What is long multiplication?
Long multiplication is the paper method for multiplying two numbers that are too big to hold in your head. You stack them, multiply the top number by each digit of the bottom number in turn, write each answer on its own row shifted one column to the left, and add the rows up.
Teachers call it the standard algorithm, or column multiplication. In England it lands in Year 5; in the US it's usually fourth and fifth grade. The method is far older than either curriculum — the same grid was being written in India and the Arab world a thousand years ago, and Fibonacci carried it into Europe in 1202.
It survives because it turns one hard multiplication into a pile of easy ones. Nobody knows 456 × 789 off the top of their head. Everybody knows 6 × 9. Long multiplication breaks the first into nine of the second plus a column of addition, and the answer comes out exact: 359,784.
The Long Multiplication Calculator at the top of this page does that on screen. It doesn't just hand you 359,784 — it draws the rows, the carries above each one, and the final sum, because the working is the part you came for. Most multiplication pages have it backwards: the answer is given away and the steps sit behind a sign-up wall or a 7-day trial. The steps are the only part with any teaching value in them.
How to use the Long Multiplication Calculator
- Type the top number — the multiplicand — into the first box. Up to 12 digits, with an optional minus sign and an optional decimal point. Thousands commas are read if you paste them in; currency symbols and fractions are not.
- Type the bottom number — the multiplier — into the second box. There's no Calculate button. The grid appears as you type and redraws on every keystroke.
- Read the grid from the top: the two numbers, a rule line, then one row for each digit of the bottom number, starting with the ones digit. Each row sits one column further left than the row below it.
- Scroll to Carries, row by row. Every partial product gets its own block with its carry digits written above the column they were carried into — the step most calculators skip, and the one people most often get wrong by hand.
- If either number has a decimal point, read the sentence under the grid. It counts the decimal places for you and says where the point lands in the answer.
- Press Copy working to take the whole layout as plain text. It's padded with spaces, so the columns stay lined up when you paste it anywhere monospace — a document, a chat message, an email to a teacher.
All of it runs in your browser. Nothing is uploaded, nothing is stored, and there's no account to make first.
How long multiplication works, row by row
456 × 789 = (456 × 9) + (456 × 80) + (456 × 700)
That single line is the whole method. Long multiplication is the distributive law written out in columns: split the bottom number into its place values — 789 is 700 + 80 + 9 — multiply the top number by each piece, then add the pieces back together.
The clever bit is the shift. You never actually multiply by 80. You multiply by 8 and slide the row one column left, which is the same thing, because moving a digit one column left multiplies it by ten. Slide two columns and you've multiplied by a hundred. That's why the columns have to line up: the position of a row is carrying its place value, so a row in the wrong column is a wrong answer even when every digit in it is right.
Here's 456 × 789 exactly as the Long Multiplication Calculator lays it out:
| Line in the grid | Where it comes from |
|---|---|
| 456 | The multiplicand — the number on top |
| × 789 | The multiplier — one row per digit, coming up |
| 4104 | 456 × 9 |
| 36480 | 456 × 8, shifted one column left |
| 319200 | 456 × 7, shifted two columns left |
| 359784 | 4,104 + 36,480 + 319,200 — the answer |
The carries live on a line above the top number, and they change for every row. Multiplying 456 by 9 starts at 6 × 9 = 54: the 4 stays in the ones column and the 5 is carried above the tens. Then 5 × 9 = 45, plus that carried 5, makes 50 — write the 0, carry 5 again. Then 4 × 9 = 36, plus 5 is 41, and with no column left to carry into, 41 becomes the front of the row. The row reads 4104 and the carry line above 456 reads 5 5.
Common multiplications, worked out
Every answer below comes straight out of this tool, not from a textbook.
| Multiplication | Rows in the grid | Answer |
|---|---|---|
| 347 × 6 | 2082 | 2,082 |
| 58 × 37 | 406, 1740 | 2,146 |
| 456 × 789 | 4104, 36480, 319200 | 359,784 |
| 1000 × 204 | 4000, 0, 200000 | 204,000 |
| 12.5 × 4 | 500 | 50 |
| 0.25 × 0.4 | 100 | 0.1 |
| -23 × 47 | 161, 920 | -1,081 |
| 123456789 × 987654321 | nine rows | 121,932,631,112,635,269 |
Four things in that table are worth a second look. A single-digit multiplier like 347 × 6 produces one row, and the grid stops there — a sum line under a single row would just repeat it. The 0 inside 204 still gets a row of its own, written as 0 rather than skipped, because skipping it is how people lose track of which column the next row belongs in.
