Long Multiplication Calculator

Long multiplication is the standard by-hand method for multiplying numbers that have more than one digit: you multiply the first number by each digit of the second number separately, shift each of those results to match its place value, then add everything together for the final product. It's called "long" because writing out every partial result takes more space on paper than a single memorized multiplication fact, but it works for numbers of any size.

A partial product is the result of multiplying the first number by just one digit of the second number, before any shifting or adding happens. A two-digit multiplier produces two partial products, a three-digit multiplier produces three, and so on - this calculator's own worked solution labels and shows every one of them for your exact numbers.

Carrying happens when multiplying a single pair of digits produces a result of 10 or more: you write down only the ones digit of that result and carry the tens digit over to be added into the next digit's multiplication. For example, multiplying 5 by 9 gives 45 - you'd write 5 and carry the 4, then add that 4 into the next digit's product before writing the next digit. This calculator's worked solution shows exactly where each carry happens for your own numbers, not just the final digits.

The shift accounts for the actual place value of the digit you just multiplied by. The tens digit of a number isn't really that digit alone - it represents that digit times 10 - so its partial product needs to be ten times larger, which on paper means shifting it one place to the left (equivalent to writing a placeholder 0). The hundreds digit shifts two places, the thousands digit shifts three, and so on, matching the digit's own position in the second number.

The sign is handled completely separately from the digits: multiply the two numbers' magnitudes (their values without any sign) using the ordinary long multiplication method, then apply a sign to the result afterward. If exactly one of the two original numbers was negative, the product is negative; if both were negative, or both were positive, the product is positive - the digit-by-digit multiplication work itself never changes based on sign.

Ignore the decimal points at first and multiply the two numbers as if they were whole numbers, then count how many total decimal places appeared across both original numbers and place the decimal point that many positions from the right in the final answer. For 1.25 x 2.4, multiplying 125 by 24 gives 3,000, and since 1.25 contributes 2 decimal places and 2.4 contributes 1 (3 total), the decimal point lands 3 places from the right, giving 3.

Yes, for any realistic input this calculator computes an exact result - numbers with up to roughly nine digits each multiply together with no loss of precision at all, verified directly against the underlying engine. Beyond that range, extremely large products can begin to lose precision or display in scientific notation, a genuine limitation of standard numeric computation rather than something specific to this calculator, and not a concern for typical classroom or homework-sized numbers.

A basic calculator app gives you only the final number, with no visibility into how it was reached. This tool instead walks through the same standard algorithm taught in school - every partial product, every carry, and the place-value shift for each digit of the second number - using your own actual numbers, so you can follow (or check) the reasoning, not just the result.

Not currently - this v1 multiplies exactly two numbers, matching the standard long multiplication method taught by hand, which is itself always a two-number-at-a-time process. To multiply three or more numbers, multiply the first two here, then use that result as one of the two numbers in a second calculation.