I'm Ellie, a 21 y/o student studying graduate math and hoping to someday be a mathematician! Please hit me up to talk about math, transness, or really anything! I'm bi, transfem, and have been in a relationship for 4 years :>
What do you recommend for beginners to improve their math skills on their own?
Allow yourself to be curious!! When you have a question about something, don’t just leave it there, try and find an answer on your own, and if you can’t, then try and look and see if anyone has done the problem before to see what you should do to figure it out! Just taking questions and ideas you have about math and not just stopping there is a lot more than a lot of people do with math.
Other than that, it really depends on what you mean by beginner (beginner at math could mean anything from learning high school algebra, to learning calculus, to starting to learn proofs)
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pressing space to interact so I can get the math loredump (ik vectors are really just ordered pairs but arrow and they do things unlike ordered pairs which have executive dysfunction and do nothing)
As you might know, mathematicians like to generalise things.
Also unordered pairs are kind of useful. You use the axiom of pairing in order to construct ordered pairs in ZF.
(Also, fun fact. You can construct the Axiom of Pairing from just the Axiom of Replacement and either Power Set or Infinity, and you do not need extensionality.)
But a space of ordered pairs is not the same thing as a vector space. A vector space is a module on a field.
Info dump below. Sort of rambly and surface level but if you are interested I can get more formal.
Preface: Patterns and Generalisations
In mathematics we often like to generalise things. We're usually studying some form of (for lack of a better word) "patterns", and trying to predict or discover things about them.
And we like keeping track of which facts about our we used to prove something. For example, if all we proved something for the real numbers, and all we used was the fact that there are infinitely many reals between any two real numbers, then we can say that the fact can be reapplied to the rationals, since that's also true for the rationals.
We formalise this with the idea of axioms. Things we will assume to be true, and then prove things out of. Then if anything satisfies these axioms, then the things we proved apply.
A very simple example are the Peano Axioms, which are used for studying the natural numbers.
So, for example, we might want to study the notion of symmetry. And from there create the axioms for groups, and group actions. The main properties you would like to hold for symmetries. You have seen one of my posts on this matter. Groups are a set equipped with a closed 'multiplication' that is associative, invertible, and unital. Now instead of proving something on individual symmetries and groups, you can start thinking about everything that has the same kind of "patterns" that symmetries do, all at once.
That leads us to the concept of Rings.
Rings
A ring is something you can do polynomials on. We really like polynomials, they are useful, and we would like polynomials to behave a certain way.
Well, in theory all we need for polynomials are addition and multiplication. All powers of things have to be natural numbers in a polynomial, so you can express exponentiation as repeated multiplication.
So we need to be able to add things. Add like terms. Cancel things. The usual basics. I will leave it as an exercise to you to think very closely of what might be necessary for polynomials, but we came up with the following definition for a thing called a "Ring".
A ring is a set R equipped with a pair of closed operations + and •, and with constants 0 and 1.
As a shorthand I will write ab to mean a•b.
(R, +, •,0,1)
Basic canceling properties that are nice:
∀x x+0=x
∀x∃(-x) x+-x = 0
Properties so that the order of addition does not matter:
∀x∀y x+y=y+x
∀x∀y∀z (x+y)+z = x+(y+z)
Basic multiplicative properties:
∀x 1x=x
∀x x1=x
∀x∀y∀z (xy)z=x(yz)
And distributive properties:
∀x∀y∀z x(y+z)=xy+xz
∀x∀y∀z (x+y)z=xz+yz
Notably, the multiplication is not necessarily commutative, which is why we have to do it on both sides each time. When multiplication is commutative, we have a commutative ring. We don't specify that rings must be commutative because there are a lot of useful things that fit the properties we want of a ring, except commutativity, but we still want polynomials of them: For example: rotations in 3d space, quaternions, and matrices.
We can also add and multiply polynomials themselves, and you can check that the set of all polynomials of a ring are also a ring! This also could lead to a conversation about what an Adjoint is.
Modules
Skipping over a lot of things in rings, now that we defined rings, we can consider modules, which are basically just "things a ring can act on". Kind of like actions for a group. This isn't super in depth, but the idea is that they have some properties:
A left R-module M is an abelian (abelian means commutative in this case) group M and a ring R with an action between (taking an an element from R and one from M to an element form M) them that has these properties:
For any r, s in R, and x,y in M:
r(x+y)=rx+ry
(r+s)x=rx+sx
(rs)∘x=r∘(s∘x)
1∘x=x
An example of this are the Real Numbers on vectors, taking a∘(v1,v2,v3) to (av1,av2,av3), or the Integers on the Imaginary Integers taking a∘bi to abi.
