Decimal Calculator – Add, Subtract, Multiply and Divide Decimals Exactly

Most calculators quietly round behind your back — that is how 0.1 + 0.2 turns into 0.30000000000000004. This one counts digit by digit with whole-number maths, so addition, subtraction and multiplication are exact to the last digit, division shows its remainder honestly, and the long-arithmetic working is laid out step by step.

The working, step by step — and the float trap

On the left, the long arithmetic behind your answer, exactly as it would be done on paper: the decimal points are lined up, the numbers are scaled into whole digits, and the point is put back at the end. On the right, the trap this page avoids.

  0.1
+ 0.2
─────
  0.3

Scaled to 1 decimal place: 1 + 2 = 3, then the decimal point goes back 1 place — the answer is exact, digit for digit.

Keep exploring

Decimals rarely travel alone — these neighbours handle the rounding, the fractions and the very large and very small.

How it works in three steps

How exact decimal arithmetic is worked out

Going deeper

Repeating decimals: why some divisions never end

Division is the one operation on this page that cannot always be exact in decimal form, and the reason is arithmetic, not a limitation of the tool. A fraction terminates exactly when its denominator, reduced to lowest terms, contains only the prime factors 2 and 5, because the decimal system is built on 2 × 5 = 10. Denominators with any other prime factor produce a repeating block, and the calculator shows that honestly: it prints the result to your chosen places and states that a remainder is left, instead of pretending the decimal ends.

Notation matters because a repeating decimal cannot be written out in full. Mathematicians put a bar over the repeating block, writers use an ellipsis, and this page uses words: it tells you the result is rounded to your chosen places and that the exact value continues past the last digit. The ellipsis and the bar mean the same thing, and neither is an approximation in disguise: 0.333… with the bar is exactly one third, while 0.3333333333 without it is an approximation that is short by one part in ten billion. The long-division working the main calculator shows makes the mechanism visible: dividing 1 by 3, every step is 10 ÷ 3 = 3 with remainder 1, so the next digit is always 3 and the remainder never clears. Dividing 1 by 7 cycles through six remainders before returning to 1, which is why its period is six digits long, and the same six digits, rotated, give two sevenths, three sevenths, and so on.

Choosing decimal places for a repeating result is a judgement, not a correction. Ten places, the default in the example above, is ample for money, measurement and school work, and the remainder note is what keeps it honest: it says the printed digits are the first ten of an unending sequence, not the whole number. For engineering work that needs more, the exact fraction is the right object to carry forward, and the decimal is only its display form.

Rounding modes: three ways to settle a tie, and where each belongs

Rounding to a chosen number of places is unambiguous except on an exact tie, when the digit to be dropped is 5 followed by nothing. Three conventions settle ties differently, and the Advanced options panel lets you apply each one to any number, so the table below is not a claim to take on trust: every cell can be reproduced there.

Half up is the school rule: on a tie, round away from zero, so 2.5 becomes 3 and −2.5 becomes −3. It is easy to teach and slightly biased upward in absolute value, because ties always move away from zero rather than splitting evenly. Half even, also called banker’s rounding, rounds a tie to the nearest even digit, so 2.5 becomes 2 and 3.5 becomes 4: over many ties the ups and downs cancel, which is why statistics, banking systems and the IEEE floating-point standard use it. Truncation is not rounding at all: it simply drops the extra digits, moving toward zero always, so 2.9 becomes 2 and −2.9 becomes −2, and it systematically understates magnitude. The precision select on the main calculator controls how many places division shows; the rounding mode in Advanced options controls how a number you supply is settled to a chosen places count, and the two settings answer different questions.

One caution belongs with the 1.005 row, because it is the classic trap. In binary floating point, 1.005 is stored as slightly less than 1.005, so a floating-point rounder can return 1.00 where decimal arithmetic returns 1.01 under half up. The Advanced panel here parses the digits you typed, not a binary approximation of them, so 1.005 half-up to two places is 1.01, as the table states. The same digit-exact parsing is what makes the main calculator’s 0.1 + 0.2 equal 0.3: the digits are the data, and no binary detour is allowed to corrupt them.

Percent, power and square root, worked exactly where exactness is possible

Percent is a division by one hundred in disguise, and the Advanced panel does it with the same exact arithmetic as the main calculator. Fifteen percent of 240 is 36, because 15 × 240 = 3,600 and 3,600 ÷ 100 = 36, with no rounding anywhere. Forty-five as a percentage of 180 is 25%, because 45 ÷ 180 = 0.25 and 0.25 × 100 = 25. A price rise works the same way: a $80 item raised by 20% gains $16 and costs $96, and a $96 item reduced by 20% loses $19.20 and costs $76.80, which is why a rise and a fall by the same percentage do not cancel: the second percentage acts on the new, larger or smaller base.

