
What's the difference between 0/0 and 1/0? - korethr
https://blog.plover.com/math/division-by-zero.html
======
codethief
The author's explanation is based on how the rationals are defined explicitly
but this leaves the question open whether the rationals couldn't be extended
somehow, so that the inverse of 0 is suddenly defined in this extension. More
generally, couldn't there be other fields where 0 has a multiplicative
inverse?

Of course the answer to both questions is negative but it's the consequence of
a much more general fact: In any field the additive zero element can never
have a multiplicate inverse. (If it did, the field's other algebraic
operations would suddenly lead to contradictions.) And in this sense, there is
also no difference between the expressions 1/0 = 1 · 0⁻¹ and 0/0 = 0 · 0⁻¹
because 0⁻¹ simply doesn't exist.

I've always found this explanation much more illuminating. _No matter what you
do_ , you can never have 0⁻¹!

~~~
daveFNbuck
You can have a multiplicative inverse of 0 if 0 = 1. This results in a field
with only a single element.

~~~
gizmo686
Unless your definition of field include the non-triviality axiom that 0 != 1.

------
bahhh
George Boole proposed something similar in The Laws of Thought: he considered
a definition of c = a/b where a = c AND b (even if he didn't write it like
that).

Thus, 0/0 was an indeterminate value, and 1/0 was seen as an impossibility.

Boole's example: if m means "is mortal" and h means "is human", then saying
"there is no immortal humans" is h AND NOT m = 0, and Boole explains that from
it you could deduce: m = h + 0/0 and not h meaning "mortals are humans and
something indeterminate which is not human"

here it's still George Boole trying to cram normal arithmetics into logic, but
still.

------
lucasgonze
I'm inspired by how clean his reasoning is. Something basic but fuzzy which we
refer to all the time - "divide by zero is undefined" \- actually contains
meaning! This makes me happy.

------
ikeboy
>When f(0)≠0, we can state decisively that there is no such Q

This is false. This relies on an assumption not stated - that f is continuous.

For a simple counterexample:

f(x) = 1, x=0

f(x) = x, x!=0

g(x) = x

Limit of the ratio as x approaches 0 is 1.

------
LoSboccacc
as most of these discussion they go from "these number are impossible!" to
"but wait! there's algebra!" without really providing a framing context of
what's going on.

this is what's going when the article say "slides smoothly in toward 2"

[https://www.google.com/search?q=(x^2%2B2x)/x](https://www.google.com/search?q=\(x^2%2B2x\)/x)

and this is another common example, which "slides" toward a different number
(which is why when you see the division alone is "undefined" and only acquires
value in context)

[https://www.google.com/search?q=sin(x)/x](https://www.google.com/search?q=sin\(x\)/x)

but beyond the "just apply limits" party trick it's the implication however
that's the most interesting part, and I'll use the words from here[1]:

> Things that appear to be zero may be nonzero in a different dimension (just
> like i might appear to be 0 to us, but isn’t)

[1] [https://betterexplained.com/articles/why-do-we-need-
limits-a...](https://betterexplained.com/articles/why-do-we-need-limits-and-
infinitesimals/)

------
Rhinobird
For x⋅0=a (i.e. 1/0) no number fits. It's like a database search that finds
nothing. It's a null. A point that is not in the set.

For x⋅0=0 (i.e. 0/0) any number fits. Is there such a thing as an anti-null?
In the infinite set, 0/0 is infinite.

If a/0 isn't any number if you exclude 0, then a/0 should be zero. Tada!

...of course zero wasn't in the set, which makes it a null

------
meuk
This can be explained clearly in two sentences:

Division is the inverse of multiplication (a * b / b = a / b * b = a).
However, multiplication by 0 is not injective (we have x * 0 = 0 for every x),
so multiplication by zero can't have an inverse operation, and both 0/0 and
1/0 are undefined.

~~~
EForEndeavour
This explanation correctly explains why 1/0 and 0/0 are not numbers, but
doesn't answer the question of what makes 1/0 and 0/0 _different_. And they
aren't both undefined.

1/0 is undefined because _no_ value of x satisfies 1/0 = x.

0/0 is _indeterminate_ because _any_ value of x satisfies 0/0 = x.

~~~
throwawaymath
0/0 is only indeterminate if your axioms allow or require it to be. Strictly
speaking you can define 0/0 to mean something, but you will lose a lot of
useful properties along the way. The only time 0/0 is not indeterminate is
when you're dealing with something exotic and obscure like an algebraic wheel.
It's correct to say 0/0 is undefined because in the vast majority of cases
where someone doesn't explicitly call out the algebraic setting, they're
working with a field.

