Credits, debits, and what a negative balance actually means

0:00Here is an account, opened with everything saved over the month before. A hundred. That number is a balance, and a balance is a position — what the account is holding right now. So it can be marked on a scale, exactly the way a floor number is marked on the wall of a lift shaft. Zero is marked, and the balance is sitting a hundred above it. Over the next few days, four things happen to this account, and every one of them moves that marker.

0:32One of them moves it somewhere a lot of people find hard to believe. On the first day, sixty is paid in. Wages, for work done. When money goes into an account, the record calls that a credit. A credit is money arriving, and on the scale it is a movement upward. A hundred, plus sixty, is a hundred and sixty. On the second day, thirty goes out, for an electricity bill.

1:03When money leaves an account, the record calls that a debit, and on the scale it is a movement downward. A hundred and sixty, minus thirty, is a hundred and thirty. Credit up, debit down. That is the entire vocabulary, and everything after this is arithmetic. On the third day, a hundred and fifty goes out, for a purchase — something bought for a small business. The balance standing there is a hundred and thirty. The purchase is a hundred and fifty.

1:38Watch the marker on the scale. It comes down, it reaches zero, and then it does not stop there. It carries straight on past zero and settles twenty below. The balance is now minus twenty. A lot of people meet that line and simply refuse to believe it, which is a completely reasonable thing to do — so let us take the objection seriously. The objection goes like this. You cannot take a hundred and fifty out of a drawer that is holding a hundred and thirty.

2:14And that is perfectly true. A drawer cannot do it. But an account is not a drawer, and that difference is the whole point. Minus twenty does not mean the account is holding a small amount of money. It means the account is holding nothing at all, and that twenty is owed on top of the nothing. Zero is empty. Below zero is borrowed. Those are two different conditions, and the sign is the only thing separating them.

2:47So why would a bank ever let that happen? Some banks do allow it, up to a limit and for a short while, and they are not doing it as a kindness. They charge for it, and the charge has a name you have probably heard used the other way round. Interest on money you have saved is money the bank pays you. Interest on a balance below zero is money you pay the bank.

3:14Same word, opposite direction, and it is the sign of the balance that tells you which one is meant. So sitting below zero costs something. Hold on to that, because the last question in this video turns on it. The fourth day, and this is the day the story was built around. Two hundred is paid in, earned because that purchase was made. The balance is minus twenty. Two hundred arrives.

3:44The marker climbs, it crosses zero, and it keeps climbing. It settles at a hundred and eighty. So the account finishes the week at a hundred and eighty, which is eighty above where it opened. The purchase cost a hundred and fifty and brought back two hundred, so it paid for itself, and fifty over. That is the answer to the question this whole story was told to raise, and we will come back to it at the very end.

4:16Now look at the whole week written out in one place. A record like this normally has two columns. Money in on one side, money out on the other, kept carefully apart. Money in: sixty on the first day, and two hundred on the fourth. Money out: thirty, and a hundred and fifty. And here is the move that makes negative numbers worth having at all. Put both columns into a single column, and give every entry a sign.

4:47Plus sixty. Minus thirty. Minus a hundred and fifty. Plus two hundred. Two opposite directions of money, sharing one column, told apart by nothing except the sign written in front. Once they share a column, working out the closing balance stops being a procedure. It is one addition. A hundred, plus sixty, minus thirty, minus a hundred and fifty, plus two hundred. A hundred and eighty. Which is exactly the number we got by walking through the week one day at a time.

5:25Two completely different routes to the same figure, and it is worth saying out loud why they had to agree. A balance is a position. A transaction is a movement. Add the movement to the position and you get the next position. That is the same sentence as a lift moving between floors, and the same sentence as a step along a number line. A bank balance is a floor number in different clothes.

5:56Two more accounts to close, and both of them start at zero. The first takes in thirty, then forty, then fifty. Then it pays out forty, then fifty, then sixty. A hundred and twenty in, altogether. And a hundred and fifty out, altogether. So it closes at minus thirty. Thirty owed. And notice something about the way it got there. This account stayed above zero right up until the very last line, and only then finished below.

6:29The closing balance tells you where an account ended. It tells you nothing at all about where it has been. The second account is stranger. It pays out one. Then two. Then four. Then eight, sixteen, thirty-two, sixty-four, and a hundred and twenty-eight. Every payment is exactly twice the one before it. By the end of those eight, the account owes two hundred and fifty-five. Then a single payment in, of two hundred and fifty-six.

7:03Take a moment before you answer, because there are two tempting wrong answers waiting here. It is not two hundred and fifty-six, and it is not a hundred and twenty-eight. The money coming in meets a debt, and what survives is the difference. Two hundred and fifty-six against two hundred and fifty-five leaves one. There is something worth noticing about all of this, and it is the only genuinely surprising thing in the video.

7:35Take those four transactions from the first story and shuffle them into any order you like. There are twenty-four ways to arrange them, and every single one of them closes at a hundred and eighty. The closing balance does not care what order the money moved in. But now ask a different question. Did the account ever go below zero? Eleven of those twenty-four orders dip into the red. Thirteen never do. Same money, same closing balance, and a completely different history.

8:12Which brings us to the last question, and it is not really an arithmetic question at all. A positive balance is generally better than a negative one, and the reason is the one from earlier. Below zero costs you something. So when is a balance below zero worth having anyway? Look back at the third day. That account went twenty into the red in order to make a purchase that brought back two hundred.

8:42It was below zero for a single day, and it came out fifty ahead. That is the whole judgement. A balance below zero is not a disaster and it is not free. It is a cost you accept when the thing it buys is worth more than the thing it charges.