CalculatorOnline.xyz
Name the rule before rounding
The rounding calculator takes one finite decimal, one target place, and one explicit rule. It shows the neighboring multiples, the exact discarded distance, whether the value is a tie, and why a particular candidate was selected.
Rounding is not one universal action. Half away from zero and half to even differ exactly at ties. Toward zero, floor, and ceiling differ for negative values. Compare all displays those consequences side by side without declaring one rule appropriate for every context.
Convert the place into a unit
Each target place has a power-of-ten unit. Ones uses 1. Tenths uses 0.1. Hundreds uses 100. To round value x to unit u, the engine studies exact scaled value x/u, chooses an integer according to the rule, then multiplies by u.
Lower candidate is the greatest place multiple less than or equal to the value. Upper candidate is the least place multiple greater than or equal to it. For a value already on the place grid, they coincide and every rule returns the same value.
Preserve result zeros affects text only. Rounding 1.2 to hundredths can display 1.20 when Yes is selected or 1.2 when No is selected. The rational value is unchanged.
Worked tie example: negative two and a half
Enter -2.5, choose Ones, and select Compare all. The neighboring integer candidates are -3 and -2. The exact value lies halfway between them, so Tie status is Yes.
Half away from zero chooses -3 because it increases absolute magnitude at a tie. Half to even chooses -2 because -2 is the even integer candidate. Toward zero chooses -2. Floor chooses -3, the greatest integer not greater than -2.5. Ceiling chooses -2, the least integer not less than -2.5.
The table therefore reads: half away -3; half even -2; toward zero -2; floor -3; ceiling -2. For positive 2.5, half away is 3, half even is 2, toward zero is 2, floor is 2, and ceiling is 3.
The example shows why a page should not describe every rule as “look at the next digit and round up.” Direction and tie policy matter.
Half away from zero
This is the default display rule elsewhere on CalculatorOnline.xyz. Values nearer the lower or upper candidate choose the nearer one. An exact tie selects the candidate farther from zero. 1.25 to tenths becomes 1.3; -1.25 becomes -1.3.
The default is a product convention, not a claim that it fits currency, taxation, measurement, standards, or statistical reporting. Those contexts may prescribe another method.
Half to even
Half to even also chooses the nearest candidate except at an exact tie, where the scaled integer candidate with even parity wins. It is sometimes called banker’s rounding, but that informal label does not make it universally required for banking.
The engine determines parity on the exact scaled integer candidates. It does not inspect a floating approximation or a displayed digit string.
Directed rules
Toward zero removes the fractional portion of the scaled value. Floor always selects the lower candidate. Ceiling always selects the upper candidate. For positive values, toward zero and floor match; for negative values, toward zero and ceiling match.
These rules are directional, not nearest. Rounding -2.01 to ones under floor gives -3 even though -2 is closer. The result panel states that behavior.
Exact ties and finite input
Because entered finite decimals become rationals, the engine can test equal distance to both candidates exactly. It does not decide a tie from an imprecise binary float. A value such as 2.5000 remains the exact same rational tie as 2.5.
Input and BigInt sizes are bounded, and only listed places are available. Scientific notation and custom units are outside this tool. Moving left or right changes the target place and marks old output stale.
The stale marker prevents a prior rule decision from appearing current after the target place changes.
Rounding is not accuracy
Fewer displayed digits do not correct uncertain input. More digits do not add evidence. A rounded result can be inappropriate if the underlying formula, unit, measurement precision, or regulatory rule is wrong.
Use a reviewed policy for financial statements, taxes, medicine, engineering tolerances, grading, and scientific significant figures. This page only applies a named numerical rule.
Rounding questions
What counts as a tie?
The exact value must be equally distant from the two neighboring place multiples.
Is half to even always better?
No. It is one convention; suitability depends on the governing method.
Does Preserve zeros change the result?
No. It changes visible place formatting only.
Are comparisons saved?
No. The table exists only in current page memory.
Enter the finite decimal, target place, and rule, then choose Inspect the rounding decision. Compare neighbors, exact distance, tie state, and signed direction before carrying the rounded value into another task.