The subject in plain language
What is long multiplication?
Long multiplication is the standard written method for multiplying numbers one digit at a time. Instead of solving the entire calculation at once, it breaks the work into smaller partial products.
Each partial product is shifted according to place value, and the completed rows are added to obtain the final product. This makes the method useful when numbers are too large to multiply comfortably in your head, when decimals must be placed carefully, or whenever you want to verify every stage of the calculation.
The familiar written method is built on base-ten positional notation: multiplying by a tens digit, for example, produces a row shifted one place to the left.
Meaning before algorithm
Long multiplication distributes one factor by place value
Each digit of the multiplier represents a different power of ten. Long multiplication forms one partial product for each of those digits, shifts it to the correct place, and adds the rows.
- a
- multiplicand
- b
- multiplier
- p
- product
Area and place value
Why partial products add to the answer
For 23 × 14, split 23 into 20 + 3 and 14 into 10 + 4. The four rectangles represent every distributed product.
2003 × 10
3020 × 4
803 × 4
12
Adding 200 + 30 + 80 + 12 gives 322. The written algorithm combines the same pieces more compactly.
Worked example
Multiply 347 by 26
- Ones row: 347 × 6 = 2,082, including the carries within that row.
- Tens row: 347 × 20 = 6,940. The zero shift records that 2 means 20.
- Add the partial products: 2,082 + 6,940 = 9,022.
Decimal reasoning
Restore the combined decimal scale
For 123.45 × 6.7, temporarily multiply 12,345 × 67. The inputs contain two plus one fractional places, so the product receives three: 827.115. This is a place-value transformation, not a rounding rule.
12,345 × 67 = 827,115 → 827.115
Verification and boundaries
Check magnitude, last digit, and inverse division
347 is near 350 and 26 is near 25, so a result near 8,750 is reasonable. The last digit must match 7 × 6 = 42, so it ends in 2. Finally, 9,022 ÷ 26 = 347.
Shift every row
A tens digit contributes tens, not ones; omitted place shifts are a common error.
Keep carries local
Each carry belongs to the next multiplication within the same partial-product row.
No early rounding
Decimal inputs remain exact throughout the calculation.
Input scope
Non-negative inputs may contain up to 200 digits and 100 decimal places.