Definition of the approx. symbol

Question

Take an unending number, say e.g $π$. If we want to show $π$'s value, should we use the approximately notation or equal sign when writing:

$π = 3.14...$ or $π ≈ 3.14...$

This might be a really simple question, i don't know, but when i really started thinking about this i couldn't really decide. Technically we can't assign $π$ a "pure exact" value so does that restrict us to the approx. symbol? Or can we say actually say that $π = 3.14...$?

One thing for sure. $π ≈ 3.14$ is at least correct.

Thanks in advance!

Answer 1

The statement $$ \tag{1} \pi = 3.14\ldots $$ tells us the first three digits in the decimal expansion of $\pi$, while acknowledging that there are more digits which we have not written explicitly. Statement (1) is valid and correct.

Answer 2

I believe the implication with ellipsis is that "there's more to this number we're not posting here for whatever reason; just pretend it's all somehow here." Like we wouldn't say, as a simpler example,

$$\frac{1}{3} \approx 0.33333...$$

because it's pretty clear what goes in that "..." - an infinite amount of $3$'s. And, technically, since every number has an infinite decimal representation, with some numbers (I'll skip the details on which numbers those are) ending entirely $0$'s or $9$'s, we likewise wouldn't say

$$1 \approx 0.9999999999... \;\;\; \text{or} \;\;\; 1 \approx 1.000000000...$$

It looks a little sillier when we do it with these simpler numbers, doesn't it? So in that sense, I think to say

$$\pi \approx 3.14159...$$

would be equally silly since we're basically saying with the ellipsis "the rest of the digits are here." We would be saying

$$\frac{1}{3} = 0.333... \;\;\;\;\;\; 1 = 1.000... = 0.999... \;\;\;\;\;\; \pi = 3.14159...$$

Granted, this is moreso a matter of how I've generally seen them used anecdotally and how I myself use them. I don't believe there's even a convention for this sort of thing since people are going to look at each representation essentially the same way - people aren't going to nitpick about "wait this text said "approximately" here."

The key point with a good notation is ease of communication, I think we could agree on that much. I guess our notation isn't perfect since it lends itself to ambiguities like these, but the notation also lends itself to people not really caring either way in these sort of "fringe" cases since people get the point regardless.