These seem awfully like vectors. And that's mostly because they are! Modules were defined based on a generalisation of the idea of vectors, but to an arbitrary ring. So vector spaces are all modules. But there are weirder modules that you can also study.
The Atrocity
Another example would be quaternions acting on a list of quaternions in the same way as the vector example. It would be, for all intents and purposes, a vector.
But the definition of a vector space isn't a module on a ring. It is a module on a Field.
What's a field? It's a commutative ring you can do division on. Like the Real Numbers or the Rationals. Not the Integers. This idea turns out to 'kind of' be the same as the ordered pairs idea, as long as you have the Axiom of Choice. It is the generalisation of that to include things with infinite dimensions and other weird cases. Just the bare patterns required for a vector space.
Quaternions are not commutative in multiplication. pq≠qp. And that's something we have put into the definition of a vector space. So a module over quaternions doesn't count as a vector space!
Which is sad because quaternions were what vector spaces originally came from!
I had a reply typed out then my phone imploded (death of the battery)
my understanding of a ring thus far is basically a set of the universe, the fundamental forces, and subatomic particles but designed around shoving polynomials into as little of a box as possible
funny story I had to look up the silly upside down letters and then afterwards when I went to reread that part i just started filling "such that" in the long spaces by default
edit: more questions: why is it called a ring? is it because it's closed (unknown if this is true)?
edit 2: trying to comprehend modules what's the circle of joy and whimsy mean
At an intuitive level, any set we can do polynomials that behave nicely on is a ring. So for example, the integers mod n form a ring.
Here is a glossary of math notation post!
It is nowhere near being comprehensive, (and I had kind of hoped other people would add to it) but it does explain a good chunk of math notation. It's a useful thing that can save you a bunch of writing!
Hmm. There are a few names for these, but the properties that we find interesting or useful about polynomials are the ones that made us choose the definition, so relaxing the restrictions on behaviour leads us to things I have not studied as much.
I happen to know one, called a rng, which is a ring without an identity (which is why we took out the letter 'i' from ring.)
There are also semirings, (and quasi-rings?) etc. which also give us weird polynomial-like things.
These all belong to a much less restricted class of thing called an algebraic structure. Which is anything you can "kind of" do some algebra on.
IIRC polynomials don't really act nicely over all rings, only commutative rings. If multiplication in a ring isn't commutative, then (ax³)(bx²) doesn't necessarily equal abx⁵.
Actually, there are a few ways of defining polynomials over rings, you can take a general polynomial which differentiates f(x)=ax and f(x)=xa, or you can assume the variable term commutes until it is evaluated, and at that point have a rule for where to evaluate it at (usually just on the left or the right of the constant), but both of these are well defined even in noncommutative settings!
A big reason this is important is because we can use the set of Laurent series’ (with the variable treated as commuting) on a noncommutative division ring to create another distinct noncommutative division ring! So polynomials can be well defined over noncommutative settings if you’re careful about it and decide ahead of time how you want to handle the variables :)
I think the construct is more clever than I am, or at least I am incapable of putting this together on the fly.
Also, question, how is or is this construction different from just taking the ring adjoint with the free monoid for a variable? They seem the same to me at the moment but I am not certain enough in my understanding to claim that.
Could you explain an example with the Laurent series?
I don’t know much about the hat the ring adjoins with the free monoid for the variable is so I can’t help there until I look into that, sorry!
But the example with the formal Laurent series is basically just that if you have an infinite power series with finitely many negative terms with the variable commuting and elements in a division ring, then of course it forms a ring under multiplication and addition, but it specifically forms a division ring for the same reason that formal Laurent series’ of fields forms a field, every element does have a unique inverse element!
pressing space to interact so I can get the math loredump (ik vectors are really just ordered pairs but arrow and they do things unlike ordered pairs which have executive dysfunction and do nothing)
As you might know, mathematicians like to generalise things.
Also unordered pairs are kind of useful. You use the axiom of pairing in order to construct ordered pairs in ZF.
(Also, fun fact. You can construct the Axiom of Pairing from just the Axiom of Replacement and either Power Set or Infinity, and you do not need extensionality.)
But a space of ordered pairs is not the same thing as a vector space. A vector space is a module on a field.
Info dump below. Sort of rambly and surface level but if you are interested I can get more formal.
