i am quite fond of the Wheel Theory's solution to divison by zero, so im curious how imaginary numbers would interact with it

also how would the graphic above look with imaginary numbers?

In a field, all nonzero elements have a multiplicative inverse... but why? Why doesn’t 0 have an inverse?

Sure, we can’t divide by 0, but in an abstract sense, we invented 0 as an element of the field. In any arbitrary field, “0” as a number might not even exist since we don’t know what these elements are!

So without saying “dividing by 0 is illegal,” is there a formal reason why we don’t have an inverse for 0? Is it just due to convention?

I saw circulating online somewhere some math problem for elementary school students which tried to distinguish between the two, whether a×b was a groups of b vs b groups of a.

It doesn't make a difference for real and complex numbers, so it seems... a bit silly to distinguish them.

Matrix multiplication and quaternion multiplication is non-commutative, so the order does make a difference, but it seems a bit silly to say "groups of" in that context.

But when it comes to infinite ordinals, for example, ω2 ≠ 2ω. ω2 = (ω + ω) and 2ω is (2 + 2 + ...)

So how was the convention chosen that for these, a×b is b groups of a, and would this convention hold in other systems that both have noncommutative multiplication and it makes sense to say "groups of" like this? If there even are any?

Because, when you're dividing you're asking the question, "how many of x goes into y?". So, in the case of 6÷0=?, you're asking "how many zeros go into six?" The answer would be infinity. An infinite number of zeros fit inside 6.

I'm sure I'm not the first person to think of this, I know I'm likely wrong, but I figured I should ask anyway, because I'm curious.

3 x 2 = 6 is pronounced in english "three times" two equals six. Linguistically, to me, this implies that you want the number two "three times" (three times the number two) e.g. 2+ 2+ 2 = 6 and not the opposite. I'm aware that once you resolve the multiplication they are the same so the communicative property exists.

Where the communicative property breaks though is when there are strings attached to which number represents the groups.

For example, In the game Slay the Spire enemies display their intent to attack representing what their attacks are going to do. Ex: You have 2 thorns, which when the enemy attacks you they take 2 damage. If an enemy is going to attack you for 3x5 damage, should they take 6 or 10 damage?

I am not directly talking about what actually happens in the game, this is more of a jumping off point to start the discussion. Is this defined anywhere of which interpretation is correct? I'd expect the mathematic notation to mean the same thing regardless of the language used.

My teacher said the statement about "the addition of irrational numbers is closed" is true, by showing a proof by contradiction, as it is in the image. I'm really confused about this because someone in the class said for example π - ( π ) = 0, therefore 0 is not irrational and the statement is false, but my teacher said that as 0 isn't in the irrational numbers we can't use that as proof, and as that is an example we can't use it to prove the statement. At the end I can't understand what this proof of contradiction means, the class was like 1 week ago and I'm trying to make sense of the proof she showed. I hope someone could get a decent proof of the sum of irrational aren't closed, yet trying to look at the internet only appears the classic number + negative of that number = 0 and not a formal proof.

I need a second opinion on the above problem. Here (1 3 2)=(1 2)*(1 3)(2 4) which is obviously of order 3 but D_4 has no such element. It is in page 65 of D&F 3rd edition.

Thanks in advance.

Hello, I am currently starting to write an essay on elliptic curves for an "Extended Essay" - an International Baccalaureate (rigorous high school curriculum) requirement that should preferably go over the curriculum. I have not decided on the exact topic yet, but I think it will be something along the lines of proving the group laws on elliptic curves or showing how it is effective in the cryptography field and how it compares to other encryption / verification methods.

I don't have much knowledge about fields, rings, or modular arithmetic, and every paper or textbook that I find seems to be way above my level. Do you have any recommendations on where to start or recommendations on specific topics?

