
1+1 = 0 - pavel_lishin
https://plus.google.com/117663015413546257905/posts/gvzrNKQqqnV
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cperciva
_Math gets simpler in a world where 1+1=0, but it doesn 't become self-
contradictory and explode into nothing. We call this number system the field
with 2 elements or F₂._

There are lots of fields in which 1+1=0, not just F₂...

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hardmath123
Full article:
[https://golem.ph.utexas.edu/category/2016/01/integral_octoni...](https://golem.ph.utexas.edu/category/2016/01/integral_octonions_part_12.html)

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pyed
the mother of all click-baits

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gridspy
Better title is 2 mod 2 == 0

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catnaroek
In the study of rings and fields, one works with operations called “addition”
and “multiplication”, typically written “+” and “*”, but which needn't have
anything to do with number addition or multiplication. “Addition” and
“multiplication” have identity elements called “0” and “1”, respectively, but
again, they needn't have anything to do with the numbers 0 and 1.

And, in a field of characteristic 2, there's no notion of 2, since “1 + 1” is
by definition “0”.

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gridspy
Putting "The sky is black" as a title seems incorrect to me, even though the
statement "The sky is black on the Moon" is true.

~~~
catnaroek
+, × (or ·), 0 and 1 are standard notation in algebra for an arbitrary ring's
addition, multiplication, additive identity and multiplicative identity. In
that context, 1 + 1 = 0 isn't automatically false. It's just a statement that
holds in some rings (e.g., F_2), but not in others (e.g., Z).

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jMyles
This is just a bit too thick for me; I imagine I can grok it with just a bit
more breakdown. Is there such a thing?

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catnaroek
The author's post is written in an abstract setting, where 0 and 1 don't mean
the usual numbers 0 and 1, but rather the additive and multiplicative elements
of a field. The most basic examples of fields are the rationals, the reals and
the complex numbers, all of which have characteristic 0. But, for any prime
number p, there exist fields of characteristic p as well. In a field of
characteristic p, the result of repeatedly adding any element p times is the
field's additive element. In particular, in a field of characteristic 2, “x +
x = 0” for any x.

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pash
I wouldn't say they don't mean the usual numbers 0 and 1 in this setting. F_2,
also called GF(2) in another standard notation, can readily be interpreted as
the set containing only the natural numbers 0 and 1, together with the
operations of addition modulo 2 and multiplication. In other words, everything
has its usual meaning, except that addition wraps around so that you remain in
the two-element set.

Finite fields are kind of fun. The body of knowledge about them is called
Galois Theory and makes a pretty good entrée into the world of abstract
algebra. Interested readers might want to check out the short, $8 book by Émil
Artin from Dover [0] for a good introduction.

0\. [https://www.amazon.com/Galois-Theory-Delivered-University-
Ma...](https://www.amazon.com/Galois-Theory-Delivered-University-
Mathematical/dp/0486623424)

~~~
catnaroek
> I wouldn't say they don't mean the usual numbers 0 and 1 in this setting.

I was talking about arbitrary fields of characteristic 2, not necessarily F_2.

> F_2, also called GF(2) in another standard notation, can readily be
> interpreted as the set containing only the natural numbers 0 and 1, together
> with the operations of addition modulo 2 and multiplication.

The question whether the additive elements of two distinct fields are the same
mathematical object is evil[0], since its answer isn't invariant under field
isomorphism. So I'd rather consider meaningless the very idea of comparing
elements of different fields.

[0]
[https://ncatlab.org/nlab/show/principle+of+equivalence](https://ncatlab.org/nlab/show/principle+of+equivalence)

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pash
The OP specifically calls them elements of F_2. There's no harm in thinking of
the elements of F_2 as the ordinary numbers 0 and 1, as they retain all the
properties of those numbers that are pertinent in the restricted setting of a
two-element field.

One needn't ensure that an equivalence is well founded in order to invoke it
for pedagogical purposes. We don't even need to specify which structure we're
talking about when we refer to "the ordinary numbers 0 and 1". It's an
imprecise statement that helps beginners grasp the basic ideas, which is all
the situation calls for.

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nilkn
> There's no harm in thinking of the elements of F_2 as the ordinary numbers 0
> and 1, as they retain all the properties of those numbers that are pertinent
> in the restricted setting of a two-element field.

No one viewpoint is going to be pedagogically valuable for all listeners.
catnaroek was providing an expanded breakdown which might be helpful to those
who struggle with the (technically poorly founded) equivalence you're putting
forward here.

> One needn't ensure that an equivalence is well founded in order to invoke it
> for pedagogical purposes.

At the beginning of "The Road to Reality," Roger Penrose has a great piece
about how he believes that informal use of "equivalences" in mathematics for
teaching purposes is responsible for preventing a lot of otherwise very smart
people from grasping basic things like fractions.

His example is that some children who struggle with normal presentations of
fractions do so because on some intuitive level they understand that "1/2" is
actually an equivalence class of fractions, but it's very difficult for a
child to reconcile that intuition with how fractions are taught to them in
many schools.

I think the lesson here is that some people learn in different ways, and for
some people using informal false equivalences or hiding formal true
equivalences from sight is counterproductive.

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cttet
This is just a demonstration of how math is not a rigid language and
consequence of misuse of dynamic typing...

If we define symbol '1' to be integer typed, then there won't be any F_2
related confusion. And 1 + 1 is always 2.

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jcoffland
The title should be:

    
    
        1 + 1 ≡ 0 (mod 2)

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viach
Great idea for a title /sarcasm

