Linear growth and linear decay

0:00Two situations that look like opposites. A journey whose cost climbs the further you go. A tank of water whose depth falls the longer the summer runs. One goes up. One goes down. And they are the same rule. Not similar. The same shape, with one character changed. Start with the journey. There is a charge of a hundred before the wheels have turned at all, and then every kilometre adds sixty.

0:31So write it. A hundred, plus sixty for each kilometre travelled. A hundred plus sixty d, where d is the distance. At zero kilometres, sixty times zero is nothing, so the cost is a hundred. At one kilometre, a hundred and sixty. Then two hundred and twenty, two hundred and eighty, three hundred and forty, four hundred. And the gaps: sixty, sixty, sixty, sixty, sixty. The same every time, which is the fingerprint.

1:06Notice that this table begins at zero, so the opening charge is sitting right there as the first entry. Not every table is written that way, and we will meet one that is not. The rule gets a name, and the name has brackets in it. C of d. That is not C multiplied by d. It is the cost at a distance of d. C is the name of the rule, and d in the brackets is what you feed it.

1:36So C of two is the cost of two kilometres, which we already have: two hundred and twenty. And now the table can be pushed as far as we like, because the rule does not stop where the table stops. Six kilometres, four hundred and sixty. Then five hundred and twenty, five hundred and eighty, six hundred and forty, and seven hundred. Ten entries added by arithmetic, not by measuring anything. That is what having a rule buys you.

2:10Two questions, one in each direction. First: what does fifteen kilometres cost? Feed the rule fifteen. Sixty fifteens are nine hundred, plus the opening hundred, is one thousand. Second, and harder: how far can seven hundred take you? Now the answer is named and the input is missing. A hundred plus sixty d is seven hundred. Take the opening charge off both sides: sixty d is six hundred. So d is ten.

2:44Ten kilometres, which is exactly where we stopped the table a moment ago. The rule answers in both directions, and that is the difference between a rule and a list. Now the tank. At the start of summer the water stands three metres deep, and every month it drops by half a metre. Three, minus a half for each month. Three minus nought point five t, where t is the number of months.

3:15At zero months, three metres. Then two and a half, then two, then one and a half, then one. The gaps: minus a half, minus a half, minus a half, minus a half. Constant again. Negative, but constant, and that is what the fingerprint asks for. So at the end of five months the water is half a metre deep. Now put the two rules next to each other. A hundred plus sixty d. Three minus nought point five t.

3:46Line up the pieces. Constant above constant: a hundred, and three. Coefficient above coefficient: sixty, and minus a half. The letters differ, because one counts kilometres and the other counts months. The constants differ. The sizes of the coefficients differ wildly. The units differ. And exactly one of those differences makes one of them climb and the other fall. It is the minus sign. Everything else is decoration. Which is worth pinning down, because there is an obvious wrong answer sitting next to the right one.

4:28The wrong answer says: growth starts from something positive, decay starts from something negative. Look at where the rule begins. But the tank begins at three. Three is positive, and the tank empties. The sign that decides the direction is not on the constant. It is on the coefficient, and only there. Take every rule you can build from a handful of constants and a handful of coefficients, both signs of each, and check thirty-six of them one at a time.

5:03The coefficient's sign predicts the direction in every single case, and the constant never once overrules it. A rule starting at minus seven and gaining fifty a year climbs. It spends its first year below nothing and it is still growth. So here are the two definitions, and there is a phrase inside them doing more work than it looks. Linear growth: the quantity gains the same amount over each equal interval. Linear decay: it loses the same amount over each equal interval.

5:41Over each equal interval. Cover that phrase up and watch what walks in. One. Twenty-five. Forty-nine. Twenty-five minus one is twenty-four. Forty-nine minus twenty-five is twenty-four. The same gain twice. But those are the squares of one, five and seven. The input stepped by four, and then by two. Squaring is not linear growth and never will be. It qualified only because nobody checked the intervals were equal. With the phrase back in, it is thrown out immediately.

6:21And the same phrase explains a coincidence in the tank table. The coefficient is a half, and the water drops by a half each row. Those two halves look like the same fact told twice. They are not. They agree only because the table steps one month at a time. Read the tank every two months instead. Three, then two, then one, then zero. Now the drop is a whole metre, and the coefficient is still a half.

6:51The change per row is the coefficient times the interval you chose to read at. Choose a different interval and the number in the table changes while the rule does not. Which is why a rate quoted without its interval is not a claim yet. Sixty is meaningless. Sixty per kilometre is a fact. Now the word that has been quietly carrying everything: linear. Because plenty of things fall without falling like this.

7:24Two things worth ten thousand. The first loses eight hundred every year. The second loses a tenth of whatever it is currently worth. The first: ten thousand, nine thousand two hundred, eight thousand four hundred, seven thousand six hundred. The second: ten thousand, nine thousand, eight thousand one hundred, seven thousand two hundred and ninety. Both fall. Only one of them has equal gaps. And look where they part. Not somewhere down the line: at the very first step. A tenth of ten thousand is a thousand, which is more than eight hundred, so the second one starts by falling faster.

8:08Then its losses shrink, because a tenth of a smaller number is a smaller loss. A thousand, then nine hundred, then eight hundred and ten. Follow both far enough and something worth seeing happens. The one losing eight hundred a year keeps losing eight hundred a year, no matter how little is left. So it runs out. Ten thousand minus eight hundred t is zero when t is twelve and a half years.

8:39Not a whole number of years, and that is fine. The moment a rule hits zero does not have to land on one of your rows. The other one never runs out at all. A tenth off leaves nine tenths behind, and nine tenths of something positive is positive. After a thousand years it is still worth something. So they cross. In year six the second one, which started by falling faster, becomes the more valuable of the two, and stays that way forever after.

9:14Losing a fixed amount and losing a fixed fraction are not two flavours of the same thing. Which brings us back to the tank, and one last question the situation asks and the rule does not. When is the water gone? Set the depth to zero. Three minus a half t is zero. So a half t is three, and t is six. Six months. Mark it. Now feed the rule seven months. Minus a half. Feed it eight. Minus one metre.

9:48There is nothing wrong with the rule. Its gaps are still exactly minus a half, forever, and it will go on answering politely for as long as you keep asking. It is the situation that stopped. A tank has no minus one metre of water, and past six months the rule is describing something that is no longer there. So three questions, every time. What is the constant, and does the table even show it. What is the coefficient, and over what interval. And where does the model stop being about the world.