Preface: Patterns and Generalisations
In mathematics we often like to generalise things. We're usually studying some form of (for lack of a better word) "patterns", and trying to predict or discover things about them.
And we like keeping track of which facts about our we used to prove something. For example, if all we proved something for the real numbers, and all we used was the fact that there are infinitely many reals between any two real numbers, then we can say that the fact can be reapplied to the rationals, since that's also true for the rationals.
We formalise this with the idea of axioms. Things we will assume to be true, and then prove things out of. Then if anything satisfies these axioms, then the things we proved apply.
A very simple example are the Peano Axioms, which are used for studying the natural numbers.
So, for example, we might want to study the notion of symmetry. And from there create the axioms for groups, and group actions. The main properties you would like to hold for symmetries. You have seen one of my posts on this matter. Groups are a set equipped with a closed 'multiplication' that is associative, invertible, and unital. Now instead of proving something on individual symmetries and groups, you can start thinking about everything that has the same kind of "patterns" that symmetries do, all at once.
That leads us to the concept of Rings.
Rings
A ring is something you can do polynomials on. We really like polynomials, they are useful, and we would like polynomials to behave a certain way.
Well, in theory all we need for polynomials are addition and multiplication. All powers of things have to be natural numbers in a polynomial, so you can express exponentiation as repeated multiplication.
So we need to be able to add things. Add like terms. Cancel things. The usual basics. I will leave it as an exercise to you to think very closely of what might be necessary for polynomials, but we came up with the following definition for a thing called a "Ring".
A ring is a set R equipped with a pair of closed operations + and •, and with constants 0 and 1.
As a shorthand I will write ab to mean a•b.
(R, +, •,0,1)
Basic canceling properties that are nice:
∀x x+0=x
∀x∃(-x) x+-x = 0
Properties so that the order of addition does not matter:
∀x∀y x+y=y+x
∀x∀y∀z (x+y)+z = x+(y+z)
Basic multiplicative properties:
∀x 1x=x
∀x x1=x
∀x∀y∀z (xy)z=x(yz)
And distributive properties:
∀x∀y∀z x(y+z)=xy+xz
∀x∀y∀z (x+y)z=xz+yz
Notably, the multiplication is not necessarily commutative, which is why we have to do it on both sides each time. When multiplication is commutative, we have a commutative ring. We don't specify that rings must be commutative because there are a lot of useful things that fit the properties we want of a ring, except commutativity, but we still want polynomials of them: For example: rotations in 3d space, quaternions, and matrices.
We can also add and multiply polynomials themselves, and you can check that the set of all polynomials of a ring are also a ring! This also could lead to a conversation about what an Adjoint is.
Modules
Skipping over a lot of things in rings, now that we defined rings, we can consider modules, which are basically just "things a ring can act on". Kind of like actions for a group. This isn't super in depth, but the idea is that they have some properties:
A left R-module M is an abelian (abelian means commutative in this case) group M and a ring R with an action between (taking an an element from R and one from M to an element form M) them that has these properties:
For any r, s in R, and x,y in M:
r(x+y)=rx+ry
(r+s)x=rx+sx
(rs)∘x=r∘(s∘x)
1∘x=x
An example of this are the Real Numbers on vectors, taking a∘(v1,v2,v3) to (av1,av2,av3), or the Integers on the Imaginary Integers taking a∘bi to abi.
These seem awfully like vectors. And that's mostly because they are! Modules were defined based on a generalisation of the idea of vectors, but to an arbitrary ring. So vector spaces are all modules. But there are weirder modules that you can also study.
The Atrocity
Another example would be quaternions acting on a list of quaternions in the same way as the vector example. It would be, for all intents and purposes, a vector.
But the definition of a vector space isn't a module on a ring. It is a module on a Field.
What's a field? It's a commutative ring you can do division on. Like the Real Numbers or the Rationals. Not the Integers. This idea turns out to 'kind of' be the same as the ordered pairs idea, as long as you have the Axiom of Choice. It is the generalisation of that to include things with infinite dimensions and other weird cases. Just the bare patterns required for a vector space.
Quaternions are not commutative in multiplication. pq≠qp. And that's something we have put into the definition of a vector space. So a module over quaternions doesn't count as a vector space!
Which is sad because quaternions were what vector spaces originally came from!
I had a reply typed out then my phone imploded (death of the battery)
my understanding of a ring thus far is basically a set of the universe, the fundamental forces, and subatomic particles but designed around shoving polynomials into as little of a box as possible
funny story I had to look up the silly upside down letters and then afterwards when I went to reread that part i just started filling "such that" in the long spaces by default
edit: more questions: why is it called a ring? is it because it's closed (unknown if this is true)?
edit 2: trying to comprehend modules what's the circle of joy and whimsy mean
At an intuitive level, any set we can do polynomials that behave nicely on is a ring. So for example, the integers mod n form a ring.
