Long multiplication is the standard written method for multiplying numbers with more than one digit. You multiply the top number by each digit of the bottom number, shift each result to the correct place value, and add the rows. This calculator lays out every partial product with its placeholder zeros, explains the carries in each row, and handles decimals and negative numbers.
How to use the long multiplication calculator
- Enter the multiplicand (top number) and the multiplier (bottom number). Decimals and negative signs are allowed; commas are ignored.
- The product appears on the tape with the number of partial products and decimal places.
- The layout shows the factors, each partial product (gray zeros are placeholders), the addition line and the product.
- Show the work describes the carries digit by digit for every row.
The standard algorithm
- Write the numbers right-aligned, one above the other.
- Multiply the top number by the ones digit of the bottom number, right to left, carrying as you go.
- Multiply by the tens digit, writing one placeholder zero first. Continue with each digit, adding one more placeholder zero each time.
- Add the partial products.
- For decimals, place the point: count the decimal places in both factors and count that many places from the right of the product.
Here b0, b1, b2 are the ones, tens and hundreds digits of the bottom number — long multiplication is the distributive property written out in columns.
Worked example: 347 × 26
Row 1 — ones digit 6: 7 × 6 = 42, write 2 carry 4; 4 × 6 + 4 = 28, write 8 carry 2; 3 × 6 + 2 = 20, write 20. Partial product: 2,082.
Row 2 — tens digit 2: write a placeholder 0, then 7 × 2 = 14, write 4 carry 1; 4 × 2 + 1 = 9; 3 × 2 = 6. Partial product: 6,940.
Add: 2,082 + 6,940 = 9,022.
If the problem were 3.47 × 2.6, the same digits give 9,022, and with 2 + 1 = 3 decimal places the product is 9.022.
Zeros in the multiplier
When a digit of the bottom number is 0, its partial product is all zeros. Some people skip that row, but then the next row must be shifted two places instead of one. The calculator writes the zero row so the shifting stays visible: in 347 × 206 the rows are 2,082, 0 and 69,400, which add to 71,482.
Checking the answer
- Estimate: 347 × 26 is close to 350 × 25 = 8,750, so 9,022 is in the right range.
- Swap the factors: 26 × 347 should give the same product.
- Divide back: 9,022 ÷ 26 = 347. The long division calculator shows that check step by step.
The lattice method produces the same digits in a grid, which some students find easier because all the carrying happens at the end. If single-digit facts are slowing you down, practice them with the multiplication table.
Frequently asked questions
What is a partial product?
It is the result of multiplying the top number by one digit of the bottom number, shifted to that digit's place value. In 347 × 26, the partial products are 347 × 6 = 2,082 and 347 × 20 = 6,940. Their sum is the final product.
Why do you add a zero when multiplying by the tens digit?
The tens digit is worth ten times its face value, so its partial product must be ten times larger. Writing a placeholder zero in the ones column shifts every digit one place left. The hundreds digit gets two placeholder zeros, and so on.
How do you multiply decimals using long multiplication?
Ignore the decimal points and multiply the whole numbers. Then count the total number of decimal places in both factors and place the decimal point that many places from the right of the product. For 3.47 × 2.6: 347 × 26 = 9,022, and 2 + 1 = 3 places gives 9.022.
Which number should go on the bottom?
Either order gives the same product, but putting the number with fewer digits on the bottom means fewer partial products to add.