Unit 2 · Topic 02 · Operations
Kabir had to work out 1000 minus 456.
He looked at the ones column: 0 take away 6. He wrote 6.
He had just decided that zero minus six is six, and nobody had told him he could not.
The class was given a stock-check problem: a shop starts with 1000 boxes and sells 456. How many are left?
Kabir set it out neatly, 1000 above 456, right-aligned, exactly as he had been taught. Then he hit the ones column.
Zero minus six. He knew six minus zero was six. He decided this was near enough and wrote 6. Then tens: zero minus five. He wrote 5. Hundreds: zero minus four. He wrote 4. Thousands: 1.
1456. More boxes after selling 456 than the shop started with.
"You cannot end up with more than you began," said Ms. Rao. "So before we fix the arithmetic — what does that answer tell you?"
"That it's wrong," said Kabir.
"That it is wrong in a particular way. You took the small digit from the big one each time, because that was the subtraction you knew how to do. Subtraction does not let you choose the order."
She drew the 1000 as a single bundle of a thousand straws. "You want to take six straws. They are all tied up in one bundle of a thousand. What do you do first?"
"Untie it," said Anaya.
"Into what?"
"Ten hundreds."
"And you still cannot take six straws from a hundred-bundle. So untie one of those into ten tens, and one of those tens into ten ones. Now you have nine hundreds, nine tens, and ten ones sitting loose."
Kabir wrote it out: the 1000 became 0 thousands, 9 hundreds, 9 tens, 10 ones.
Ones: 10 − 6 = 4. Tens: 9 − 5 = 4. Hundreds: 9 − 4 = 5.
544.
"That is why every zero you borrow across turns into a nine," said Ms. Rao. "It gave one away to the right and took one from the left. Nine is what is left in the middle."
Kabir checked it the honest way: 544 + 456 = 1000.
Subtraction takes the bottom number from the top one, in that order. You may not swap a column round because the bottom digit happens to be larger — that is the single most common subtraction error.
When the top digit is too small, you BORROW from the column to its left. One of that column's units is exchanged for ten of yours, because each column is worth ten of the column to its right.
Borrowing across a zero looks harder but follows the same rule. The zero cannot lend, so it borrows in turn from ITS left. Every zero you pass through ends up as a 9 — it received ten and immediately gave one away.
Every subtraction can be checked by addition. If the answer plus the number you took away does not return the number you started with, something has gone wrong.
Pick any round number like 1000, 5000 or 10,000 and subtract your house number from it.
Then add your answer back to your house number. If you do not land exactly on the round number, find the column where a borrow went missing.
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