The h Bar

2:00 PM

This helps me to e.g. grasp the inner product when it is not the standard one,, because you are changing some parameters or properties of the maths object

I once spent a week trying to derive a mathematical object that can incoprate division by zeor, simply because I am curious and for no other reason

only to be more convinced why division by zero is not useful in the end, so I learnt something in the process, I guess...?

bolbteppa

@Secret I think what you're doing is just playing with the fact that the additive group of integers remains a group when you go to the quotient group mod 2, more specifically just looking at the fact that $(\mathbb{Z},+)/~ = (\{0,1\},+_{\mod 2})$ is closed...?

Secret

so 0 is the identity here?

ACuriousMind

@Secret The assignment of "odd" and "even" to numbers is formally called a $\mathbb{Z}/2\mathbb{Z}$-grading (see $G$-graded algebras‌)

bolbteppa

Sorry, can't get the ~ to show up in that

Secret

@ACuriousMind I see, guess I learnt something new today

bolbteppa

2:06 PM

That is just a way to split $\mathbb{Z} = Odd+Even$, in this case it's the same thing as just putting even into the $\overline{0}$ class etc...

Secret

ok that makes sense (glad they taught us modular arithimetic back in my linear algebra course)

bolbteppa

Modular arithmetic written as $a = nm + b$ is incomprehensible, written as as $a = b + mn$ it makes perfect sense, I hope I'm smart :\

Secret

can we actually have non abelian modular arithmetic?

ACuriousMind

@bolbteppa Indeed, but the nice thing about formulating it as a Z/2Z-grading means that this gives you an immediate way to generalize the concept of "even" and "odd" to e.g. Grassmann algebras.

bolbteppa

Sorry the important thing was switching $a = nm + b$ into $a = b + nm$

Good point

ACuriousMind

2:11 PM

@Secret "Arithmetic" would rather by definition be something over Z, so abelian, but of course you can divide out arbitrary normal subgroups out of groups. The notion of quotient group generalizes the idea of "integers modulo n" to arbitary groups in the sense that the integers modulo n are just the quotient of Z by the subgroup consisting of multiples of n.

Secret

@Acuriousmind 2nd comment: makes sense

@ACuriousMind So using what you said, Z/3Z will be basically consist of "even", "odd" and<insert name for the third even/odd behaving thing that is not even nor odd>?

bolbteppa

I wonder what's even:

ACuriousMind

@Secret Well, no, it makes no sense to call things in Z/3Z even or odd since 1+1=2 mod 3, so 'odd' times 'odd' would be neither even nor odd, which destroys the former defining properties of 'odd' and 'even'

Secret

ok

bolbteppa

@skullpetrol does he think it's okay to divide by zero?

skull petrol

2:17 PM

No @bolbteppa but he can't explain why not.

Secret

My brain wiring must have accidentalyl crossed somehow...
I tend to treat a maths problem like a physics problem and a physics problem like a maths problem

1) Maths problem

$$blah+blah=djdfjh$$
$$r*r*e=e$$

My impression: O look, there's an interesting phenomenon here, and it obeys the rules and properties as shown

Manipulating some algebriac expression

My impression: Let's see what happens if I throw in this, and take out that

2) Physics problem:

for example:

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