Long Multiplication Calculator
Multiply numbers with the standard algorithm, shown step by step — partial products, place-value shifts, decimal-point placement, and the sign rule.
What is long multiplication?
Long multiplication breaks a hard product into easy single-digit multiplications. You multiply the top number by each digit of the bottom number, write each result as a partial product shifted into the correct place, and add the partial products.
The steps
- Multiply the top number by the ones digit of the multiplier.
- Multiply the top number by the tens digit, and shift that partial product one place left.
- Continue for every digit of the multiplier.
- Add the partial products, then place the decimal point and sign.
Worked example: 234 × 56
- 234 × 6 — 6×4 = 24 (write 4, carry 2), 6×3 = 18 + 2 = 20 (write 0, carry 2), 6×2 = 12 + 2 = 14. First partial product: 1404.
- 234 × 5 — gives 1170, shifted one place left to 11700.
- Add: 1404 + 11700 = 13104.
So 234 × 56 = 13,104.
Multiplying decimals: 45.2 × 0.21
Drop the points and multiply 452 × 21 = 9492. The two factors have 1 + 2 = 3 decimal places, so place the point three digits from the right: 9.492.
The sign rule
Same signs make a positive product; different signs make a negative one:
- −12 × 13 = −156
- −12 × −13 = 156
Common mistakes
- Forgetting the shift (or the placeholder 0) on later partial products.
- Miscounting decimal places when placing the point.
- Losing a carry inside a partial product.
Long multiplication is the standard-algorithm twin of lattice multiplication — same answer, different bookkeeping — and it relies on long addition to total the partial products.
Frequently asked questions
- What is long multiplication?
- Long multiplication (the standard algorithm) multiplies a number by each digit of the second number in turn, writes each result as a shifted partial product, and adds the partial products together. It works for any size of number without a times table beyond ten.
- What is a partial product?
- A partial product is the result of multiplying the whole top number by a single digit of the bottom number. For 234 times 56, the partial products are 234 times 6 equals 1404 and 234 times 5 equals 1170, shifted one place left to become 11700.
- Why do I shift each partial product to the left?
- Because each digit of the multiplier stands for a larger place value. The 5 in 56 means 50, not 5, so 234 times 5 is really 234 times 50 equals 11700. Shifting one place left multiplies by ten and keeps the place values lined up.
- How do I multiply decimals?
- Ignore the decimal points and multiply the numbers as whole numbers. Then count the total number of decimal places in both factors and place the decimal point that many digits from the right of the answer. 45.2 times 0.21 has three decimal places, so 9492 becomes 9.492.
- What is the sign rule for multiplication?
- Two numbers with the same sign give a positive product, and two numbers with different signs give a negative product. So negative 12 times 13 is negative 156, while negative 12 times negative 13 is positive 156.
- Which number goes on top?
- It does not change the answer, but placing the number with more digits on top usually means fewer partial products to add. This calculator arranges the digits for you and echoes your original inputs.
- How is long multiplication different from lattice multiplication?
- They give the same product. Long multiplication mixes multiplying and carrying in each partial product, while lattice multiplication separates every single-digit product into a grid and adds along diagonals at the end, which some learners find easier to track.