Long Multiplication Calculator

Type two numbers and the calculator multiplies them the way it is taught in school, with the standard long multiplication algorithm. It shows each partial product, the carries, the place-value shift for each row and the final sum, both as a written explanation and laid out in columns. It handles whole numbers of up to 30 digits and decimals, and checks the answer with exact arithmetic. Use it to check homework, to see where a mistake crept in, or to explain the method to a learner.

a × b = a × (ones digit of b) + a × (tens digit of b) × 10 + a × (hundreds digit of b) × 100 + …

1. Write the larger number on top and the smaller underneath, lining up the right-hand digits.
2. Multiply the top number by the ones digit of the bottom number, right to left, carrying into the next column whenever a product is 10 or more.
3. Multiply by the tens digit and write the result shifted one place left, which is the same as multiplying by 10. Continue for each digit.
4. Add the partial products.

First, 347 × 6: 6 × 7 = 42, write 2 and carry 4; 6 × 4 + 4 = 28, write 8 and carry 2; 6 × 3 + 2 = 20, giving 2082. Next, 347 × 2 in the tens place gives 694, written one place left as 6940. Adding them, 2082 + 6940 = 9022.

Ignore the decimal points, multiply as whole numbers, then count the decimal places in both numbers and place the point that many digits from the right of the answer. For 12.5 × 0.4, multiply 125 × 4 = 500; there are two decimal places in total, so the answer is 5.00, which is 5. Estimating first, 12 × 0.4 is about 5, is a quick check that the point is in the right place.

A zero digit gives a partial product of zero, which you can skip as long as the next row is shifted correctly. In 908 × 307, the tens digit is 0, so the rows are 908 × 7 = 6356 and 908 × 300 = 272400, adding to 278756. Forgetting the extra shift after a zero is one of the most common mistakes.

• Adding the carry before multiplying instead of after: carry digits are added to the product of the next column.
• Forgetting to shift each new row one place left (or to write a placeholder zero).
• Misaligned columns when adding the partial products; squared paper helps.
• Placing the decimal point by lining up the decimals, as in addition, instead of counting decimal places.

Two quick checks catch most errors. First, estimate by rounding: 347 × 26 is close to 350 × 25, about 8,750, so an answer near 9,000 is plausible while 902 or 90,000 is not. Second, check the last digit: the ones digit of the product must match the ones digit of the product of the two ones digits, here 7 × 6 = 42, so the answer must end in 2. Casting out nines is a traditional check that works with digit sums, though it cannot catch every mistake.

Some classrooms teach the area or box method, which splits each number into tens and ones and multiplies every pair, or the lattice method, which records partial products in a grid. They produce the same partial products in a different layout. The standard algorithm is the most compact and is the method named in the Common Core standards for fifth grade.