------
ddxxdd
Someone needs to copy this poster's simplistic style, and use it to explain to
me why Laplaces equation of electrostatic potential is solved with Bessel
functions times an exponential times a sine/cosine wave (edit: in cylindrical
coordinates).

~~~
codethief
I suppose you've already looked at:

[https://en.wikipedia.org/wiki/Cylindrical_harmonics](https://en.wikipedia.org/wiki/Cylindrical_harmonics)

?

~~~
ddxxdd
No, but that page has all the info that Jackson's Classical E&M textbook
already explains.

I was being slightly facetious, but any average person will tune out a
complete derivation after 15 integral signs, 50 terms, and 100 factors unless
there is a pleasing narrative woven throughout the discussion.

~~~
codethief
Haha, I feel you!

------
tzs
Well, 1/0 - 0/0 = (1-0)/0 = 1/0, so the difference is 1/0\. :-)

~~~
aijony
or -1/0

~~~
ngcc_hk
Or whatever/0

For the article what it try to do is using limit to see whether the difference
converge to a finite number

diff(x) = one(x)/zero’(0) - zero(x)/zero’’(0)

Where one and all zero function approach 1 and 0, smooth etc

Find diff(x) when lim x-> 0

------
jchook
I read a relevant article about how the semantics of division by Zero
ultimately rest on arbitrary decisions of the mathematics community.

Trying to dig this up I found these:

\- [https://www.1dividedby0.com/](https://www.1dividedby0.com/)

\- [https://www.hillelwayne.com/post/divide-by-
zero/](https://www.hillelwayne.com/post/divide-by-zero/)

------
gizmo686
>But to really understand the difference you probably need to use the calculus
approach

I am probably showing my bias as an algebraist here, but there is a purely
algabraic way of seeing the difference between 0/0 and 1/0, which is (in my
view) more intuitive.

The standard way of constructing the field of fractions is to start with the
integers (or your choice of base ring), and define an equivalence relation on
the set:

    
    
        Q = { (x,y) | x,y are integers, y!=0 } 
    

Given by:

    
    
        (a,b) ~ (c,d) iff ad = bc
    

Intuitively, the pair (a,b) is the fraction a/b, and the equivalence
relationship is defining that: a/b = c/d is equivalent to ad =bc (assume b,d
are non-zero). From that you can define multiplication and division in the
expected way and verify that the resulting structure is a field that behaves
as you would expect the rationals to behave.

One thing to notice in the above view is that the prohibition on dividing by 0
is entirely artificial. By removing 6 characters from the above definition, we
arrive at the following structure:

    
    
        W = { (x,y) | x,y are integers } 
        (a,b) ~ (c,d) iff ad = b
    

Let [W] be the set of equivalence classes on W, and [Q] be the set of
equivalence classes on Q.

If we define addition and mulitplication on [W] and [Q] in the "obvious" way,
then [Q] behaves identically to the field of rational numbers (indeed, this is
a common construction.

Further, [W] contains [Q] as a subset, so we can view [W] as an extension to
the rational numbers.

We can verify that [W] contains exactly two elements not present in [Q], which
I will denote:

    
    
      ⊥ = [(0,0)] = 0/0
      ∞ = [(1,0)] = 1/0 = a/0 for a!=0 and a!=⊥
    

The question now becomes, is "division" meaningful on [W]. If we keep the same
definition of multiplication as with [Q], it is clear that we cannot define an
inverse operation [0].

However, we can define a unary operation called the reciprical, given by:
/[(x,y)] = [(y,x)]

And define a division operation as multiplication by the reciprical. So:

[(a,b)] div [(x,y)] = [(a,b)] * (/[(x,y)]) = [(a,b)] * [(y,x)] = [(ay,bx)]

And note that, on the subset [Q], this operation is identical to our standard
notion of division.

With this, we have extended [Q] into a structure, [W], for which division by 0
is defined.

Further, by doing so, we added exactly 2 elements: 0/0 and 1/0\. So we can see
that 0/0 and 1/0 are distinct (in this structure) in a way that 1/0 and 2/0
are not. (as 1/0 ~ 2/0 in the same way that 1/1 ~ 2/2)

In fact, the behaviour of ⊥ ∞ should not be suprising to this forum.

⊥ behaves essentially like NaN in that it "absorbs" the other number in all
operations. So:

    
    
        x + ⊥ = ⊥ and 
        x * ⊥ = ⊥.
    

∞ behaves mostly like you would expects:

    
    
        0 * ∞ = 0, 
        x * ∞ = ∞ | x != 0 and x != ⊥
        x + ∞ = 0 | x != ⊥
    

The main "weirdness" here is that ∞=-∞

In fact, this [W] I have been describing is an established structure refered
to as a Wheel [1], and can be constructed over the real or complex numbers
just as easily (it can also be viewed as an extension of the more well known
Riemann sphere by adding ⊥)

[0] I will note that the lack of division by 0 means that we cannot define an
inverse operation for multiplication in [Q] either.

[1]
[https://en.wikipedia.org/wiki/Wheel_theory](https://en.wikipedia.org/wiki/Wheel_theory)