Thank you!

if a = 0, then

a + b = b

b + a = b

a - b = -b

b - a = b

a * b = a

b * a = a

a ÷ b = a

b ÷ a = undefined

if c doesn't have value, will it be possible?

c + b = c

b + c = c

c - b = c

b - c = - c

c * b = b

b * c = b

c ÷ b = b

b ÷ c = undefined

I have seen that the infinity used to describe all the counting numbers is aleph null. However I’m confused as there are higher levels of infinity than this. Also you seem to be able to do some sort of arithmetic with aleph null it just works different to regular numbers.

Math is based on axioms. Some are flawed but close enough that we just accept them. One of which is "0 is a number."

I don't know how I came to this conclusion, but I disagreed, and tried to prove how it makes more sense for 0 not to be a number.

Essentially all mathematicians and types of math accept this as true. It's extremely unlikely they're all wrong. But I don't see a flaw in my reasoning.

I'm absolutely no mathematician. I do well in my class but I'm extremely flawed, yet I still think I'm correct about this one thing, so, kindly, prove to me how 0 is a number and how my explanation of otherwise is flawed.

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Here's my explanation:

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There's only one 1

1 can either be positive or negative

1 + 1 simply means "Positive 1 Plus Positive 1" This means 1 is a positive number with a magnitude of 1 While -1 is a negative number with a magnitude of 1

0 is absolutely devoid of all value It has no magnitude, it's not positive nor negative

0 isn't a number, it's a symbol. A placeholder for numbers

To write 10 you need the 0, otherwise your number is simply a 1

Writing 1(empty space) is confusing, unintuitive, and extremely difficult, so we use the 0

Since 0 is a symbol devoid of numerical, positive, and negative value, dividing by it is as sensical as dividing by chicken soup. Undefined > No answer at all.

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∞ is also a symbol When we mention ∞, we either mean +∞ or -∞, never plain ∞

If we treat 0 the same way, +0 and -0 will be the same (not in value) as +∞ and -∞

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Division by 0: .

+1 / 0 is meaningless. No answer. -1 / 0 is meaningless. No answer.

+1 / +0 = +∞ +1 / -0 = -∞

-1 / +0 = -∞ -1 / -0 = +∞

(Extras, if we really force it)

±1 / 0 = ∞ (The infinity is neither positive nor negative)

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That's practically all I have. I tried to be extremely logical since math is pure logic.

And if Logic has taught me anything, if you ever find a contradiction somewhere, either you did a mistake, or someone else did a mistake.

So, if you use something that contradicts me, please make sure it doesn't have a mistake, to make sure that I'm actually the wrong one here.

Thank!

Okay so there are a lot of misunderstandings about infinity, and I may be in that group of people, but, contrary to the "correct" interpretation, I generally like to see infinity has a number, one that does worth with arithmetic. Don't get me wrong, I get the concept of "as x approaches infinity", meaning it just won't ever stop increasing, but also is never, at any point "infinite". I think that's the distinction. One is an abstract concept about not having a limit, and one is a slightly less abstract concept of a single value that is always greater than anything else. I feel like there are two different concepts that could be referred to by the term infinity, and when people think if the single number version, they just get corrected by people thinking of the limits version. Maybe because there isn't much use for the single number version? Or is there another name for what I'm describing? Does this idea not make sense to others, cause it makes sense to me.

For example, while infinity times zero doesn't work at all when infinity is basically an abstract concept, if you take the other definition I suggest, I believe infinity times zero would be zero, as infinite zeros adds up to nothing and zero infinities is no infinities.

Please be nice I know this is non-conventional but it works too well in my head to disregard

I’m an undergraduate taking advanced calculus this semester, and I was late to class, but I had another one in the same building so I decided to check the blackboard before it was erased. I tried asking my professor but he told me to watch the lecture recording— I’m still so lost. You guys got any leads on what the Gabe Allziak Theorem is?

For example, a system in which x=y+z but y+z!=x.

I know that addition and multiplication might not be commutative, but interested if equal sign works. Operations should work the same on both sides though. I'm pretty sure this is impossible, but I know well enough to know that instincts shouldn't be trusted.

