Comment by ndriscoll

Comment by ndriscoll 3 months ago

'10/3 = 3' is also false, and is something you put forward as true. Meanwhile, '10/3 ≈ 3' and '10/3 ≈ 3.0' are both equally true, as is '10/3 ≈ π' if you're in a pinch. Also true is that math is full of conventions, and it makes sense to use the conventions you feel are appropriate for what you're doing. Sometimes that might be significant figures, which I suppose you're alluding to. Other times, it might be propagation of uncertainty. Other times error tracking is not even relevant; you might just round the thing but also want to have all of your expressions be of the same type. For that matter, you may have 3: ℝ = 3.0: ℝ by definition. The other poster never gave any indication of whether or why some particular convention should apply.

Teachers not having the time to muse about such ideas and instead needing to package everything into a presentation appropriate for an entire room full of children is one of the more obvious failure modes of industrialized education.

vel0city 3 months ago

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ndriscoll 3 months ago

'10 / 3 = 3' is either bad notation or wrong. It's not true under any usual definition of 10, 3, / or =. '3 = 3.0' on the other hand is perfectly reasonable in many circumstances. If you think 10/3 can equal 3 but not 3.0, you are either confused or confusing or both. What you mean to write is '≈', and when you do that, it's obvious that 3 and 3.0 are both usable in that sentence.
It is perfectly reasonable to define 3: ℕ = succ(succ(succ(zero))). It's also perfectly reasonable to define 3: ℝ as the image of succ(succ(succ(zero))): ℕ under the canonical embedding. Or you can define 3: ℚ with the obvious element. You can also define 3.0: ℚ or 3.0: ℝ as the obvious elements. If you were really a deviant, I suppose you could even define 3.0: ℕ, and people would roll their eyes, but everyone would understand you. Obviously, there are reasonable ways to define things so that `3 = 3.0` is a meaningful sentence (typechecks) and also literally true.
Again, different conventions are used in different contexts. The "user" of mathematics should pick the conventions and notations that make sense for what they're doing to communicate what they're trying to say. That itself is an important lesson. The sigfig convention you learned in middle school isn't the word of God.
Not being aware of these things to be capable of musing about them is I suppose another issue with our education system.

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- vel0city 3 months ago
  
  If I ask for someone for 3 of something and they give me 3.001 of it, it's whatever. If I ask someone for 3.000 of something and they give me 3.001 of it, it's out of spec.
  
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  ndriscoll 3 months ago
  
  Admittedly I only did this in school and it's been over 10 years, but I recall when doing engineering drawings, we'd specify ± (or separate lower/upper tolerances in some situations). Using decimal points to indicate uncertainty was not a thing I believe I did after high school. Does any actual professional use decimal places and not explicit ±?
  Similarly, we calculated those ± values using the chain rule/uncertainty propagation, not with the simple decimal place rules you learn as a kid. I assume no one serious uses the child rules when CAD software can just as easily use the real ones.
  
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  vel0city 3 months ago
  
  > we'd specify ± (or separate lower/upper tolerances in some situations)
  > we calculated those ± values using the chain rule/uncertainty propagation
  Yes, that's common in detailed engineering documents. It still doesn't change the fact if I ask for 3.000 and you give me 3.001 I'm not going to consider that in-spec despite not giving a ±. It's assumed if I wrote it out to that decimal point I'm caring about that level of precision.
  > Using decimal points to indicate uncertainty was not a thing I believe I did after high school
  Well, I'd imagine since the topic of lesson was understanding whole numbers at a basic level this was probably a high school or lower class, probably more like elementary or middle school. You know, in that time when you did use decimal places to indicate precision. This person wasn't talking about losing points at their engineering job.
  
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JohnHaugeland 3 months ago

> 10 / 3 can = 3, depending on the expected levels of precision.
>
> 10 / 3 will never = 3.0.
You should read what they wrote again. They wrote with `≈`, which is a different operator than `=`.
What they wrote is correct.
.
> It is sad you still haven't learned this lesson after many decades.
>
> I hope I'm never on a bridge you build or plane built to the specs you write if you truly think 10 / 3 = 3.00000.
HN doesn't allow this sort of behavior.

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