Long Multiplication — Definition, Formula & Examples
Long multiplication is a written method for multiplying numbers with two or more digits by breaking the problem into smaller multiplications based on place value, then adding the results together.
Long multiplication is an arithmetic algorithm in which each digit of one factor is multiplied by every digit of the other factor, with partial products aligned according to their place value positions and then summed to produce the final product.
How It Works
Write one number above the other, lining up the digits by place value. Multiply the top number by each digit of the bottom number, starting from the ones place. Each time you move to the next digit of the bottom number, shift the partial product one place to the left (or add a zero at the end) to account for its place value. Finally, add all the partial products together to get the answer.
Worked Example
Problem: Multiply 34 × 12 using long multiplication.
Step 1: Multiply 34 by the ones digit of 12, which is 2.
Step 2: Multiply 34 by the tens digit of 12, which is 1 (representing 10). Write a 0 in the ones place to shift left, then multiply.
Step 3: Add the two partial products together.
Answer: 34 × 12 = 408
Why It Matters
Long multiplication is the foundation for multiplying any size numbers by hand. You will rely on this skill when working with area calculations, unit conversions, and later when multiplying polynomials in algebra.
Common Mistakes
Mistake: Forgetting to shift partial products to the left when multiplying by the tens digit, hundreds digit, etc.
Correction: Each new row must be shifted one more place to the left. A helpful trick is to place a zero at the end of each new partial product row: one zero for the tens digit, two zeros for the hundreds digit, and so on.