Power with a whole-number exponent is repeated multiplication, and for exponents from 0 to 20 the Advanced panel computes it exactly, digit for digit. Two point five cubed is 15.625, because 2.5 × 2.5 = 6.25 and 6.25 × 2.5 = 15.625. One point two to the tenth power is 6.1917364224, the factor by which a quantity growing 20% per period grows over ten periods, and the exact decimal is worth having because a rounded growth factor, compounded ten times, drifts far from the true one. Negative and fractional exponents are deliberately outside the panel’s range: a negative exponent is a reciprocal, and a fractional exponent is a root, and both deserve their own tools rather than a silently rounded answer.

Square root is the opposite case: an exact decimal answer usually does not exist, so the panel says so and gives ten places. The square root of 2 is 1.4142135624 to ten places, and the digits continue without repeating, because 2 is not a perfect square. The square root of 0.25 is exactly 0.5, because 0.5 × 0.5 = 0.25, and perfect squares and perfect-square decimals are the exception that proves the rule. Use the root for geometry, for the diagonal of a square, for standard deviation inputs, and wherever a squared quantity must be undone, and remember that the tenth place is a display choice, not the value itself.

Chained operations in the Advanced panel are evaluated strictly left to right, as written, except where brackets group a step first, as in the last row. That order is a stated convention, not the usual precedence rule: ordinary algebra would do multiplication before addition, so 0.1 + 0.2 × 0.3 under precedence is 0.16, while left-to-right it is 0.09. The panel’s convention matches how a running total is built on a simple calculator, keystroke by keystroke, and the panel says so on its face. For anything where precedence matters, group with brackets and evaluate each group with the main calculator, which is exact at every step, rather than trusting a chain to guess your intent.

Decimal places in money, measurement and school work

How many places to show is a decision about the reader, not about the mathematics. Money is shown to two places in most currencies, because the smallest coin is a hundredth of the main unit, and a price computed to more places must be rounded to cents before it can be paid: a bill of $19.995 cannot be settled in cash, and the half-up rule settles it to $20.00 while truncation settles it to $19.99, a difference the customer notices. Measurement follows the instrument: a ruler marked in millimetres justifies one decimal place in centimetres, and quoting a length to four places from such a ruler invents precision the measurement never had. School work follows the question: if the inputs carry two significant figures, an answer carried to ten places is not more accurate, it is merely longer, and the honest answer matches the precision of its inputs.

The precision select on the main calculator is about division display, and it defaults to a sensible middle: enough places to see the pattern of a repeating decimal, few enough to read at a glance. For addition, subtraction and multiplication the places take care of themselves, because exact arithmetic keeps every digit the inputs supplied: 1.20 plus 1.30 is 2.50, and trailing zeros are kept in the working even when a display might trim them, because 2.50 and 2.5 are the same value with different claims about precision. When a result must be shortened for a report, a price or a label, use the rounding panel in Advanced options and say which mode you used: a rounded number without its mode is a claim without its method, and two people rounding the same tie by different modes will honestly disagree by one in the last place.

Comparing decimals: place value decides

Two decimals are compared one place at a time, from the left. The first place where they differ decides, whatever follows it.

Line the numbers up by their decimal points and compare digit by digit, starting with the largest place. For 0.45 and 0.5, the tenths differ at once: 4 tenths against 5 tenths, so 0.5 is larger, and the 5 hundredths in 0.45 cannot rescue it, because one tenth outweighs any number of hundredths. The same rule orders 0.59, 0.6, and 0.605: write them as 0.590, 0.600, and 0.605, and the order reads off the thousandths — 0.59, then 0.6, then 0.605. Adding trailing zeros to the shorter number changes nothing about its value; it only makes the comparison visible, which is why it is the first step in every comparison done by hand.

Trailing zeros are the most common source of false comparisons. The number 0.5, the number 0.50, and the number 0.500 are the same number, because zeros after the last non-zero digit in the fractional part add no value, just as zeros before the first non-zero digit in the whole part add none. The zeros matter only as a statement of precision: 0.50 claims measurement to the hundredth, while 0.5 claims only the tenth, and in money, measurement, and school work the number of places stated is part of the answer, which is why this page keeps the places you choose rather than trimming them. For value alone, strip the trailing zeros and compare what remains; for a reported result, keep the places the setting requires, and let the trailing-zero behaviour described in the rounding section above guide how the result is shown.

On a number line, the rule is visible rather than verbal: 0.45 sits halfway between 0.4 and 0.5, 0.5 sits on the 0.5 mark, and 0.59 sits just short of 0.6, and the left-to-right order on the line is the order of the numbers. The line also shows why a decimal between 0 and 1 is always less than 1 however many digits it carries: 0.9999 is still short of 1, because every place after the point is a fraction of the place before it, and no finite string of them sums to a whole. Comparing decimals is therefore a place-value exercise, not a digit-counting one: the number with more digits is not the larger number, the number with the larger digit in the leftmost differing place is, and the calculator above settles any close call exactly, without the floating-point approximation that makes some calculators report 0.1 plus 0.2 as 0.30000000000000004.

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