Written arithmetic

Nobody writes out $7 + 5$ — that is a sum for the head. But $478 + 356$ is already a calculation you can slip on, and $8124 - 2758$ even more so. That is why large numbers are handled in writing: set one under the other, digit under digit, and worked through column by column.

All four written algorithms rest on one idea: a hard calculation is broken down into a sequence of operations on single digits, whose results we know by heart.

The shared principle: a column is a place value

Writing numbers in a column, we line them up so that units sit under units, tens under tens, hundreds under hundreds. The columns are not decoration — each one is a place value, which is why they must never be shifted.

Addition, subtraction and multiplication run right to left; division alone runs left to right.

Column addition

Add column by column from the right. When a column total comes out greater than $9$, its units digit stays in place and the ten travels into the next column as a carry — written as a small digit above it.

Step by step: units $8 + 6 = 14$, so we write $4$ and carry $1$. Tens $7 + 5 = 12$, plus the carried $1$ makes $13$ — write $3$ and carry $1$ again. Hundreds $4 + 3 = 7$, plus $1$ makes $8$. The answer is $834$.

Column subtraction

Subtraction also runs from the right. When the top digit is smaller than the one below it, we borrow one unit from the column on the left: the current digit gains $10$ and the neighbouring one drops by $1$.

Units: $4 < 8$, so a borrow — $14 - 8 = 6$, and the tens fall from $2$ to $1$. Tens: $1 < 5$, another borrow — $11 - 5 = 6$, and the hundreds fall from $1$ to $0$. Hundreds: $0 < 7$, one more borrow — $10 - 7 = 3$, and the thousands fall from $8$ to $7$. Thousands: $7 - 2 = 5$. The answer is $5366$.

The check is the same as for mental subtraction: $2758 + 5366$ must give $8124$.

Column multiplication

By a single-digit number we multiply in one pass. By a multi-digit one we multiply once per digit, and each further partial product shifts one place to the left, because by then we are multiplying by tens rather than by units.

First $347 \cdot 6 = 2082$. Then $347 \cdot 2 = 694$ — but that $2$ stands for tens, so the product is written shifted one place left, which means $6940$. Finally the two products are added in a column: $2082 + 6940 = 9022$.

That shift is the whole point of the algorithm rather than a drawing convention: multiplying by $26$ is broken into multiplying by $6$ and by $20$.

Long division

Division alone runs from the left. Four steps repeat until the digits of the dividend run out: divide, multiply, subtract, bring down.

1. Divide — how many times does $4$ go into $4$? Once. Write $1$ above the line.
2. Multiply — $1 \cdot 4 = 4$, written under the digit being divided.
3. Subtract — $4 - 4 = 0$.
4. Bring down the next digit of the dividend, the $9$: we now have $09$.

And round again: $9 : 4$ is $2$ (write $2$ above the line), $2 \cdot 4 = 8$, $9 - 8 = 1$, bring down the $2$ and we have $12$. Then $12 : 4 = 3$, $3 \cdot 4 = 12$, $12 - 12 = 0$. The digits are gone and the remainder is $0$, so $492 : 4 = 123$.

Whatever is left once the last digit has been brought down is the remainder, always smaller than the divisor. The definition of the remainder and the formula $a = b \cdot q + r$ are in the lesson on division.

Common mistakes

• Shifted columns — units must sit under units; one column out is an error of a whole place value.
• A lost carry — the carried one has to be added into the next column, not left hanging above it.
• Borrowing without reducing the neighbour — if the digit on the left is not dropped by $1$ after a borrow, the answer comes out too big.
• An unshifted partial product — multiplying by the tens digit shifts the result one place left; without that, $347 \cdot 26$ gives $2776$ instead of $9022$.
• A missing zero in the quotient — if the divisor does not fit into the figure after a digit is brought down, write $0$ in the quotient and bring the next digit down; skipping that step loses a digit from the answer.