~~~
cygaril
Assuming the definition of ~ is meant to be the same on Q and W (I think
there's a typo), then it's no longer transitive and therefore no longer an
equivalence relation: (0, 0) ~ (1, 2) and (0, 0) ~ (2, 3) but not (1, 2) ~ (2,
3).

~~~
gizmo686
I'm passed the window where I can edit my post, so I will correct it here.

You are correct that my original post is mistaken. I wish I could attribute
that to a typo, but it was really me just working from memory without checking
my work. A correct equivalence relation for a wheel over integers is as
follows:

    
    
        (a,b) ~ (x,y) iff there exists s1,s2 (both non-zero integers) such that:
            (s1 * a, s1 * b) = (s2 * x, s2 * y)
    

When b,y are non zero, this is the same as the equivalence relationship on
fractions.

------
tim333
Well, in javascript 0/0 == NaN while 1/0 == Infinity.

An interesting thing is they have the value -0 so you can set a=-0 and 1/a
returns -Infinity.

~~~
dylan604
This should be part of an update to the Destroy All Software's Wat video:
[https://www.destroyallsoftware.com/talks/wat](https://www.destroyallsoftware.com/talks/wat)

~~~
garmaine
Except that is defined, standard IEEE 754 behavior and is true on nearly all
programming languages.

------
cairo_x
Nothing has an infinite number of nothings in it.

One has zero nothings in it...? It also has an infinite number of somethings
in it. And each something has another infinity of divisible somethings within
that.

Therefore anything that isn't nothing (in the universe of abstraction which
doesn't exist in reality, but whatever) is infinite.

OBVIOUSLY.

NOTE: Hasn't read the article and is mathematically challenged.

------
8note
my calculus answer is that usually you see 0/0 when looking for limits

1/0 definitely has no real value 0/0 might have one, but you've framed the
question poorly -- if you rewrite the problem in a different way, you might
find the value of 0/0 for your case eg, by applying L'Hôpital's rule

------
murkle
Extended number line is one way to formalise this
[https://en.m.wikipedia.org/wiki/Extended_real_number_line](https://en.m.wikipedia.org/wiki/Extended_real_number_line)

------
graycat
To answer the question, move to the _extended real numbers_ as in, say, H.
Royden, _Real Analysis_.

In short, the extensions work and at times are convenient but work less well
than the rules of the real numbers; the extensions are a bit tricky and have
to be careful.

------
askvictor
Also fun: 0^0 (not an emoji; 0 to the power of 0). Windows Calculator gives an
answer of 1; Excel throws a number error, while Android hedges its bets (shows
"Undefined, or 1").

------
pugworthy
In a kind of both joking way, and in a serious as a programmer way, either way
is the same - and not exceptional. I mean beyond that it will raise an
exception. Either way, you divided by zero.

------
yuriko
I'm definitely joking, but

$ node

> 1.0/0.0

Infinity

> 0.0/0.0

NaN

------
saagarjha
> Last year a new Math Stack Exchange user asked What's the difference between
> 0/0 and 1/0? I wrote an answer I thought was pretty good, but the question
> was downvoted and deleted as “not about mathematics”.

Never change, Stack Exchange.

~~~
dreamcompiler
Almost every useful answer I've ever found on SE or SO has been marked down
with some snarky judgmental horseshit meta-comment. At this point, if I don't
see such markdowns, I assume the answer isn't very useful until proven
otherwise.

~~~
ultrarunner
I wonder if this is a repeatable phenomenon wherein a universally-useful
resource is more likely to be accessed by a wider gamut of people, thus
essentially guaranteeing a judgmental response. That would mean those snarky
responses _really are_ a mark of a good and useful question.

~~~
wallace_f
After the world solved the scarcity problem with respect to everyone's ability
to survive a normal, healthy, dignified lifespan, most of the economy
continued to shift towards conspicuous consumption.