Here is a glossary of math notation post!
It is nowhere near being comprehensive, (and I had kind of hoped other people would add to it) but it does explain a good chunk of math notation. It's a useful thing that can save you a bunch of writing!
Hmm. There are a few names for these, but the properties that we find interesting or useful about polynomials are the ones that made us choose the definition, so relaxing the restrictions on behaviour leads us to things I have not studied as much.
I happen to know one, called a rng, which is a ring without an identity (which is why we took out the letter 'i' from ring.)
There are also semirings, (and quasi-rings?) etc. which also give us weird polynomial-like things.
These all belong to a much less restricted class of thing called an algebraic structure. Which is anything you can "kind of" do some algebra on.
IIRC polynomials don't really act nicely over all rings, only commutative rings. If multiplication in a ring isn't commutative, then (ax³)(bx²) doesn't necessarily equal abx⁵.
Actually, there are a few ways of defining polynomials over rings, you can take a general polynomial which differentiates f(x)=ax and f(x)=xa, or you can assume the variable term commutes until it is evaluated, and at that point have a rule for where to evaluate it at (usually just on the left or the right of the constant), but both of these are well defined even in noncommutative settings!
A big reason this is important is because we can use the set of Laurent series’ (with the variable treated as commuting) on a noncommutative division ring to create another distinct noncommutative division ring! So polynomials can be well defined over noncommutative settings if you’re careful about it and decide ahead of time how you want to handle the variables :)
It represents the fact that there is a doubly-basis dependent definition for homomorphisms between right division ring modules as a left division ring module which induces an isomorphism with the module of matrices!
Basically it lets us treat homomorphisms as matrices multiplying Cartesian vectors if we have bases for our modules :)
Anya is live and ready to show you everything. Watch her strip, dance, and perform exclusive shows just for you. Interact in real-time and make your fantasies come true.
✓ Live Streaming✓ Interactive Chat✓ Private Shows✓ HD Quality
Anya is LIVE right now
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Free to watch • No registration required • HD streaming
pressing space to interact so I can get the math loredump (ik vectors are really just ordered pairs but arrow and they do things unlike ordered pairs which have executive dysfunction and do nothing)
As you might know, mathematicians like to generalise things.
Also unordered pairs are kind of useful. You use the axiom of pairing in order to construct ordered pairs in ZF.
(Also, fun fact. You can construct the Axiom of Pairing from just the Axiom of Replacement and either Power Set or Infinity, and you do not need extensionality.)
But a space of ordered pairs is not the same thing as a vector space. A vector space is a module on a field.
Info dump below. Sort of rambly and surface level but if you are interested I can get more formal.
Preface: Patterns and Generalisations
In mathematics we often like to generalise things. We're usually studying some form of (for lack of a better word) "patterns", and trying to predict or discover things about them.
And we like keeping track of which facts about our we used to prove something. For example, if all we proved something for the real numbers, and all we used was the fact that there are infinitely many reals between any two real numbers, then we can say that the fact can be reapplied to the rationals, since that's also true for the rationals.
We formalise this with the idea of axioms. Things we will assume to be true, and then prove things out of. Then if anything satisfies these axioms, then the things we proved apply.
A very simple example are the Peano Axioms, which are used for studying the natural numbers.
So, for example, we might want to study the notion of symmetry. And from there create the axioms for groups, and group actions. The main properties you would like to hold for symmetries. You have seen one of my posts on this matter. Groups are a set equipped with a closed 'multiplication' that is associative, invertible, and unital. Now instead of proving something on individual symmetries and groups, you can start thinking about everything that has the same kind of "patterns" that symmetries do, all at once.
That leads us to the concept of Rings.
Rings
A ring is something you can do polynomials on. We really like polynomials, they are useful, and we would like polynomials to behave a certain way.
Well, in theory all we need for polynomials are addition and multiplication. All powers of things have to be natural numbers in a polynomial, so you can express exponentiation as repeated multiplication.
So we need to be able to add things. Add like terms. Cancel things. The usual basics. I will leave it as an exercise to you to think very closely of what might be necessary for polynomials, but we came up with the following definition for a thing called a "Ring".
A ring is a set R equipped with a pair of closed operations + and •, and with constants 0 and 1.
