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Long Multiplication Calculator
A long multiplication calculator that shows the standard algorithm, each shifted partial product, and the box method.
The top number, up to 6 digits
The bottom number, up to 6 digits
23 times 45 equals 1035.
Product
23 × 45 = 1035
Standard algorithm
Indigo zeros hold the shift. Each one moves that partial product a place to the left.
Steps
- 5 is in the ones place. 3 × 5 = 15, write 5 and carry 1. 2 × 5 + 1 = 11, write 1 and carry 1. Write the remaining carry 1. The partial product is 115.
- 4 is in the tens place. 3 × 4 = 12, write 2 and carry 1. 2 × 4 + 1 = 9, write 9. Shift one place to the left. The partial product is 920, which is 23 × 40.
- Add the partial products: 115 + 920 = 1035.
Box method
23 = 20 + 3
45 = 40 + 5
Place values of the multiplicand run down the side. Place values of the multiplier run across the top.
| × | 40 tens | 5 ones |
|---|---|---|
20 tens | 800 20 × 40 | 100 20 × 5 |
3 ones | 120 3 × 40 | 15 3 × 5 |
Sum of the partial products
800 + 100 + 120 + 15 = 1035
This long multiplication calculator shows the standard algorithm for two whole numbers of up to 6 digits. It multiplies by each digit of the multiplier from the right, carries within the row, shifts the next partial product one place, and adds the partial products. A second panel uses the box method, splitting both numbers into place values and adding every cell. 23 × 45 is 1035, and 108 × 16 is 1728.
How the partial products are written
Long multiplication writes the two numbers in a stack and multiplies the top number by each digit of the bottom number, starting at the ones place. When a product is 10 or more, write the ones digit and carry the rest to the next column. The next digit shifts that row one place to the left, and each of those rows is a partial product. For 23 × 45 the rows are 115 and 920, and 115 + 920 = 1035.
The box method breaks each number into place values and multiplies every part by every other part. 23 is 20 + 3 and 45 is 40 + 5, so the boxes are 20 × 40 = 800, 20 × 5 = 100, 3 × 40 = 120, and 3 × 5 = 15. Their sum is the same 1035. A place that holds 0, such as the tens place of 108, is left out of the grid because that partial product would be 0.
Frequently Asked Questions
How do you multiply with the standard algorithm?
Line up the ones places. Multiply the top number by the ones digit of the bottom number, then by each digit to its left, shifting one place each time. Add those partial products. 108 × 16 has partial products 648 and 1080, and 648 + 1080 = 1728.
What is a partial product?
A partial product is one row of the work: the top number times a single digit of the bottom number, moved to that digit's place. In 23 × 45 the partial products are 115, which is 23 × 5, and 920, which is 23 × 40. Adding them gives 1035.
How is the box method different from long multiplication?
The box method writes every place-value product in its own cell before any carrying. For 23 × 45 the cells are 800, 100, 120, and 15, then those are added. Long multiplication folds the carries into each row and lines the rows up by shifting. Both methods reach the same product.
Why does 999 × 999 equal 998001?
Each digit is 9, so each row carries. 999 × 9 = 8991, the tens row is 89910, and the hundreds row is 899100. Adding those partial products gives 998001. The page accepts whole numbers from 0 through 6 digits, and it leaves out decimals and negative signs.