If you look honestly at humanity, the motivations of the average person are
selfish and unenlightened. An internet mod enjoys authority and an elevated
sense of self worth by putting other people down.

There really are some people genuinely enthused by virtues such as
intellectual discovery and shared dignity, but they're in short supply and
likely to be outnumbered in the population clamoring to become internet mods.

I've had a few posts removed from a SO site after I corrected someone who
happened to be a mod there. And the Innocence Project fights to give innocent
people their lives back because some prosecutors don't want a scratch on their
resume.

I think it would be better if we acknowledged these realities more-often
rather than pretending all the evil people in the world died in Nazi Germany,
or wherever, and they weren't just normal people.

I think this is also why I always retreat into nerdy or intellectual
endeavors. Not because I am particularly intelligent, but because the real
world is so ugly.

~~~
ineedasername
I agree with everything you said, except that first part about solving the
scarcity problem. I'd need you to elaborate there, because it doesn't seem
like that's been done in many parts of the world.

~~~
wallace_f
Humanity currently possesses the capability to provide everyone their basic
human needs (food, safe water, housing, sanitation, essential healthcare) and
hunan rights, but not the willingness. People die preventable deaths because
they lack access to basic, affordable healthcare while other people are
getting plastic surgery. People are homeless all over LA and SF while QE
printed trillions for Wall St. Doctors without Borders hospitals are bombed
while Wolf Blitzer says ending war there is a "moral issue because it will
cost US defense contractors jobs."

~~~
ineedasername
Ah, capability to feed, health care, etc., yes. We _should_ be in a post-
scarcity civilization with respect to basic needs. I seem to recall reading
studies though that showed on a per-dollar basis, it was much better in the
long term to boost the local economy rather than just give handouts in the
form of food aid etc. However, my thought in response to that has always been
that it frames the situation as though there were a choice between the two. My
thinking is that if people are living in misery, poverty, disease, and death,
why make this a choice? We can do both. It's a multi-decade project, but not
intractable. It's also probably hopelessly optimistic, but I'm okay with
failures that move the ball forward a bit.

~~~
wallace_f
I also think it is true that local economies have been devastated by hand
outs. It's a complex issue but it's also not.

There is no real excuse here as far as human cognitive ability goes. The
problem is with morality.

I think these examples show that as humanity goes forward I think moral
integrity/intelligence/bravery is more important to develop than our purely
technological and scientific capabilities.

------
TomMckenny
>I wrote an answer I thought was pretty good, but the question was downvoted
and deleted as “not about mathematics”.

Why post on some else's site? Especially quality work that took real effort.
I'm guilty of it too (although nothing so cool is this article).

We complain about the web becoming dangerously centralized compared to the
past. Much of this is because of free high grade labor that adds value to an
already dominant organization.

~~~
jobigoud
The usual answer to that is "discoverability". You can either post your
content to someone else's website and have it read/viewed/watched, or post it
where virtually no one will see it so you might as well not post it.

~~~
TomMckenny
Yes but discoverability to what end? In one case the reader then thinks and
says "I learned this on Stack Exchange" rather than a possibility of "I
learned this from Mark".

In one case an organization that is much complained about becomes even more of
a necessity and so has less incentive to fix itself. In another the reverse
happens.

In hyperbole: in one case you may as well not post it, in the other case you
may as well have not written it.

------
ggm
The one thing his article didn't say for the arithmetically challenged person
(me) is very much about 1/0 - he certainly talked about anything/0 and
function()/0 but dived off why 1/0 and 0/0 have subtle difference.

It's a good read but he missed his clear mission. One extra paragraph?

~~~
mannykannot
The middle section is the author's answer: they are different because they
fail to yield a definite answer in different ways and for different reasons.
You will find some alternative ways of looking at it in the comments here.

------
marknadal
The main argument I've seen against division by zero is because:

"Then 2 + 2 could equal 7!"

If you go through the examples of how they derive that, you can actually fix
it quite easily by assigning each 0/0 pair to different variables. This
prevents the underhand commutative rule violation.

Then all that you are left with is linear algebra. Whether that equation is
solvable depends upon how many constraints you have. But it at least doesn't
cause math to violate itself.

~~~
throwawaymath
That doesn't work. You're changing the setting of the problem from scalars to
vectors in order to avoid breaking the uniqueness properties of fields and
rings. The solution set for a system of linear equations is not unique (if it
is at all) in the same sense that e.g. additive and multiplicative identity
elements are unique.