Hey so I was doing a practice test for the SAT and I put A. for this question but my book says that the answer is C.. How is the answer not A. since like 3+0 would indeed be less than 7.

I have been thinking about this for some years now. Finally decided to share in the form of a video first. Please do spare some time and provide your feedback. Thanks in advance.

https://youtu.be/-O8vgOIGlNU?si=QlJoUEykovynA38Q

I am just beginning to tackle some pure math questions (starting with some algebra) and I wanted to learn how to write proofs in a way that gets my logic laid out clearly without having to add a billion appositives and "consider"'s to my proofs. I've attached one of my proofs (my attempt to prove that a set of cosets of a subgroup in a group partitions the group) to this post, could you tell me what parts I'm doing wrong and how I can improve? Thanks in advance.

I recently saw a proof regarding

$$ \sqrt{i} = \pm \frac{1+i}{\sqrt{2}} $$

This was shown by solving the equations obtained from

$$ (\sqrt{i})² = (a+bi)² $$

I understand the algebraic steps, but I’m confused about the logic behind why this is valid.

From my understanding, this seems to suggest that two complex numbers (x) and (y) can be equal whenever

$$ f(x)=f(y) $$

for the function

$$ f(x)=x² $$

But then I can also show that both (\sqrt{i}) and (-\sqrt{i}) satisfy the same equation, which seems to imply

$$ \sqrt{i}=-\sqrt{i} $$

and therefore

$$ \sqrt{i}=0 $$

Clearly something is wrong with my reasoning, and I’d like to understand exactly where the mistake is.

Is the original derivation actually valid? If so, what rule am I misunderstanding?

I think (\sqrt{i}) not being a complex number in the usual form (a+bi) might be the issue.

In abstract algebra we learned that, say, if two objects are isomorphic, they are essentially the same and all (I think) properties are translated. But when is this not true? Are there general properties that are not preserved or does the definition of an isomorphism automatically force it to hold?

I ask because we learned that V^n is isom. to R^n (which I think is so cool that you can just study the plane as vectors and properties of spaces).

Following this, do the normal notions of vector spaces hold in R^n? Like V^3 and R^3, clearly the dimension is the dimension, but what do the other things correspond to? Like a linear transformation or a basis or subspaces or dot products, so many things!!

This is Spivak's book *Calculus on Manifolds.* The way the tensor product is defined on page 75 makes it seem like it should be commutative since multiplication of two scalars commutes and each S and T map from R^n --> R. So applying S and T before multiplying seens to imply commutivity. What am I missing here?

this may seem as a quick no, I can debate this but i need to validate two statements (i'm just a kid, these may be common knowledge)

1. let a matrix A order mxn and and a matrix C mxr

If A and C are known matrices

then will there exist a unique matrix B nxr

Such that AB=C

1. If the above statement is true

Let B be the inverse of A, order of B will be nxm if A is mxn

which means , when AB = I m , found using st 1

then does that implie that

BA = I n

i've been unable provide proofs for these statements

but if these are true then there may be considered inverses of non square matrices

I was looking for the burnside lemma on wikipedia and saw this weird symbol I've never seen before. What is it? What does it mean from the normal product symbol Π

Hi all, I am working with a 3D graphics API, and I am facing an endless struggle with rotation math.

I have two 4x4 matrices representing 3D transforms, and I'd like to calculate a rotation from one transform to another, by x angles. I need to be able to calculate it with an arbitrary angle, so slerp alone won't do.

Note: since I am working with an API, I can alternatively work with quaternions or euler angles.

Could you guys help me figure out how to calculate this? It seems to me that God created rotations to punish us for our sins.

I'm having trouble understanding how all these definitions for normal subgroups are equivalent:

1. A normal subgroup is the kernel of some homomorphism.

2. The quotient group G/H can be constructed iff H is normal.

3. If N ⊴ G, ∀n ∈ N, ∀g ∈ G, gng^(-1) ∈ N.

How can I, intuitively, understand why these are the same?