As a shorthand I will write ab to mean a•b.
(R, +, •,0,1)
Basic canceling properties that are nice:
∀x x+0=x
∀x∃(-x) x+-x = 0
Properties so that the order of addition does not matter:
∀x∀y x+y=y+x
∀x∀y∀z (x+y)+z = x+(y+z)
Basic multiplicative properties:
∀x 1x=x
∀x x1=x
∀x∀y∀z (xy)z=x(yz)
And distributive properties:
∀x∀y∀z x(y+z)=xy+xz
∀x∀y∀z (x+y)z=xz+yz
Notably, the multiplication is not necessarily commutative, which is why we have to do it on both sides each time. When multiplication is commutative, we have a commutative ring. We don't specify that rings must be commutative because there are a lot of useful things that fit the properties we want of a ring, except commutativity, but we still want polynomials of them: For example: rotations in 3d space, quaternions, and matrices.
We can also add and multiply polynomials themselves, and you can check that the set of all polynomials of a ring are also a ring! This also could lead to a conversation about what an Adjoint is.
Modules
Skipping over a lot of things in rings, now that we defined rings, we can consider modules, which are basically just "things a ring can act on". Kind of like actions for a group. This isn't super in depth, but the idea is that they have some properties:
A left R-module M is an abelian (abelian means commutative in this case) group M and a ring R with an action between (taking an an element from R and one from M to an element form M) them that has these properties:
For any r, s in R, and x,y in M:
r(x+y)=rx+ry
(r+s)x=rx+sx
(rs)∘x=r∘(s∘x)
1∘x=x
An example of this are the Real Numbers on vectors, taking a∘(v1,v2,v3) to (av1,av2,av3), or the Integers on the Imaginary Integers taking a∘bi to abi.
These seem awfully like vectors. And that's mostly because they are! Modules were defined based on a generalisation of the idea of vectors, but to an arbitrary ring. So vector spaces are all modules. But there are weirder modules that you can also study.
The Atrocity
Another example would be quaternions acting on a list of quaternions in the same way as the vector example. It would be, for all intents and purposes, a vector.
But the definition of a vector space isn't a module on a ring. It is a module on a Field.
What's a field? It's a commutative ring you can do division on. Like the Real Numbers or the Rationals. Not the Integers. This idea turns out to 'kind of' be the same as the ordered pairs idea, as long as you have the Axiom of Choice. It is the generalisation of that to include things with infinite dimensions and other weird cases. Just the bare patterns required for a vector space.
Quaternions are not commutative in multiplication. pq≠qp. And that's something we have put into the definition of a vector space. So a module over quaternions doesn't count as a vector space!
Which is sad because quaternions were what vector spaces originally came from!
May I ask why? As a part of the abstract algebra course I need to explain them somehow to undergraduate students, and I can't think of an elementary reason why they are beautiful (I don't want to brush them off as yet another curiosity). I mean, hyperkähler geometry is great and all, but I cannot go into that, and I definitely don't want to digress on non-pure-math applications which I know nothing about. I also don't have topological tools to explain the classification of division R-algebras... And same for octonions, of course.
Quaternions in my experience are so beautiful because they (and other quaternion division algebras or Laurent series’ built off them) are the only intuitive noncommutative division algebras which can be easily communicated without going on for 10 minutes defining things.
On top of that, looking at modules over the quaternions, you can define similar ideas to inner product and normed vector spaces which cannot be done for any other modules over noncommutative division rings! It’s a very interesting set of values
Anya is live and ready to show you everything. Watch her strip, dance, and perform exclusive shows just for you. Interact in real-time and make your fantasies come true.
✓ Live Streaming✓ Interactive Chat✓ Private Shows✓ HD Quality
Anya is LIVE right now
FREE
Free to watch • No registration required • HD streaming
It's so so important to think about how to think about math while keeping your relationship with it healthy and I love that the math textbook is acknowledging it
Girls we need to see more love for associative division algebras and division rings, they don’t get the love they need, we aren’t even taught that these are the same thing viewed from two different angles :,>
Right, considering the current state of corporate politics on this site, and that it seems that only those affected seem to be actively speaking on the matter, it is up to I, the only fucking cishet on tumblr, to drag this out to a wider audience.
REBLOG IF YOUR ACCOUNT IS A TRANSFEM SAFE SPACE.
We need to show these higher ups how much we truly value them.
is your blog a safe space for freaks, deviants, cringe failgirls, transfems you find personally annoying, transfems who are Loud and Opinionated, is it safe for all of us ?
