How to use the long multiplication step-by-step calculator
- Enter the number to multiply and the number it is multiplied by. Each can have up to 9 digits, and commas are fine.
- Read the standard method table: one row for each digit of the second number, with the partial product and the carries used.
- Read the box method table: place values of the first number across the top and of the second number down the side, with each box holding one product.
- Compare the total at the bottom of each table with the product at the top. They always agree.
Formula
partial product for digit d at place 10^j = (first number) x d x 10^j
product = sum of the partial products
box method: box = (digit x 10^i of the first number) x (digit x 10^j of the second number)
product = sum of all boxes
A carry is the tens digit of a single-digit product plus any carry brought in. The page also checks the result against exact big integer multiplication.
4,075 times 12
Start with the ones digit, 2: 4,075 x 2 = 8,150. The ones place 5 x 2 = 10 writes 0 and carries 1, and 7 x 2 + 1 = 15 writes 5 and carries 1. Next, the tens digit, 1: 4,075 x 10 = 40,750. Add them: 8,150 + 40,750 = 48,900.
The box method splits 4,075 into 4,000, 0, 70 and 5 and 12 into 10 and 2. The boxes are 40,000, 0, 700, 50 for the 10 row and 8,000, 0, 140, 10 for the 2 row, and they also add up to 48,900. Another example: 346 x 27 = 2,422 + 6,920 = 9,342.
Two views of the same work
Long multiplication and the box (area) method use the same partial products. The standard layout stacks them in rows and uses carries to keep each digit small, which is quick on paper. The box layout keeps every place value visible, which makes it clear why shifting a row left by one place multiplies it by ten. Seeing both side by side helps when you teach or learn the method.
For division, see the long division calculator. For a different paper method that lays the digits on a diagonal grid, see the lattice multiplication worksheet generator, and for practice sheets, the math worksheet generator.
Checking your own work
Estimate first: 4,075 is a little over 4,000 and 12 is a little over 10, so the product should be a little over 40,000 and under 50,000. A quick check with the last digit also helps: 5 x 2 ends in 0, so the product ends in 0. Compare your steps with the rows above. If a partial product differs, check the carries first, because a missed carry is the most common slip.
Assumptions and limits
- Both numbers are whole numbers of 0 or more with at most 9 digits.
- Negative numbers and decimals are not handled. Multiply the digits as whole numbers and place the decimal point afterward.
- The carries column lists the nonzero carries that pass into the next place for that row, not the final carry out of the last digit.
- Results are exact. Large numbers are shown with thousands separators.
Frequently asked questions
Why does each row shift to the left?
Each digit of the second number stands for a power of ten. The tens digit multiplies by ten times its value, so its partial product ends with a zero, and the hundreds digit with two zeros, and so on.
What is a carry in long multiplication?
When a digit-by-digit product is 10 or more, you write the ones digit and carry the tens digit to the next place, where it is added to that place product. It keeps every written digit between 0 and 9.
How is the box method different?
It breaks each number into place values and multiplies every pair, filling a grid of boxes. Nothing is carried while you fill the boxes, and you add everything at the end.
Why is there a limit of 9 digits?
Nine digits keep the box table readable and the product within 18 digits. Larger numbers fit in the calculation, but the table would be too wide to read.