Order is the point: a sequence carries position as well as value

0:00Here are four lists of numbers, and you have met all four before. One, two, three, four, five, six. The counting numbers. One, three, five, seven, nine, eleven. The odd numbers. One, three, six, ten, fifteen, twenty-one. And one, four, nine, sixteen, twenty-five, thirty-six. Each ends in three dots, and those dots are a promise rather than an apology. They do not mean that a few more numbers follow. They mean this list has no last number at all.

0:35Ask the odd numbers for the term in position four hundred and the answer is seven hundred and ninety-nine. That is what makes such lists worth studying. They never run out, and they are still predictable. Let me do something small and destructive. Take one, four, nine, sixteen, and write them in a different order. Sixteen, four, nine, one. As a collection it is the very same thing: nothing added, nothing removed.

1:08But look at the steps between them. In order the steps were three, five, seven, and anyone can carry that on. Scrambled, the steps are minus twelve, five, minus eight, and there is nothing there to carry on. So I laid the first five square numbers out in every order there is. There are a hundred and twenty. Two of those orders have steps that settle into a pattern. The other hundred and eighteen have nothing that settles at all.

1:39The numbers were never the pattern. The order was. So what are we actually studying? Not the numbers on their own. Write two rows instead of one. Along the top, the positions. One, two, three, four, and onwards for ever. Along the bottom, the values that sit at them. A sequence is that pairing, and each value in it is called a term. The top row never changes; it is the same for every sequence there has ever been.

2:12All of the content is in the bottom row, and in which value lands underneath which position. Throw the pairing away and a question about position stops meaning anything, because a collection has no positions to ask about. Since positions are doing the work, they need names. We write a letter with a small number underneath. T one is the term in position one, t two the term in position two.

2:42For the odd numbers, t one is one, t two is three, t three is five, and t four is seven. Say that last one out loud as the term in position four. The small number is an address, not a multiplication. T four is not t times four. That is the easiest slip in this topic, and worth deciding now never to make. When we want two sequences in view at once, we simply reach for another letter.

3:13S one, s two, s three for a second list. U one, u two, u three for a third. Different letters, identical idea. Here is the first thing you can do once positions exist. Take each term, subtract the one before it, and write those differences in a row underneath. For the counting numbers, every gap is one. For the odd numbers, every gap is two. Both rows come out flat, and a flat row of gaps is the easiest pattern to continue.

3:49Now the third list. One to three is two, three to six is three, six to ten is four, then five, then six. Those gaps are not flat. They grow. And the fourth list gives three, five, seven, nine, eleven, growing by two each time. Notice what just happened there. The gaps of a list are themselves a list. So difference them again. Underneath two, three, four, five, six we get one, one, one, one. Flat.

4:22Underneath three, five, seven, nine, eleven we get two, two, two, two. Also flat. All four of our lists go flat after at most two rounds. That is the whole trick of predicting what comes next: keep taking gaps until a row settles, then walk back up. But look at what every step of that rested on. The phrase, the one before. A gap is not a property of a term. It is a property of a term together with its predecessor.

4:55Take the order away and the very first subtraction has nothing to subtract from. There is a second thing positions let you do, and it runs the other way. Instead of the step from the previous term, take everything up to and including this one. One. One and two is three. One and two and three is six. Then ten. Then fifteen. Those are the running totals of the counting numbers, and they are exactly our third list.

5:27You can see why if you put dots down. One dot. Add a row of two underneath and you have three, in a triangle. A row of three and you have six. A row of four, ten. A row of five, fifteen. Each new row is the next counting number, and the triangle holds the total so far. That is what a triangular number is. The name is a picture of an addition.

5:55Do the same with the odd numbers and something better happens. One. One and three is four. One and three and five is nine. One and three and five and seven is sixteen. Those are the square numbers. Here is the reason. Take an array of counters, eight rows by eight columns, sixty-four in all. Peel it from one corner outwards, in L-shaped shells. The innermost shell is a single counter. The next holds three. Then five, seven, nine, eleven, thirteen, and fifteen.

6:32Eight shells, and every one of them odd, because each L adds a new row, a new column, and the corner where they meet. Add the eight shells and you have sixty-four, which is eight times eight. A square number is the running total of the odd numbers, laid out flat. Look at what we now have. Gaps take a list and hand back its steps. Running totals take a list and hand back its sums.

7:01And those two operations undo each other. Difference the triangular numbers and the counting numbers come back. Total the counting numbers and the triangular numbers come back. The same holds for the odds and the squares, and for every other list here. Two operations pointing opposite ways, both built out of nothing but position. One asks what has changed since the term before. The other asks what has piled up so far.

7:33Neither question exists without an order to ask it in. Not every sequence carries on for ever. Six, twelve, twenty-four, forty-eight, ninety-six. Five terms and a full stop. There is no position six here. Asking this list for a sixth term is not a hard question. It is a meaningless one. That is a finite sequence. It has a last term, so the whole of it can be written down.

8:03The lists we began with have no last term, and the three dots are how we say so. The difference is not how big they are. It is whether the pairing runs out. One more distinction, and it is the sharpest. The terms may be anything at all. One, a half, a third, a quarter. A sequence that shrinks for ever without ever reaching zero. Its gaps are minus a half, then minus a sixth, then minus a twelfth. Negative, and shrinking themselves.

8:38Or minus seven, minus three, one, five, nine, which begins below zero and steps over it four at a time. Fractions, negatives, whole numbers, whatever you like. The bottom row is free. But the top row is not. There is no position zero, and no position ninety-four point six. Both of our operations need the term before, and ninety-four point six has nothing before it. A sequence hands one value to every counting number.

9:13Its order is not decoration. The order is the whole of its content. Two operations live on that order. The gap, which looks back one step, and the running total, which looks back all the way. And everything ahead of us is a question about position. What is the term in position fifty-three? For the squares it is two thousand eight hundred and nine. Is a hundred and thirty-seven anywhere in this list? For the squares and the triangles alike, no.

9:47Neither could be asked of a bag of numbers. From here on, a rule has exactly one job. You hand it a position, and it hands you back the term that sits there.