Calculate with Long Multiplication
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What is a long multiplication calculator?
Long multiplication is a place-value algorithm for multiplying whole numbers. It multiplies the multiplicand by each digit of the multiplier, shifts each partial product to match that digit’s place, and adds the partial products to obtain the final product.
Important terms
- Multiplicand
- The whole number being multiplied.
- Multiplier
- The whole number whose digits determine the partial products.
- Partial product
- The result of multiplying the multiplicand by one digit of the multiplier, shifted to the correct place value.
- Product
- The final result after all aligned partial products are added.
When this method is useful
- Checking written multiplication homework step by step.
- Multiplying integers too large for comfortable mental arithmetic.
- Teaching how the distributive property and decimal place value work together.
- Verifying exact products without floating-point rounding.
Formula and variables
- a
- the multiplicand
- b
- the multiplier
- dₖ
- the multiplier digit in the 10ᵏ place
- P
- the product
Writing the multiplier as a sum of digits times powers of ten applies the distributive property. Each term a × dₖ × 10ᵏ is one aligned partial product; their sum is a × b.
Domain note: This method accepts non-negative whole numbers containing up to 30 digits each. Use the regular Multiplication Calculator for signed or decimal values.
How to use this calculator
- 1
Enter the multiplicand and multiplier as whole numbers.
- 2
Starting with the multiplier’s ones digit, multiply it by every digit of the multiplicand and handle carries within that row.
- 3
Repeat for each multiplier digit, shifting the next partial product one place farther left each time.
- 4
Add the aligned partial products.
- 5
Check that the displayed sum equals the exact product.
Step-by-step examples
Example 1
Problem: Multiply 347 by 26.
Substitution: 347 × 6 = 2,082; 347 × 20 = 6,940
Result: 2,082 + 6,940 = 9,022
The tens-row partial product is shifted one place because the 2 in 26 represents 20, not 2.
Example 2
Problem: Multiply 405 by 32.
Substitution: 405 × 2 = 810; 405 × 30 = 12,150
Result: 810 + 12,150 = 12,960
The zero in 405 remains a place-value placeholder; it does not remove the tens place.
Rules, edge cases, and related ideas
Distributive property
For example, 347 × 26 equals 347 × (20 + 6), so it equals 347 × 20 plus 347 × 6.
Place-value alignment
A partial product must begin beneath the place represented by its multiplier digit; otherwise its value changes by a power of ten.
Commutative check
Reversing the factors gives the same product, although the number and shape of written rows may change.
Inverse check
For a nonzero factor, dividing the product by that factor should recover the other factor with remainder zero.
Edge cases and limitations
- Multiplying by zero gives zero, although the working still records the zero partial product.
- Leading zeros do not change a number’s value and are removed from the displayed result.
- Inputs longer than 30 digits are rejected to keep the working readable; the accepted range is still calculated exactly.
How this differs from a related concept
Long multiplication and lattice multiplication calculate the same product using the same one-digit facts. Long multiplication arranges shifted partial-product rows, while lattice multiplication places tens and ones in cells and sums diagonal bands.
Understand the result
Every displayed row is a genuine partial product. A zero multiplier digit produces a zero row, and a digit in the tens, hundreds, or higher place adds the corresponding number of trailing zeros before the rows are added.
Common mistakes
- Forgetting to shift the second and later partial products left.
- Dropping a carry during a one-digit multiplication row.
- Adding partial products with misaligned place values.
- Treating a zero inside a factor as if the place did not exist.
Frequently asked questions
Why does the second partial product end in zero?
It comes from the tens digit of the multiplier, so its one-digit product is multiplied by 10 and shifted one place left.
Does the order of the factors matter?
The product is unchanged because whole-number multiplication is commutative, but choosing the shorter factor as the multiplier often produces fewer written rows.
Can this calculator multiply decimals?
No. This page explains whole-number long multiplication. Use the regular Multiplication Calculator for decimal or negative factors.
How can I check a long multiplication answer?
Estimate the magnitude, reverse the factors, or divide the product by one nonzero factor to see whether the other factor is recovered.