Long Multiplication Calculator

Enter two numbers to see the full long multiplication breakdown — every partial product by place value, the final product, and a casting-out-nines accuracy check.

Quick Facts

Standard algorithm
Multiply by each digit, then add
Multiply the top number by each digit of the bottom number (ones, tens, hundreds, ...), shifting each partial product one place left, then sum every partial product.
Decimal placement
Count total decimal places
Multiply as whole numbers, then place the decimal point that many digits from the right in the final product.
Casting out nines
(A mod 9) x (B mod 9) mod 9 = product mod 9
A fast error check based on remainders — if the two sides disagree, there is an arithmetic mistake.

Your Results

Calculated
Product
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Multiplicand x multiplier
Partial products
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One per multiplier digit, shifted by place value
Digit multiplications
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Single-digit multiplications performed
Casting-out-nines check
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Quick remainder-based verification

Ready

Enter a multiplicand and multiplier, then press Calculate.

How Long Multiplication Works

Long multiplication is the standard written method for multiplying multi-digit numbers by hand. It works because of the distributive property: any number can be broken into a sum of place values (thousands, hundreds, tens, ones), and multiplying two numbers is the same as multiplying every place-value piece of one number by every place-value piece of the other, then adding all those pieces together. The written algorithm is a shortcut for doing that expansion one digit of the multiplier at a time instead of one digit of each number at a time.

The algorithm, step by step

Write the multiplicand on top and the multiplier underneath. Starting with the rightmost (ones) digit of the multiplier, multiply it by the entire top number to get the first partial product. Move to the next digit left (the tens digit), multiply it by the entire top number, and shift that partial product one place to the left (equivalent to multiplying by 10) — in the classic layout this is shown by writing a placeholder 0 in the ones column. Continue for the hundreds digit (shift two places), and so on for every digit of the multiplier. Finally, add all the partial products together in a single column addition to get the final product. For example, 3247 x 58: the ones digit 8 gives 3247 x 8 = 25,976; the tens digit 5 gives 3247 x 5 = 16,235, shifted left one place to 162,350; adding 25,976 + 162,350 = 188,326.

Multiplying decimals with the same method

Long multiplication handles decimals by temporarily ignoring the decimal points: strip them out, multiply the two numbers as if they were whole numbers using the steps above, then count how many digits appeared after the decimal point in each original factor. Add those two counts together and place the decimal point that many digits from the right in your product. For instance, 3.2 x 1.5 has one decimal place in each factor (two total), so 32 x 15 = 480 becomes 4.80.

Common mistakes

  • Forgetting the placeholder shift: every partial product after the first must be shifted one additional place left, or the digits will be added in the wrong columns.
  • Losing track of carries: when a single-digit multiplication produces a two-digit result, the tens digit must be carried into the next column, just like in addition.
  • Misplacing the decimal point: count decimal places in both factors, not just one, and count them in the original numbers, not in an intermediate result.
  • Sign errors: the product of two negative numbers is positive; the product of one negative and one positive number is negative.

Checking your result with casting out nines

Casting out nines is a quick error check based on the fact that a number and its digit sum always leave the same remainder when divided by 9. Find the remainder of each factor mod 9, multiply those two remainders together and reduce the result mod 9, then compare it to the remainder of your product mod 9. If the two don't match, there's an arithmetic mistake somewhere in the long multiplication. A match doesn't guarantee the answer is perfect (it can miss digit-swap errors that happen to preserve the remainder), but it catches the vast majority of common slips.

Frequently Asked Questions

What is long multiplication?
Long multiplication is the standard pencil-and-paper method for multiplying multi-digit numbers. You multiply the top number by each digit of the bottom number separately (starting from the ones digit), shift each of these partial products one place left for every place value the multiplier digit represents, then add all the partial products together to get the final answer.
How do I multiply decimal numbers using long multiplication?
Ignore the decimal points and multiply the digits as whole numbers. Then count the total number of decimal places in both factors combined, and place the decimal point that many digits from the right in the product. For example, 3.2 x 1.5 has 1 + 1 = 2 decimal places, so 32 x 15 = 480 becomes 4.80.
Why do I shift the partial product when multiplying by the tens digit?
Each digit in the multiplier represents a place value (ones, tens, hundreds, and so on). Multiplying by the tens digit really means multiplying by that digit times 10, so the partial product is shifted one place to the left (equivalent to appending a zero) before it is added to the others. A hundreds digit shifts two places, and so on.
How can I check a long multiplication answer without redoing the whole problem?
Use casting out nines: find the remainder when each factor is divided by 9 (the digit sum, reduced repeatedly, works the same way), multiply those two remainders together and reduce mod 9, then compare it to the remainder of your product divided by 9. If they don't match, there is an arithmetic error somewhere; if they match, your answer has passed a quick consistency check (though it cannot catch every possible mistake).

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