Long Multiplication

You know your times tables up to 12 \times 12. But what about 47 \times 36? That is not a fact you can pull from memory, and it is far too much to juggle in your head all at once. This is the moment long multiplication was invented for: a method that chops any big multiplication into a handful of easy single-digit steps you can always rely on. Humans have used versions of it for centuries, to build pyramids, run empires, and balance the books.

The whole idea rests on one move you already trust: break each number into its tens and ones, multiply the easy pieces, and add them back up. Let's see it two ways — as a picture first, then as the compact method people scribble on paper.

The grid method: break the rectangle apart

The friendliest way in is the grid method, which is just the area model of a big rectangle chopped into smaller rectangles. Take 23 \times 14. Split 23 into 20 + 3 and 14 into 10 + 4. Now multiply every part of one by every part of the other — four easy products:

(20 \times 10) + (20 \times 4) + (3 \times 10) + (3 \times 4) = 200 + 80 + 30 + 12 = 322

Each piece is a little rectangle of the big 23 \times 14 rectangle; add the four areas and you have the whole. The place-value labels do all the hard thinking for you — because they are written on the grid, you can't forget them.

See it built

Lay the parts out in a grid. The columns are headed 20 and 3; the rows are headed 10 and 4. Each cell holds the product of its row and column heading — then add all four.

The column method: the same sum, folded up

On paper people often write it more compactly as column multiplication. Stack the numbers, lined up by place value, and multiply in two passes. First multiply 23 by the ones of 14 (that is 4): 23 \times 4 = 92. Then multiply by the tens (that is 10): 23 \times 10 = 230, which you write shifted one place to the left. Finally add the two rows:

\begin{array}{r} 23 \\ {} \times 14 \\ \hline 92 \\ 230 \\ \hline 322 \end{array}

Notice the four grid products are hiding in here: 92 = 80 + 12 (the ones row) and 230 = 200 + 30 (the tens row). Same parts, same total — the grid method and the column method are two layouts of one idea.

Worked example — 47 × 36

Now the sum from the very start. We multiply 47 by the ones (6), then by the tens (30), then add.

Pass 1 — the ones. 47 \times 6: think 7 \times 6 = 42 (write 2, carry 4), then 4 \times 6 = 24, plus the carried 4 makes 28. So 47 \times 6 = 282.

Pass 2 — the tens. Write a placeholder 0 in the ones column first (because we are really multiplying by 30, not 3), then 47 \times 3 = 141. So this row is 1410.

\begin{array}{r} 47 \\ {} \times 36 \\ \hline 282 \\ 1410 \\ \hline 1692 \end{array}

Add the two rows: 282 + 1410 = 1692. That placeholder 0 is the hero of the whole method — the next card explains why.

Worked example — a three-digit number: 213 × 24

Bigger numbers use the exact same recipe; there are just more single-digit products to do. For 213 \times 24, multiply by the ones (4), then by the tens (20, so a placeholder 0), then add:

\begin{array}{r} 213 \\ {} \times 24 \\ \hline 852 \\ 4260 \\ \hline 5112 \end{array}

Here 213 \times 4 = 852 and 213 \times 20 = 4260, and 852 + 4260 = 5112. However many digits the numbers have, the plan never changes: multiply each part, shift for place value, and add.

When you multiply by the tens digit, you are not multiplying by 3 — you are multiplying by 30. You must account for that place value, either by writing a 0 in the ones column first or by shifting the whole row one place to the left. Forgetting this placeholder is the single most common long-multiplication mistake, and it makes your tens row roughly ten times too small:

\begin{array}{r} 47 \\ {} \times 36 \\ \hline 282 \\ 141\ \, \color{red}{✗} \\ \hline 423\ \, \color{red}{✗} \end{array} \qquad\text{vs}\qquad \begin{array}{r} 47 \\ {} \times 36 \\ \hline 282 \\ 1410\ \, \color{green}{✓} \\ \hline 1692\ \, \color{green}{✓} \end{array}

A quick sanity check saves you every time: 47 \times 36 is roughly 50 \times 36 = 1800, so an answer of 423 is obviously far too small — but 1692 looks just right. And take care with carry digits too: write the carry small above the next column and remember to add it in.

Very nearly. Your phone and computer multiply in binary — numbers written with only 0s and 1s — but they use precisely this algorithm: multiply by each digit, shift left for place value, and add the partial products up. Because the only digits are 0 and 1, "multiply by a digit" is trivially easy, so the whole thing becomes just shift and add. The method you are learning by hand is, at heart, the one running billions of times a second inside every machine around you.

And the grid method has deep roots too. It is a rediscovery of the ancient lattice multiplication used by medieval merchants, and championed in Fibonacci's famous 1202 book Liber Abaci — the book that helped bring these "Hindu–Arabic" numerals, and easy multiplication with them, to Europe. When you draw a grid, you are drawing something a 13th-century trader would recognise at a glance.