Decimals get multiplied as whole numbers and the point is moved at the end. For 12.5 × 4 the grid shows 125 × 4 = 500; 12.5 has one decimal place and 4 has none, so the point moves one place left and the answer is 50. For 0.25 × 0.4 the grid shows 25 × 4 = 100, and with two places plus one place that's three places altogether: 0.1. Negatives keep their signs out of the columns entirely — -23 × 47 shows 23 and 47 in the grid, and the minus goes on the total.
The last row is the interesting one. 123,456,789 × 987,654,321 is exactly 121,932,631,112,635,269, and a calculator doing that multiplication in ordinary floating point returns 121,932,631,112,635,260. The last digit is simply gone. This tool multiplies character by character on the digits you typed, the way you would on paper, so nothing is rounded at any point along the way.
Where long multiplication goes wrong
Almost every wrong answer in column multiplication comes from one of four places. A row written in the wrong column, which is silently wrong — all the digits are right and the total is off by a factor of ten. A carry that got dropped, or added into the wrong column. A zero in the multiplier that got skipped instead of given a row. And a decimal point moved the wrong number of places, usually because only one of the two numbers got counted.
The fastest defence is an estimate. Round both numbers and multiply in your head before you start: 456 × 789 is roughly 500 × 800 = 400,000, so 359,784 is believable and 35,978 is not. That one check catches nearly every misplaced row and every misplaced point.
The Long Multiplication Calculator has limits of its own, and they're deliberate. It reads up to 12 significant digits and 12 decimal places per number; go past that and it says so rather than quietly truncating. It multiplies exactly two numbers, not a chain of them. Half-typed input — a lone minus sign, a trailing point, a stray letter — hides the result instead of flashing an error, because those are states you pass through while typing. And a comma is only read as thousands grouping when it's unambiguous: 1,234 reads as 1234, while 12,5 is rejected rather than guessed at as 12.5.
Related calculators
The obvious companion is the long division calculator, which walks through the other half of the same school syllabus — quotient, remainder and every intermediate step. If the numbers you're multiplying are fractions rather than decimals, the fraction calculator multiplies them and simplifies the result for you.
Multiplying a number by itself over and over is what the exponent calculator handles, and it's happy with negative and decimal powers too. Going the other direction — pulling a number apart into what was multiplied to make it — is the prime factorization calculator, which draws the factor tree as well as the answer. And when the multiplication you actually need is "what's 18% of this", the percentage calculator is the shorter route.
One more worth knowing about: the Long Multiplication Calculator never rounds, so a product with a long decimal tail comes back with every digit intact. If what you want is that answer tidied to two places, or to the nearest hundred, hand it to the rounding calculator, which will also tell you which rounding rule it applied.
Frequently asked questions
What is a partial product?
It's one row of the grid: the top number multiplied by a single digit of the bottom number, before it's shifted. In 456 × 789 the partial products are 4,104, 3,648 and 3,192. Once shifted into their columns they're written 4104, 36480 and 319200, and adding those three gives 359,784. Some curricula call the whole method "partial products" for this reason.
Which number should go on top?
Either works — the answer is identical. But putting the number with fewer digits on the bottom gives you fewer rows to write and fewer chances to slip. 789 × 456 needs three rows just like 456 × 789, but 1234 × 56 needs two rows while 56 × 1234 needs four. The tool doesn't reorder your numbers, so the grid matches whatever you typed.
How many rows will the grid have?
One per digit of the bottom number, including its zeros. A three-digit multiplier always gives three rows. The one exception is a single-digit multiplier: you get one row and no sum line under it, because that row already is the total.
Is there a shortcut for multiplying by 10, 100 or 1,000?
Yes — add that many zeros to the end, and long multiplication is just the long way of proving it. 1000 × 204 gives rows of 4000, 0 and 200000, which add to 204,000, exactly what you'd get by writing 204 and tacking on three zeros. The shortcut is safe. It's worth doing the grid once to see why it works.
My handwritten answer doesn't match the calculator. What went wrong?
Compare row by row rather than checking the totals. Type the same numbers in, then look at your first row against the tool's first row. If the rows match and only the total differs, it's an addition slip. If a row is wrong, open Carries, row by row and find the column where your digit and the tool's digit part company — that's almost always a carry that got left behind.
What does the Long Multiplication Calculator cost?
Nothing, and there's no account, no trial and no paywall over the working. This is the full tool, not a preview of one. Microapp makes money from membership rather than from gating steps, and 10% of every dollar it earns goes to charity, off the top, audited quarterly.