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Comment by tikhonj | Hacker News Reader

The worst part is that they are both, in a critical sense, wrong. There is no absolute 100% universal "right" answer to how division works—it's just a matter of convention! Merely getting into the argument of whether 1/0 "is" 0 is wrong.

There are practical reasons why we leave division by 0 undefined in "normal" arithmetic but we could, just as easily, operate in a world where 1/0 is defined. In some contexts, 1/0 = 0 is a totally sensible convention! In others, we can define it to be ∞ or, hell, something else entirely.

This approach of teaching math as a bunch of universal facts that are either true or false is actively harmful—but super convenient for top-down education and assessment!—because it teaches people the wrong way to think about math or even about conceptual models in general.

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Yet we can still plainly declare that the teacher is incorrect, unless we missed the part of the story where they were discussing floating point arithmetic, or the extended complex plane, or some other deliberate definition unrelated to the arithmetic of the reals.

> 1/0 = 0 is a totally sensible convention

Except that for all other numbers we have 1/x * x = 1, while 0*0 = 0. Defining division by zero this way breaks existing rules.

0 is always going to be a special case for division in real numbers. But how we handle that special case is a matter of convention.

Of course it’s a matter of convention. But it’s a convention without which the rest of mathematics taught to school children throughout the world falls apart.

Calling the color of the sky blue is just a matter of convention as well, but if one random teacher starts calling it green, she’s doing a terrible disservice to her students. It doesn’t matter that there exists some languages without a difference between blue and green.

The "convention" is inconsistent with division by small quantities that get closer and closer to zero.

  1/1000000 = 0.000001
  ...
  1/10 = 0.1
  1/1 = 1
  1/0.1 = 10
  1/0.0001 = 1000
  1/0.0000001 = 1000000
  ...
  1/0 = ... 0?  WTF

What is the application for 1/0 = 0? Where is this used? Does it actually make a meaningful appearance outside of drunken barroom conversations, and classrooms staffed by clowns?

    ceil(0.000001) = 1
    ceil(0) =... 0? Wtf?

But that's a perfectly well defined function. Not every function has to be continuous.

In the first place, not every function has to be defined everywhere.

Well, if 1/x -> infinity as x -> 0, x > 0, and 1/x -> -infinity as x -> 0, x < 0, then the average of +infinity and -infinity is zero. :-)

> 0 is always going to be a special case for division in real numbers.

Right — because it’s impossible to have an inverse of zero multiplication.

Eg, 0 = 0 + 0 -> 1 = 1 + 1 -> 0 = 1 if you have a multiplicative inverse to 0, and hence you have collapsed your entire system to a single value.

These are students we’re still teaching that basic algebraic structure to — that it’s not possible to have a multiplicative inverse to zero.

Calling that a “convention” is wrong.

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Hey, can you please make your substantive points without shallow dismissals or name-calling? This is in the site guidelines: https://news.ycombinator.com/newsguidelines.html.

If you know more than others, that's great, but in that case the thing to do is to share some of what you know, so others can learn. If you don't want to do that, that's fine, but in that case please don't post. Putdowns and swipes only degrade the discussion.

https://hn.algolia.com/?dateRange=all&page=0&prefix=true&sor...

> The worst part is that they are both, in a critical sense, wrong.

I thought the worst part is that in context of "normal" mathematics taught in schools one is obviously wrong, and also unable to do research and admit own mistakes, and that one is actually teaching children.

> This approach of teaching math as a bunch of universal facts that are either true or false is actively harmful

Kids are taught not philosophy or logic but certain mathematical paradigm (that you call "normal") in which division by zero is an error. If you have a formula you decompose it to smaller formulas, if in a smaller formula you have a division by zero you simplify it to zero then you screwed up the entire thing. There's no true or false value for 1/0, the right move is not to play and not solve any equation in which division by zero appears.

While it's true that you can do things with the Reimann sphere and such, this is 3rd grade, and 1/0 should not be defined.

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