Rounded result 3.14 2 decimal places

2 decimal places 3.14 Counted from the decimal point

2 significant figures 3.1 Counted from the first nonzero digit Open this value in the sig fig tool

3.14159 3.14

Final place hundredths place

First discarded digit 1

Outcome Kept

Why this is the answer

1Set the targetThe final shown digit belongs in the hundredths place.
2Inspect the discarded partThe discarded part begins with 1 and is below halfway.
3Apply the methodThe first discarded digit is 1, immediately after the hundredths place. The discarded part is below half a unit in that place, so the retained digit stays unchanged.

Rule applied

Round half up

The first discarded digit is 1, immediately after the hundredths place. The discarded part is below half a unit in that place, so the retained digit stays unchanged.

Decimal3.14

Scientific3.14 × 10^0

E-notation3.14e0

Exact input value: 3.14159. The rounding calculation uses this decimal value directly.

How to round to decimal places

Decimal places count positions to the right of the decimal point. One decimal place ends at tenths, two end at hundredths, three end at thousandths, and so on. The count always begins at the decimal point, even when the number starts with several zeros. For example, the 4 in 0.00456 is in the thousandths place, so rounding that value to two decimal places considers the hundredths place, not the first nonzero digit.

Formal rule

For a value x and N decimal places, let q = 10^-N. The rounded answer is an integer multiple of q selected by the chosen rounding method. Under nearest half-up rounding, compute the two neighboring multiples of q, choose the nearer one, and choose the one farther from zero when x lies exactly halfway between them.

The familiar digit procedure is an efficient way to apply that rule. Keep every digit through the requested place, inspect the complete discarded part, and decide whether the retained value changes. A first discarded digit below 5 is below halfway. A digit above 5 is above halfway. A 5 is an exact tie only when every following discarded digit is zero; 5 followed by any nonzero digit is above halfway.

Locate the decimal point and count N positions to the right.
Keep the digit in that place and everything before it.
Inspect the next digit and all later digits to identify below-half, above-half, or an exact tie.
Apply the selected nearest or directed rounding method, carrying through 9s when necessary.
Append zeros when needed so the answer visibly contains exactly N decimal digits.

Worked examples

3.14159 to 2 decimal places: 3.14. The hundredths digit is 4 and the first discarded digit is 1, so half-up leaves the 4 unchanged.

12.005 to 2 decimal places: 12.01 with half-up but 12.00 with half-even. The discarded part is exactly 0.005; half-even keeps the already-even hundredths digit.

9.999 to 2 decimal places: 10.00. The discarded 9 increases the hundredths digit. The carry passes through all retained 9s, and two zeros remain visible because the requested place is still hundredths.

1.2 to 4 decimal places: 1.2000. Nothing is discarded. Three zeros are appended to state the requested number of places without changing the value.

-2.675 to 2 decimal places: -2.68 with half-up or half-even. The exact tie follows a retained 7, which is odd, so half-even also increases the magnitude to make the retained digit even.

Quick rounding-method table

Nearest methods differ only at an exact halfway value. Directed methods react to any nonzero discarded part, so the sign of the input matters.

Method	Rule	`2.5` to 0 DP	`-2.5` to 0 DP
Half up	Nearest; ties away from zero	3	-3
Half even	Nearest; ties make the retained digit even	2	-2
Half down	Nearest; ties toward zero	2	-2
Ceiling	Toward positive infinity	3	-2
Floor	Toward negative infinity	2	-3
Truncate	Toward zero	2	-2

Half-up is a common classroom convention. Half-even reduces systematic upward movement across collections containing many exact ties. Financial, laboratory, engineering, and regulatory work may name a specific method, so follow the rule required by the relevant standard rather than assuming that every context uses the same default.

Decimal places versus significant figures

Decimal places and significant figures answer different questions. Decimal places fix a position relative to the decimal point. Significant figures begin at the first nonzero digit and describe a count of meaningful digits. Rounding 0.00456 to two decimal places gives 0.00, while rounding it to two significant figures gives 0.0046. For 1234.56, two decimal places gives 1234.56 because the value is already exact at hundredths, while two significant figures gives 1200.

The side-by-side result uses the same input, count, and rounding method to make that contrast concrete. A zero-decimal-place target has no matching significant-figure calculation because a quantity cannot be rounded to zero significant figures; significant-figure counts start at one. In that case the comparison is marked undefined instead of silently substituting a different target.

Zeros, notation, and exact display

Written zeros matter in a decimal-place answer. The values 10, 10.0, and 10.00 are numerically equal, but the last form explicitly reaches the hundredths place. This calculator therefore preserves exactly N digits after the point, including when rounding creates a carry or the original number has fewer decimal digits. Scientific and E notation provide equivalent copyable values; for zero, the exponent is normalized to zero, such as 0.00 x 10^0 and 0.00e0 for two decimal places.

Do not round an intermediate result merely because a display is short. Keep guard digits through a multi-step calculation and round once at the end unless the governing procedure explicitly requires staged rounding. Repeated rounding can change an answer: rounding 2.449 first to 2.45 and then to one decimal place produces 2.5, while rounding the original directly to one decimal place produces 2.4 under half-up.

Calculation method and limits

The server reads the mantissa and exponent as text, represents the value as an integer coefficient times a power of ten, and compares discarded decimal digits directly. Carries are propagated one character at a time. The browser uses the shared exact BigInt-rational engine. Neither implementation converts the submitted value to a binary PHP or JavaScript floating-point number, avoiding familiar representation surprises around values such as 1.005 and 2.675.

Inputs are limited to one decimal or scientific-notation literal with at most 1,000 mantissa digits, an exponent from -1,000 to 1,000, and a target from 0 to 100 decimal places. Commas, fractions, arithmetic expressions, infinity, and NaN are rejected. These limits keep server output bounded while still supporting values far beyond ordinary hardware-number ranges.

Compare this result in the Round to N Significant Figures tool, study the complete rounding guide, review the core rounding rules, learn why ties differ in the half-even guide, or see rounding versus significant figures explained with more examples.

Methodology and sources

The implementation is checked against exact ties, values just above and below halfway, negative directed rounding, carries across powers of ten, scientific exponents, fixed zero padding, and the decimal-place versus significant-figure distinction. Server fixtures exercise the string engine without relying on browser formatting.

Changelog

2026-08-10 - Initial release with six exact methods, complete server rendering, fixed decimal padding, and a same-input significant-figure comparison.

Round to N Decimal Places