Written calculation

You can read number sentences from basic math notation and use the properties of operations to see why reordering or splitting sums can still give the same answer. This chapter teaches written calculation — how to add, subtract, multiply, and divide on paper by stacking digits in vertical form: one number above another, digit under digit. The ideas work for any size number; the visualizer below works best with about eight digits or fewer per line.

Why columns?

In counting and numbers you learned place value: the ones column, tens column, hundreds column. In vertical form, digits that share a place must sit in the same column. If you shift a digit left or right, you are changing its value.

Always line up the ones under the ones, then tens under tens, and so on. Draw a horizontal line under the stack before you write the answer.

Carry and borrow

In the four operations, a column sometimes gives a digit from 0 to 9 and sometimes gives 10 or more (or too small to subtract). Then you adjust the next column to the left.

• Carry — in addition, when a column sums to 10 or more, you write the ones digit of that sum in the column and pass the extra ten to the next column on the left.
• Borrow — in subtraction, when the top digit is smaller than the bottom digit, you take one ten from the column on the left, add 10 to the current column, then subtract.

Work from the right (ones) toward the left, one column at a time.

Carry: 27 + 59 = 86

¹
  27
+ 59
────
  86

In the ones column: \(7 + 9 = 16\). Write 6 in the ones place. Carry 1 to the tens column (written small above the tens). In the tens column: \(2 + 5 + 1 = 8\).

Borrow: 47 − 19 = 28

−1
  47
− 19
────
  28

In the ones column: \(7 - 9\) does not work. Borrow 10 from the tens: the 4 becomes 3, and the ones digit becomes 17. Then \(17 - 9 = 8\). In the tens column: \(3 - 1 = 2\).

Addition

Write the numbers one above the other, plus sign to the left of the bottom number, line underneath, sum below. Add each column from right to left; carry when needed.

Subtraction

Put the larger number on top. Subtract column by column. When the top digit is too small, borrow from the left (decrease that digit by 1, add 10 to the current column). Example with a zero in the middle: \(503 - 278 = 225\) needs borrowing across the tens column.

Multiplication

Write the larger number on top (often). Multiply the top number by each digit of the bottom number, one partial product per digit, shifted so ones line up with ones. Add the partial products for the final answer.

   23
 × 14
 ────
   92
  230
 ────
  322

Here \(23 \times 4 = 92\) and \(23 \times 10 = 230\); \(92 + 230 = 322\).

Division

Long division writes the divisor outside a bracket, the dividend inside, and the quotient on top. At each step: how many times does the divisor fit into the current part of the dividend? Write that digit in the quotient, multiply, subtract, then bring down the next digit. If the division is not exact, the amount left is the remainder.

To calculate and check a remainder, always use \( \text{dividend} = \text{divisor} \times \text{quotient} + \text{remainder} \), with \(0 \le \text{remainder} < \text{divisor}\). For example:

\[ \begin{aligned} 157 \div 12 &= 13 \text{ remainder } 1 \\ 157 &= 12 \times 13 + 1 \end{aligned} \]

The remainder can also be written as a fraction of the divisor: \[ \begin{aligned} 157 \div 12 &= 13 + \frac{1}{12} \end{aligned} \]

Names of the parts

Try it yourself

Use the visualizer below: enter two or more whole numbers (one per line), choose an operation, and click Generate visualization. Step through with Play or the arrow buttons to see each carry, borrow, and column update.

History

People calculated with tokens, rods, and abacuses long before everyone used pencil and paper. In many places, including China, India, and the Arab world, scribes lined up digits in columns so addition and subtraction could be done digit by digit.

Mechanical calculators (1600s–1900s) had to solve the same puzzle: when one wheel moved from 9 to 0, it had to add 1 to the wheel for the next place. Early machines often did addition first, then ran a separate carry cycle to update higher digits. Later designs passed the carry instantly along the whole machine. That idea appears again inside modern chips: a carry flag records whether the last step overflowed a column.

Derivation

Why column addition works

Split each number by place value. For example, \(27 + 59 = (20 + 7) + (50 + 9) = (20 + 50) + (7 + 9) = 70 + 16 = 86\). The ones column adds \(7 + 9\); the tens column adds \(2 + 5\) plus any carry. Each column only mixes the same place — that is why alignment matters.

Subtraction and addition

Subtraction is the inverse of addition: \(c = a - b\) means the same as \(c + b = a\). Borrowing moves one ten from the left so the subtraction in the current column matches that story.

Checkpoints

Real life

• Shopping — adding prices on a receipt column by column.
• Scores — finding the difference between two team totals.
• Sharing — dividing a total equally (e.g. \(157\) stickers among \(12\) children: quotient 13, remainder 1).

Question bank — Easy

  34
+ 25
────
  59

Intermediate

¹
  27
+ 59
────
  86

Difficult

Hardcore

References

• Wikipedia — Carry (arithmetic) and Subtraction.
• Cajori, Florian — A History of Mathematical Notations.
• Other school methods (Austrian/additions, partial differences) exist; this chapter uses column borrow as in most HK primary texts.