A GP plots as a curve, and what that curve tells you

0:00Here are two sequences that begin the same way. Both start at three, both have six second, and after that they part company. The first one adds three every time. Three, six, nine, twelve, fifteen. The second one doubles every time. Three, six, twelve, twenty-four, forty-eight. Set each out as a table: position along the top, value underneath. Now every column is a pair. One and three. Two and six. Three and nine, or three and twelve.

0:35Nothing so far is new. What is new is what happens when you plot those pairs instead of reading them. Take the doubling sequence first. Its five pairs are one and three, two and six, three and twelve, four and twenty-four, five and forty-eight. Put a dot at each one. Position across, value up. The first two dots tell you nothing, because any two dots lie on a line. The third tells you everything. Draw the line through the first two, carry it on, and the third dot is not on it.

1:12Nor the fourth, nor the fifth. They bend away upward, and get steeper as they go. Now plot the adding sequence beside it. Five dots, and they lie exactly on one straight line. Same start, same second value, different shapes. Why can the doubling one never be straight? Because of what straight means. Points on a line rise by the same amount for every step across. That is not a fact to remember about lines. It is what a line is.

1:46So look at the rises. Three to six is a rise of three. Six to twelve is a rise of six. Twelve to twenty-four is a rise of twelve, twenty-four to forty-eight a rise of twenty-four. Every rise is different, and each one is double the last. There is no single amount to rise by, so there is no line to lie on. The bend is not a wobbly drawing. It is arithmetic.

2:17There is a tidier way to put that. Divide each rise by the height it started from. Three over three is one. Six over six is one. Twelve over twelve is one. Every rise is the whole of the height already reached. That is what a constant ratio does. The share is fixed, and the share is just the ratio less one. Now try that on the adding sequence. Its rise is always three. But three over three is one, three over six is a half, three over nine a third, three over twelve a quarter.

2:55The share drains away. A fixed amount gives a shrinking share; a fixed share gives a growing amount. That is the whole difference between the families, and the bend is what a fixed share looks like. One exception, and it is worth a minute. Take seven, seven, seven, seven, seven. Each term is one times the term before it, so the ratio is a constant one. That is a geometric progression.

3:26But each term is also nothing more than the one before, so the difference is constant. That is an arithmetic progression. It is both at once. And it plots as five dots on a perfectly level line. So the rule is not that a geometric progression never plots straight. It is that a geometric progression with a ratio other than one never does. The condition matters. Now the part that gets misused, and the most important idea here.

4:01A bend is a one-way finding. Here are the square numbers. One, four, nine, sixteen, twenty-five. Plot them and they bend upward, much like the doubling ones did. So if a bend meant a constant ratio, these would have one. Divide and see. Four over one is four. Nine over four is two and a quarter. Sixteen over nine is about one point seven eight, and twenty-five over sixteen one point five six.

4:34Those ratios are falling away, not holding steady. This is not a geometric progression at all. So a bend rules the straight-line family out on sight, and it rules nothing in. To confirm a constant ratio you must still divide every consecutive pair. The picture will not do it for you. Here are two plots from a single rule. Cut a triangle at its midpoints, throw the middle piece away, and repeat on everything that survives.

5:09The piece counts run one, three, nine, twenty-seven, eighty-one, and plotted against the stage they climb clean off the top of any reasonable scale. The black area runs one, three quarters, nine sixteenths, twenty-seven sixty-fourths, eighty-one two hundred and fifty-sixths. On a chart those read one, point seven five, point five six, point four two, point three two. One climbs and the other falls. And here is the thing worth noticing: both bend the same way. Upward.

5:46The bend is reporting that the ratio is constant. It is not reporting which way the values are going. What does the falling one do in the long run? It flattens. Each stage loses a quarter of what is left, so as the values shrink the drops shrink too, and the trace lies closer and closer to the bottom. But it never touches the bottom. Every one of those areas is three quarters of a positive number, and three quarters of something positive is still positive.

6:22Carry it out two hundred stages and it is still above zero. On a chart marked in steps of two tenths, stage six is the first one whose value sits below the lowest marked line. That is what makes it look like arrival. A coarse scale flatters. The sequence has not arrived anywhere. Here is one from outside geometry. A ball is dropped from twenty-four feet, each bounce three quarters as high as the one before.

6:56Twenty-four times three quarters is eighteen. Eighteen times three quarters is thirteen point five. Then ten point one two five, then seven point five nine three seven five, and on down. Plot the bounce heights and you get exactly that flattening shape. But be careful where this progression starts. Twenty-four feet is the drop. It is not a bounce, so it is not a term. The first term is eighteen. A chart will often plot the twenty-four anyway, at bounce number zero. Sensible charting, and a trap if you read the axis literally.

7:39Be careful with the numbers written beside the dots, too. The fourth height is ten point one two five, and the label beside it reads ten point one two. The fifth is five point six nine five, and its label reads five point seven zero. One went down and the other went up. Both are written to two decimal places, just not by the same rule. Put all eight labels through three standard rounding rules. The closest fit reproduces seven and misses one.

8:14And not one of the three produces that eighth label from the working. A label no rounding rule can reach is not a rounding. It is a slip. Reading a chart is not the same as computing. Now put the picture to work. How many bounces until the ball stays below one sixth of the height it was dropped from? One sixth of twenty-four is four feet. Draw a line straight across the chart at four.

8:44The sixth bounce reaches four point two seven feet, above the line. The seventh reaches three point two zero, below it. Seven bounces. Two ways to get that wrong, and both are common. Round four point two seven to the nearest foot first and you get four; let four count as below four and you answer six. Or call the twenty-four foot drop the first term, everything shifts by one bounce, and you will answer eight.

9:16The line is at four feet. Compare the real height against it, and round nothing first. One last thing, about what a plot will not give you. A different ball. Dropped from eighty metres, rising to sixty per cent of its height each time. The bounces are forty-eight, twenty-eight point eight, seventeen point two eight, ten point three six eight, and six point two two zero eight metres. Every one is a dot on the chart, and you can read any of them straight off.

9:52Now ask for the total distance the ball travels up to its sixth landing. No dot on that chart is the answer, and there is no shortcut formula either. You walk it. Eighty metres down, then every bounce counted twice, once up and once back down. Eighty plus twice one hundred and ten point six six eight eight. Three hundred and one point three four metres. A graph shows you every term. Adding them up is still your job.