rounding (also called "round half to even" or "IEEE 754 default rounding") solves this by rounding half-way cases to the nearest even digit:
Over a large set of numbers, half the .5 cases round up and half round down, producing no net bias. This is the default rounding mode in IEEE 754 floating-point arithmetic (used by virtually all computers), Python's built-in round() function, and many financial systems.
Tax authorities specify how taxpayers should round figures on returns:
UK (HMRC): Generally, income and expenses on Self Assessment returns should be rounded to the nearest pound. HMRC guidance says to round down income figures and round up expense figures to your advantage when the pence would make a trivial difference. VAT calculations under the standard scheme are rounded down to the nearest penny per invoice.
US (IRS): The IRS allows rounding to the nearest dollar on most returns. Round down any amount less than 50 cents and round up 50 cents or more. Consistency is required — if you round on one line, you must round on all lines. The IRS Form 1040 instructions explicitly permit this.
The ceiling function (⌈x⌉) always rounds up to the next integer, regardless of the fractional part. ⌈3.1⌉ = 4, ⌈−2.9⌉ = −2.
The floor function (⌊x⌋) always rounds down to the next integer below. ⌊3.9⌋ = 3, ⌊−2.1⌋ = −3.
These are used in computer science for memory allocation, pagination, and index calculations where you always need an integer and the direction matters.
Truncation simply removes digits beyond a specified position without any rounding. 3.99 truncated to 1 dp is 3.9 (not 4.0). This is equivalent to always rounding toward zero. Truncation is fast and simple but introduces more error than proper rounding for most purposes. It is sometimes called "round towards zero" or "chop".
All computers represent real numbers using binary floating-point arithmetic. Many decimal fractions that look simple (like 0.1) cannot be represented exactly in binary, leading to tiny errors that can accumulate over many calculations. For example, in many programming languages, 0.1 + 0.2 does not exactly equal 0.3 due to floating-point representation.
This is why financial software often uses integer arithmetic (working in pence or cents, not pounds or dollars) or arbitrary-precision libraries. The Patriot missile failure in the Gulf War (1991) was caused by accumulated floating-point rounding errors over 100 hours of continuous operation.
In England, Wales, and Northern Ireland, rounding is introduced in KS2 (Key Stage 2, ages 7–11): pupils learn to round to the nearest 10 and 100. At KS3 (ages 11–14) rounding to decimal places and significant figures is introduced. GCSE Mathematics (14–16) tests rounding in nearly every exam paper. A-Level Mathematics uses significant figures for quoting answers to calculation questions. Exam boards (AQA, Edexcel, OCR) typically require 3 significant figures unless the question specifies otherwise.
In the United States, rounding is covered under the Common Core State Standards (CCSS) for Mathematics. Grade 3 covers rounding to nearest 10 and 100. Grade 4 adds rounding to 1000 and multi-digit numbers. Significant figures are introduced in middle school science and reinforced in high school chemistry and physics using the sig fig rules. The SAT and ACT specify that answers should be rounded to the nearest whole number or thousandth as required by the question.
Standard rounding rounds any digit of 5 or above up, and any digit below 5 down. So 2.5 rounds to 3, 2.45 rounds to 2.5 (to 1 dp), and 2.449 rounds to 2.4. This is the most common rounding method taught in schools in both the UK and US.
Banker's rounding (round half to even) rounds .5 cases to the nearest even number. So 2.5 → 2, 3.5 → 4, 4.5 → 4, 5.5 → 6. Over large datasets this eliminates the upward bias of standard rounding. It is the default in IEEE 754 arithmetic and Python's round() function.
Decimal places count digits after the decimal point. Significant figures count from the first non-zero digit. For large or small numbers they give very different results: 0.001234 to 2 dp is 0.00, but to 2 sf it is 0.0012. Significant figures are more useful for expressing measurement precision in science.
Divide the number by 10 (or 100 or 1000), apply standard rounding to get an integer, then multiply back. For 347 rounded to the nearest 10: 347/10 = 34.7, round to 35, multiply back to get 350. For 347 to the nearest 100: 347/100 = 3.47, round to 3, gives 300.
HMRC generally allows taxpayers to round income and expenses to the nearest pound on Self Assessment returns. The guidance suggests rounding down income and rounding up expenses when in doubt. VAT invoices under the standard scheme should round down to the nearest penny. Always check current HMRC guidance for your specific situation.
Truncation simply removes digits beyond the specified precision without any rounding — it always rounds toward zero. For positive numbers, truncation equals rounding down (floor). For negative numbers they differ: truncate(−2.7) = −2 but floor(−2.7) = −3. Rounding down always moves toward negative infinity; truncation always moves toward zero.
Use the same number of significant figures as the least precise measurement in your calculation. If you measure 4.5 cm (2 sf) and 3.12 cm (3 sf), your answer should be given to 2 sf. For A-Level and GCSE in the UK, 3 significant figures is the default unless otherwise stated. For US AP Science, follow the sig fig rules strictly.
This is due to binary floating-point representation. 0.1 and 0.2 cannot be represented exactly in binary (just as 1/3 cannot be written exactly in decimal), so tiny errors accumulate. This is normal IEEE 754 behaviour. Financial software avoids this by working in integers (pence/cents) or using decimal arithmetic libraries.
Results are for educational and general purposes only. For financial, tax, or scientific applications always verify rounding requirements with the relevant authority or standard. Floating-point arithmetic may introduce minor precision differences.