Long Multiplication Calculator
Column, grid or lattice multiplication written out with every carry.
Answer
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| Diagonal | Digits added | Sum | Write | Carry |
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About the Long Multiplication Calculator
Type two numbers to see their product worked out the way it is written in school. The column method (the standard algorithm) multiplies the top number by each digit of the bottom one — carries written above the row, a placeholder zero for each place — and then adds the rows column by column. The grid or box method splits both numbers into hundreds, tens and ones (or tenths and hundredths) and multiplies every pair. The lattice method puts each pair of digits in a box split by a diagonal and adds along the diagonals.
Every step is explained in words, and pointing at or tapping a step highlights its digits in the layout. Decimals work in every method: the tool shows how to count the decimal places, and the grid uses decimal place values directly. Copy the working as text, or print it as an answer key.
How to use it
- Type the two numbers — whole numbers or decimals, up to 30 digits each. You can also type
456 × 23in the first box. - Choose Column, Grid (box) or Lattice.
- Read the layout and the numbered steps. Point at or tap a step to highlight the digits it is about.
- Copy working copies the layout and the steps as plain text; Print prints the answer and the working without the rest of the page.
Examples
456 × 23
10,488
456 × 3 = 1,368 and 456 × 20 = 9,120 (written with a placeholder 0); 1,368 + 9,120 = 10,488.
456 × 23
10,488
400, 50 and 6 across the top, 20 and 3 down the side: 8,000 + 1,000 + 120 = 9,120 and 1,200 + 150 + 18 = 1,368.
456 × 23
10,488
The diagonals add up to 8, 8, 4, 10 (write 0, carry 1) and 0 + 1 = 1, read from the bottom right.
4.56 × 2.3
10.488
456 × 23 = 10,488, and 2 + 1 = 3 decimal places.
406 × 205
83,230
The tens digit of 205 is 0, so there is no tens row: 2,030 + 81,200 = 83,230.
Common uses
- Checking long multiplication homework one step at a time.
- Showing a child where every carry and placeholder zero comes from.
- Comparing the column, grid and lattice methods on the same numbers.
- Printing worked answers for a worksheet.
The column method
- Write the numbers one under the other, lined up on the right.
- Multiply the top number by the ones digit of the bottom number, from right to left. When a product is 10 or more, write the ones digit and carry the tens to the next column.
- Move to the tens digit. Write a 0 in the ones column first — this digit stands for tens — then multiply again. Write two zeros for the hundreds digit, and so on. A 0 digit in the bottom number gives a row of zeros, which you can skip.
- Add the rows column by column, carrying as in ordinary addition.
The grid (box) method
Split each number into its place values: 456 = 400 + 50 + 6 and 23 = 20 + 3. Draw a grid with the parts of one number across the top and the other down the side, multiply each pair (20 × 400 = 8,000) and add everything up. The grid shows why long multiplication works — every part of one number is multiplied by every part of the other — and it is the same as the area model of a rectangle 456 wide and 23 high.
The lattice method
Draw a grid with one box for every pair of digits and a diagonal through each box. Write the first number across the top and the second down the right side. In each box write the product of its two digits, tens above the diagonal and ones below. Then add along the diagonals from the bottom right, carrying into the next diagonal, and read the answer down the left side and along the bottom. All the multiplying happens before any adding, which avoids many carrying mistakes. Its Italian name, gelosia multiplication, means the grille or lattice of a window. The earliest records of it are in Arab arithmetic by the late 13th century (Ibn al-Bannāʾ), in a Latin treatise written in England around 1300 and in Chinese arithmetic by 1450 (Wu Jing); a 16th-century Indian commentary on the Līlāvatī uses it too (Wikipedia: Lattice multiplication).
Multiplying decimals
Ignore the decimal points, multiply as whole numbers, then give the answer as many decimal places as the two numbers have together. 0.2 × 0.3: 2 × 3 = 6, and 1 + 1 = 2 decimal places, so the answer is 0.06 — not 0.6. Estimate to check where the point goes: 4.56 × 2.3 is about 5 × 2 = 10, so 10.488 is right.
Limitations
- Up to 30 digits in each number for the column method and 12 for the lattice. The grid method is shown for numbers with up to 8 non-zero digits, because a bigger grid is hard to read.
- Fractions and numbers in scientific notation are not accepted: write numbers out in full, and use the fraction calculator for fractions.
- Negative numbers are multiplied without their signs; the sign is decided at the end (one negative number gives a negative product, two give a positive one).
Privacy
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Frequently asked questions
How do you do long multiplication?
Multiply the top number by each digit of the bottom number in turn, starting with the ones digit; start each new row one place further left (with placeholder zeros); then add the rows. For 456 × 23: 456 × 3 = 1,368, 456 × 20 = 9,120, and 1,368 + 9,120 = 10,488.
Why do you put a zero in the second row?
Because the second digit is in the tens place. In 456 × 23 the 2 means 20, so the second row is 456 × 20 = 9,120. Writing a 0 in the ones column first shifts the row one place to the left; the hundreds row gets two zeros, and so on.
Where does the decimal point go when you multiply decimals?
Count the digits after the decimal point in both numbers and add them: that is how many decimal places the answer has. 1.25 × 0.4: 125 × 4 = 500, and 2 + 1 = 3 places gives 0.500 = 0.5.
Which method is best?
They always give the same answer. The grid method makes place value easy to see and is often taught first; the column method is the quickest on paper; the lattice method keeps the multiplying apart from the adding, which avoids carrying mistakes.
How can I check a long multiplication?
Estimate first: 456 × 23 is close to 500 × 20 = 10,000, so 10,488 is sensible. To check exactly, divide the answer by one of the numbers — 10,488 ÷ 23 = 456 — or work it again with a different method.