How the column methods work

You already know what the four operations mean: adding joins, subtracting takes away or compares, multiplying makes equal groups, dividing shares or groups. This layer is about how we actually do them when the numbers get big, like a kirana shop's monthly sales of ₹2,48,675, or a district population in lakhs, and why each step of the method is allowed.

Every column method you will meet rests on one idea: place value. A digit's position tells you whether it counts ones, tens, hundreds or lakhs. Carrying, borrowing, shifting a row one place left in long multiplication and bringing a digit down in long division are all ways of respecting that one idea.

Mathematicians give every part of a calculation its own name. That sounds fussy, but it saves a lot of confusion. "Divide 84 by 7" and "divide 7 into 84" both mean 84 ÷ 7, and the words dividend and divisor tell you exactly which number is which, whatever order the sentence uses.

Here is the full set for the four operations.

To add large numbers, write them one under the other so that digits of the same place value stand in the same column: ones under ones, tens under tens, lakhs under lakhs. Then add one column at a time, starting from the ones on the right.

Why start on the right? Because a column can overflow. If the ones add up to 15, that is 1 ten and 5 ones. The 5 stays; the 1 ten must be carried into the tens column. If you started on the left, you would have to go back and fix columns you had already written.

Column subtraction works the same way, from right to left, one place at a time. The difficulty comes when a digit on top is smaller than the digit below it, as in 52 − 17: you cannot take 7 ones from 2 ones.

So you borrow, which really means regroup: take 1 ten from the tens column and change it into 10 ones. Now the ones column has 12, and 12 − 7 = 5. The tens column has one fewer ten, so it is 4 − 1 = 3. Answer: 35.

The number itself has not changed: 52 is still 5 tens and 2 ones, or 4 tens and 12 ones. You have just rewritten it in a form that lets the subtraction happen.

Multiplying by 10 makes every part of a number ten times bigger: ones become tens, tens become hundreds, and so on. So every digit moves one place to the left, and the empty ones place is filled with a 0.

• 347 × 10 = 3,470 (3 hundreds became 3 thousands)
• 347 × 100 = 34,700 (every digit moves two places)
• 347 × 1,000 = 3,47,000 (three places)

Dividing by 10, 100 or 1,000 is the reverse: digits move right. 45,000 ÷ 100 = 450.

To multiply 468 by 37, you do not need a 37 times table. Split the multiplier by place value: 37 = 30 + 7. Then

• 468 × 7 is one easy row (a partial product);
• 468 × 30 is another easy row: 468 × 3, shifted one place left because it is really tens;
• add the two rows.

That is the whole of long multiplication. The only things that go wrong are forgetting the shift, and forgetting to add carries.

Long division shares a big number out one place at a time, starting from the left. That is the opposite direction from the other three methods, and there is a good reason: when sharing out money, you share the biggest notes first. Share the ₹1,000 notes, change any left over into ₹100 notes, share those, and so on down to the coins.

Each place goes through the same four-step cycle.

Sometimes, after you bring a digit down, the divisor does not go into it at all. The answer for that place is 0, and the 0 must be written in the quotient. Skipping it is the most common long-division mistake, and it makes the answer ten (or a hundred) times too small.

In 8,127 ÷ 8, the right answer is 1,015 remainder 7. A learner who forgets the zero writes 115 remainder 7, and 8 × 115 + 7 is only 927, nowhere near 8,127.

Every operation has an inverse that undoes it, and that gives you a built-in way to check:

• Check an addition by subtracting: if 3,456 + 1,289 = 4,745, then 4,745 − 1,289 must be 3,456.
• Check a subtraction by adding: difference + subtrahend = minuend.
• Check a multiplication by dividing: 17,316 ÷ 37 must be 468.
• Check a division by multiplying, and adding the remainder.

That last one is so important it has its own name, the division check (in higher maths, the division algorithm).

Money and measurements usually come with two units at once: rupees and paise, kilometres and metres, kilograms and grams, litres and millilitres. The golden rule: both amounts must be in the same unit before you operate.

There are two safe ways to do this:

1. Convert everything to the smaller unit (paise, metres, grams, millilitres), do ordinary whole-number arithmetic, then convert back.
2. Work in columns of units, carrying or borrowing 100 paise, 1,000 m, 1,000 g or 1,000 mL instead of 10.

Most wrong answers in arithmetic are not random. They come from a small number of habits, and once you know them you can hunt for them. Here are the most common, each with a real wrong answer and a quick way to catch it.