Long Multiplication Calculator

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Long multiplication layout

    1 2 3 4
×       5 6
───────────
    1 2 2
    7 4 0 4
  1 1 2
  6 1 7 0 0
───────────
  6 9 1 0 4

Step by step

Add the rows: 7404 + 61700 = 69104.

How to Use the Long Multiplication Calculator

1. 1. Enter the two numbers. The first is the multiplicand and the second the multiplier, but the product is the same either way. Decimals and negative numbers are fine, up to 25 digits each.
2. 2. Choose how shifted rows look. Schools differ: some write placeholder zeros at the end of each row, others leave blank spaces and shift the row left. Pick the one your teacher uses.
3. 3. Show or hide the carries. With the box ticked, the carry from each digit is written above the next digit to its left, as it is on paper.
4. 4. Read the steps. Below the layout, every digit multiplication is written out — for instance 3 × 6 + 2 (carried) = 20: write 0, carry 2 — followed by the final addition.

How Column Multiplication Works

Long multiplication is the distributive law written in columns. To multiply 1234 by 56, you really compute 1234 × 6 + 1234 × 50: multiply by each digit of the second number, scale each result by that digit's place value, and add. Each digit of the second number gives one partial product. Within a row you multiply from right to left; when a digit product reaches 10 or more, you write the ones digit and carry the tens digit into the next column.

A digit in the tens place stands for ten times as much, which is why its row ends in a zero. Writing that zero (a placeholder) or leaving a blank and starting the row one column to the left are two ways of recording the same fact. A row for a 0 digit would be all zeros, so it is left out.

Worked Example: 1234 × 56

    1 2 3 4
×       5 6
───────────
    1 2 2
    7 4 0 4
  1 1 2
  6 1 7 0 0
───────────
  6 9 1 0 4

• Row 1, multiplying by 6: 4 × 6 = 24 (write 4, carry 2); 3 × 6 + 2 = 20 (write 0, carry 2); 2 × 6 + 2 = 14 (write 4, carry 1); 1 × 6 + 1 = 7. The row is 7404.
• Row 2, multiplying by the 5 in the tens place: the same steps with 5 give 6170, and with the placeholder zero the row is worth 61,700.
• Add the rows: 7404 + 61700 = 69104.

Checking the Answer

The calculator always shows an estimate: each number rounded to one significant figure and multiplied. Here 1234 × 56 ≈ 1000 × 60 = 60,000, so a product near 69,104 is plausible while 6,910 or 691,040 would not be. A second check is casting out nines. The digit sums of 1234, 56 and 69104 reduce to 1, 2 and 2 (1 + 2 + 3 + 4 = 10, and 1 + 0 = 1; 5 + 6 = 11, and 1 + 1 = 2; 6 + 9 + 1 + 0 + 4 = 20, and 2 + 0 = 2). Because 1 × 2 = 2, the check passes. It catches most slips but not all — swapped digits go unnoticed — so use it alongside the estimate.

Multiplying Decimals

Ignore the decimal points while multiplying, then put one back in the answer. Count the decimal places in both numbers and add them: the product has that many places. For 12.5 × 0.4, work out 125 × 4 = 500. There are one and one decimal places, two in all, so the product is 5.00, which is 5. Do the counting from the numbers you started with, not from the product, because a trailing zero can hide places.

Common Mistakes

• Dropping a carry. Add the carry to the next digit product before you split it into a ones digit and a new carry.
• Forgetting the placeholder zero. Without it the row is ten times too small, and so is the answer.
• Misaligned columns. Every row's last digit must sit directly under its place value; the layout above keeps that alignment.
• Counting decimal places in the product. Count them in the two factors and add.

Frequently Asked Questions

How do you multiply large numbers by hand?

Write one number above the other, multiply the top number by each digit of the bottom number from right to left, carrying as needed, and write each result on its own row shifted one place further left. Then add the rows. The calculator shows each of those steps for the numbers you enter.

Why do you add a zero when multiplying by the tens digit?

A digit in the tens place is worth ten times its face value, so its row is ten times larger. The zero at the end makes the row ten times larger; leaving the units column blank and shifting the row left does exactly the same thing.

How do you multiply decimals using long multiplication?

Multiply as if there were no decimal points, then count the decimal places in both factors and place the point that many places from the right of the product. For 0.5 × 0.5, 5 × 5 = 25 with two decimal places in all, giving 0.25.

Does it matter which number goes on top?

The product is the same either way because multiplication is commutative, but the work differs. It is usually shorter to put the number with fewer digits underneath, since that gives fewer rows. A number containing zeros underneath saves rows, too.

What happens with negative numbers?

Multiply the absolute values and then fix the sign: the product is negative if exactly one number is negative and positive if both or neither are. So −12 × 3 = −36 and −12 × −3 = 36.

Is lattice multiplication different from long multiplication?

Lattice (grid) multiplication writes the same single-digit products in a grid and adds along diagonals instead of carrying row by row. The digit products are identical and so is the final answer; the layout is simply another way to organise the carries.