Long Multiplication Calculator

What is Long Multiplication?

Long multiplication, also known as the standard algorithm, is a method for multiplying multi-digit numbers by breaking the calculation into smaller, manageable steps. It involves multiplying each digit of the multiplier by each digit of the multiplicand, recording partial products, and then adding those partial products together to get the final result. This systematic approach works for both whole numbers and decimal numbers and is a fundamental arithmetic skill taught worldwide. Practice related operations with our Long Division Calculator and the Long Addition Calculator.

For example, multiplying 2552 by 64 using long multiplication involves multiplying 2552 by the ones digit (4) to get 10208, then multiplying 2552 by the tens digit (6) to get 15312 (shifted one place left), and finally adding these partial products to arrive at 163328. The step-by-step process reveals exactly how each digit contributes to the final product.

How to Perform Long Multiplication

Follow these steps to multiply numbers using the long multiplication method:

• Stack the numbers: Write the larger number (multiplicand) on top and the smaller number (multiplier) below, aligning the digits by place value. For decimal numbers, ignore the decimal points initially.
• Multiply by ones digit: Starting with the rightmost digit of the bottom number, multiply it by each digit of the top number from right to left. If the product exceeds 9, carry the tens digit to the next column.
• Record the partial product: Write the result below the line, aligned with the ones column.
• Move to the next digit: Repeat the process for each digit of the bottom number, shifting each new partial product one place to the left.
• Add the partial products: Once all digits have been multiplied, add all the partial products together using long addition to get the final answer.

Long Multiplication with Decimals

Multiplying decimal numbers using long multiplication follows the same process with one additional step. First, count the total number of decimal places in both the multiplicand and the multiplier combined. Then, ignore the decimal points and multiply the numbers as if they were whole numbers using the standard algorithm. Finally, place the decimal point in the product so that it has the same number of decimal places as the total counted earlier.

For example, to multiply 45.2 by 0.21: there are 3 total decimal places (one in 45.2 and two in 0.21). Multiply 452 by 21 to get 9492, then place the decimal point to give 9.492. This method works for any combination of decimal numbers.

Multiplying Negative Numbers

When multiplying numbers that include negative values, follow the standard multiplication algorithm first, ignoring the signs. Then apply these rules: if one number is positive and the other is negative, the product is negative. If both numbers are positive or both are negative, the product is positive. The long multiplication calculator handles this automatically, showing the correct sign in the final product.