Long Multiplication Calculator

How long multiplication works

Long multiplication — column multiplication — splits one hard product into a few easy ones. It rests on a single rule of arithmetic, the distributive law: a number may be broken into its place values and multiplied piece by piece.

That is exactly what the column layout writes down. Each digit of the multiplier gives one partial product, shifted as far to the left as its place demands, and the partial products are added. The carries do the same job inside a single row: when a column passes nine, the tens move on to the next column.

Because multiplication is commutative, 16 × 3 and 3 × 16 are the same product — 48 either way. The order does change the work, though: the number with fewer digits makes the better multiplier, because it produces fewer partial products. Pages on this site are addressed with the larger number first, so 3 × 16 and 16 × 3 both lead to the same solution.

Step by step

1. Line the numbers up on the right. The multiplicand on top, the multiplier underneath, units under units.
2. Multiply by the last digit of the multiplier. Go right to left through the multiplicand, writing the units of each result and carrying the tens.
3. Move to the next digit and shift. That digit stands for tens, so its partial product is ten times bigger — write it one place further left (or put a zero in the empty place).
4. Add the partial products. Their sum is the answer.
5. Check by dividing back. The product divided by either factor must give the other one exactly.

Products people look up

Where multiplying by hand still pays

• Homework and exams: the method is taught in years 4–6, and marks are given for the working, not only for the answer.
• Shopping and quantities: 12 packs of 5, 15 boxes of 12 — the arithmetic that decides whether one trolley is enough.
• Materials: tiles, planks, sheets and seedlings are counted in rows times columns.
• Time and money: a weekly figure times 52, a daily figure times 30 — one column and the year is in front of you.
• Checking a calculator: a mistyped digit is caught by an estimate, and an estimate is a single-digit multiplication.

Where the column method stops

• Fractions and mixed numbers are not multiplied in a column: numerators multiply with numerators and denominators with denominators. That is the fraction multiplication calculator.
• Decimals go through the whole-number product first: multiply without the point, then move the point back by as many places as the two factors have together.
• A product is not an area. 16 × 16 is 256 as a number; if those are feet, it is 256 square feet, which is what the square footage calculator answers.
• The reverse operation is division. To undo a product, use long division — it is also how the answer here is checked.

Shortcuts worth knowing

• × 5 is half of × 10: 48 × 5 = 480 ÷ 2 = 240.
• × 9 is × 10 minus the number: 37 × 9 = 370 − 37 = 333.
• × 11 for two digits: add the digits and drop the sum in the middle — 24 × 11 = 2|6|4 = 264.
• × 12 is × 10 plus × 2: 15 × 12 = 150 + 30 = 180.
• Round and correct: 19 × 7 = 20 × 7 − 7 = 140 − 7 = 133.