Four Different Equals Signs
Last timeProducts, Powers and Roots
An equals sign can define a thing, claim something always true, pose a question, or record one lucky coincidence. The four read very differently.
The equals sign looks like the least interesting mark on the page. It is
actually doing four unrelated jobs, and knowing which one is in force changes
what you should do with the line.
The four jobs
Here are four lines, all with an equals sign, and no two of them are the same
kind of statement.
The first is a definition. Somebody writes that the loss is the average
squared error, and in doing so introduces a name that did not exist a moment
ago. Nothing about a definition can be wrong. The author is entitled to call
that quantity anything they like, and trying to verify the line is wasted
effort. The only sensible response is to remember the name.
The second is an identity. Somebody writes that a certain sum equals
, and that is a claim: it says the two sides agree for every value of
. It can be checked and it can be false. If it is true, somebody had to
prove it, and the fact that proving it was work is part of what the line is
telling you.
The third is an equation. Somebody writes that something equals nought and
means it as a question: which values make this true. Here the equals sign is
not asserting anything at all, it is posing the problem, and the answer is a
set of values which may be empty, may be one, may be infinite.
The fourth is a coincidence. Somebody writes that the error at step fifty
equals the error at step ninety. Two particular quantities happen to match,
nothing general follows, and the line is an observation rather than a law.
| step | the line | which job | how you know | what to do with it |
|---|---|---|---|---|
| 1 | L equals the average squared error | definition | the name on the left has not been used before | remember the name; there is nothing to check |
| 2 | the sum of 1 to n equals n(n+1)/2 | identity | both sides were already meaningful, and it claims they always agree | check a small case, then trust the proof it came from |
| 3 | x^2 - 5x + 6 equals 0 | equation | there is an unknown and the line is clearly asking something | solve it: the answer is a set of values, here two of them |
| 4 | the loss at step 50 equals the loss at step 90 | coincidence | both sides are particular numbers from a particular run | note it; nothing general follows |
| 5 | the derivative of x^2 equals 2x | identity | it holds for every x, and was proved once | use it as a rule anywhere |
| 6 | theta := theta - a*g | assignment | the colon before the equals, and the same name on both sides | read it as an instruction executed repeatedly, not a claim |
| 7 | sin(x) is approximately x | an approximate claim | the approximate sign, and a missing condition | find the condition; it is only true for small x |
| 8 | the cost is proportional to n^2 | a claim up to a constant | the proportional sign, with the constant deliberately dropped | use it to compare sizes, never to predict a running time in seconds |
Telling a definition from a claim
Some authors mark their definitions. A colon before the equals sign, a triple
bar instead of the equals sign, or the words we define, let, or set. Any of
these says explicitly that a name is being introduced.
Most authors do not mark them, and then the test is the one in the table above:
look at the left-hand side and ask whether you have seen it before on this
page. If it is new, the line is a definition. If it was already meaningful, the
line is a claim about it.
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 planThe contentsThis 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