Rounding Calculator

What are the six rounding modes?

Two calculators can disagree on 2.5 without either one being broken. They apply different tie-breaking rules. A tie, meaning a discarded part that measures exactly half a step of the rounding grid, only comes up for three of the six modes. The rows below cover an exact tie (2.5 and 2.675), a negative value that swaps ceiling and floor (-2.5) and a value nowhere near a tie (3.14159).

Half up sends a tie away from zero, half down sends it toward zero and half to even sends it to whichever neighbour ends in an even digit. The other three never consult the deciding digit: ceiling moves toward positive infinity, floor moves toward negative infinity and truncate drops the surplus digits while keeping the sign.

For a price carried to the cent or a school exercise, half up is the answer people expect. For a column of figures that will be summed, half to even keeps the total closest to the exact sum. Learning how to round on a calculator is largely a matter of knowing which of these six rules the machine at hand applies.

How do you round to the nearest tenth?

A stopwatch reading of 3.14159 seconds comes back as 3.1 at the tenth, since the hundredths digit is 4 and everything past it is discarded. Nudge that digit up to 6 and 3.16 climbs to 3.2. Set the place list to tenth and the page works as a round to the nearest tenth calculator, with the hundredths digit highlighted in the breakdown.

The same reading moves one column right for the hundredth, where the thousandths digit decides and 2.675 lands on 2.68 under half up. That column separates a cent kept from a cent lost.

What is banker's rounding?

Add 0.5, 1.5, 2.5 and 3.5 together and the exact total is 8. Round each one to a whole number with half up and the total becomes 10, because all four ties moved in the same direction. Half to even, the rule usually called banker's rounding, sends each tie to whichever neighbour ends in an even digit: 0.5 falls to 0, 1.5 climbs to 2, 2.5 falls to 2 and 3.5 climbs to 4, for the exact total of 8.

That behaviour is the default of the IEEE 754 standard, the specification every mainstream processor follows for floating-point arithmetic. Each language then picks its own rule on top of it, which is why one rounds a tie differently from the next.

• Python's built-in round() and .NET's Math.Round use half to even, so round(2.5) returns 2.
• JavaScript's Math.round sends a tie toward positive infinity, which makes Math.round(-2.5) equal to -2.
• Excel's ROUND moves a tie away from zero, so ROUND(-2.5, 0) returns -3.

Why does a calculator round 2.675 to 2.67?

The digits 2.675 have no exact equivalent in binary. A calculator stores them in a double, the 64-bit floating-point type behind almost every spreadsheet cell, so what it holds is 2.67499999999999982. That value sits below the tie, which leaves a rounding rule reading it no option but to go down. The same trap catches any round off calculator that multiplies by 100 first, turning 1.005 into 1.00 and 8.575 into 8.57.

Nothing on this page passes through a float. The engine cuts the digit string at the target position, reads the first discarded digit, then carries the increment leftward through the digits it kept, so that 9.99 rounded to one decimal comes out as 10.0 and every finite decimal stays exact.

On a single invoice line the gap is one cent. Repeat it over a few hundred lines and the reconciliation stops matching, which is the usual reason someone goes looking for a decimal type such as Python's Decimal.

What is the difference between decimal places and significant figures?

A lab reading of 0.00045678 carries eight decimal places and five significant figures, because counting starts at the first non-zero digit. Asked for three significant figures, it comes back as 0.000457, a value that still needs six decimal places to write out. That gap is what makes significant figures the working unit for very small measurements.

Rounding 9.99 to two significant figures pushes the carry through both nines and gives 10 with no trailing zero, since writing 10.0 would claim a third significant figure the measurement never had. The field accepts 1 to 15 significant figures.