Numbering Things

Last timeThe Sum Sign Is a Loop

A small number beside a letter says which one. Two of them say which row and which column, and the mark above the line means three different things.

The last lesson used $x_i$ dozens of times without ever saying what the small

number beside the letter is doing. This lesson says.

A subscript picks one item out of a collection. That is the whole of it, and

the complications are all about how many subscripts there are, what they run

over, and the fact that the mark above the line looks the same and means three

different things.

One subscript

Writing $x_i$ says that there is a collection called $x$ and this is the item

at position $i$ .

Note that the collection keeps the bare name. $x$ is all of the numbers; $x_7$

is one of them. A line that uses both is doing so deliberately, and in

particular a line that writes $x$ where you expected $x_i$ is probably talking

about the whole collection at once.

Two questions settle what the positions are, and both are answered by the

declarations rather than by the symbols. Where does the counting start, and how

many are there. Mathematical writing nearly always starts at one; almost every

programming language starts at nought; and a reader moving between a paper and

its implementation has to translate every subscript on the way. The count is

almost always a letter declared nearby, usually $n$ .

A subscript outside the range is not nought. It does not exist, in the same way

that the thousandth item of a nine hundred item list does not exist, and a

formula that allows one has an error in it rather than an edge case.

FIG 1One entry of a table, with every decoration named

: the whole table, which keeps the bare name. Nothing is being picked out until a subscript is attached
: the first subscript, which is the row. Always the row, by a convention with no reason behind it beyond agreement
: the second subscript, which is the column
: a sum over the first subscript, which means the rows are being added up. There are m rows, so m terms
: free, and so the same throughout: whichever column is being talked about. The result still mentions j, which is correct, because the answer is one number per column

The right-hand expression is the whole lesson in one line. Because the sum binds the row index and leaves the column index free, the expression is a column total. Swapping which letter sits under the sum sign changes it into a row total, and nothing else about the line would look any different.

Two subscripts

Two subscripts mean a table, and the order is row first, column second,

without exception in any writing you are likely to meet.

The order carries real content. For a general table, $a_{23}$ and $a_{32}$ are

different numbers. The entire meaning of the claim that a table is symmetric is

that those two are equal for every pair, which is a strong and useful property

and not something to be assumed because the subscripts look interchangeable.

Combining this with the previous lesson gives the one piece of reading that

unlocks a great deal of applied mathematics: look at which subscript is bound

by the sum sign. If the row index is bound, the rows are being combined and the

answer is per column. If the column index is bound, the answer is per row. The

two expressions differ by one letter and describe completely different

operations.

The lesson stops here

5 more paragraphs to go

You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.

The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.

See the plan The contents

This is the reading half

Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.

The contents

Numbering Things

One subscript

Two subscripts

5 more paragraphs to go

This is the reading half

Read alongside