Beautiful job explaining the trouble you have to go through to create the real numbers and not just saying "all the scalars" or "the decimals". The real number are actually quite complex (lol pun), and this is largely overlooked by a lot of professors.
Truly appreciate the hard work and passion behind these videos. They are the complex numbers cardinality worthier than producing a bunch of discouraging low quality content.
i have a degree in astrophysics and absolutely love watching math videos...the only problem i have is understanding how the math presented is used in physics... i guess i have to see 'what is the point' of the topic to understand the process better...i remember taking a statistical mechanics course that showed how the math can be used to represent the flow of electricity as an 'electron gas' that freaking blew me away... fyi, you do a fantastic job of presenting the topic....
Hi Socratica. Many thank fo your videos. I have aquestion that has been with me for years. What is the difference between Complex numbers and 2 dimensional vectors. Could there be a way in which 2x1 vectors behave in the same way than complex numbers? I would love to listen to your answer. I hope you can see this comment
@1:18 I'm not sure why this is phrased as "adding rational numbers to the group" instead of as "adding the division operation to the group." Should I take this as simply being your choice of phrasing and division is implicitly included like subtraction is, or is there more significance to having not included the operator? I also find simply saying that "adding a variable is how you extend the complex numbers" is not very illuminating without defining "variable" in the context of set members. Is this like adding root(2) to the member set without defining it?
What We must realize is that a field not only contains the possibility of any given field of any given type of number, yet, it additionally contains the very notion of possibility itself. What We are speaking of primarily is the very possibility of Field. This Field takes place in an abstract space. However, We must not assume what We mean by an an abstract Space. It has been my encounter with the Study of Mathematics that this "Abstract Space" that "contains" any given field may in fact takes place in the Mind. However, We must further investigate the very notion of Mind, for what We mean by "Mind" is quite a complex undertaking. It has been the Story of Western Philo-Sophical thought that Mind along with Being is either Identical or Different. This issue which disturbs every authentic Mathematician concerns the very distinction between Plato and Aristotle: Are Ideas Real? Nevertheless, What We mean by Field, and it would be quite limited to say, that It is merely a collection of elements of such and such. I understand We are speaking in terms of the System known as Abstract Algebra, which truly gets to the Heart of Math as a Whole: What is the the relation between any given possibility of what We refer to as Numbers? Thank You for Your Love of Math, Kelly Phoenix Rose
At 4:03, I don't think so. What about quaternions? We definitely can extend complex numbers to quaternions and further to even higher dimension numbers.
We have to be very careful about what we mean by "extend" here. Her answer to her question is correct - you cannot extend the complex numbers by topological closure or algebraic closure. The field of complex numbers is Cauchy complete (meaning all Cauchy sequences converge) and algebraically complete (meaning all nonconstant polynomials have a root). Of course, you can extend the complex numbers to other rings and fields, but it's not in the same ways we extended things before - we are not forcing sequences to converge or forcing polynomials to have roots. The quaternions form a ring which has the complex numbers as a subring, but the quaternions are also not a field, so that's something to keep in mind here too - we are mostly concerned with extending to fields. There is no finite-degree field extension of the complex numbers.
¿Eres de Quito Ecuador? I’ve been over 20 times for work. A wonderful country. May I ask, you have a natural born American accent. How did you acquire it.
How does this relate to the surreal numbers? They boast being the "largest" field possible. Are the surreal numbers an extension of the reals when you add omega and epsilon?
Well, by the strict definition of a field, the surreal numbers don't *technically* form a field. But it's only because the collection of surreal numbers is too big to be a set. Rather, the collection of surreal numbers is what's called a "proper class". But one can generalize the notions of algebra to proper classes, and in this generalization, the surreal numbers form a field. The surreal numbers aren't the largest field - they are the largest _ordered_ field. A field F is called an "ordered field" if there exists an ordering ≤ on the elements of F so that the ordering is a total ordering (meaning that given any two elements x and y of F, x ≤ y or y ≤ x must be true) which is compatible with the field operations. The compatibility with the field operations can be stated in the form of the following two axioms: 1. If x ≤ y, then x+z ≤ y+z for all z in F. 2. If 0 ≤ x and 0 ≤ y, then 0 ≤ xy. The rational numbers and real numbers are examples of ordered fields. The complex numbers is an example of a field that is _not_ an ordered field. One of the interesting things about fields is that they can _only_ be related to each other in terms of one being a subfield of the other. In other words, given a ring homomorphism f : F → E between two fields F and E, either f is the zero map (that is, f(x) = 0 for all x in F), or f is one-to-one/injective. In the case that the map is the zero map, we say that F and E have no relation to each other. If f is one-to-one, we can identify F with a subfield of E, since F is isomorphic to the image of f, which is a subfield of E. When we say that the surreal numbers S are the "largest" ordered field, what we mean is that, given any ordered field F, there exists a ring homomorphism f : F → S which is one-to-one and compatible with the ordering (in other words, whenever x ≤ y, f(x) ≤ f(y)). So every ordered field is isomorphic with a subfield of the surreal numbers. So, in some sense, the surreal numbers are an extension of the real numbers. But you would need to add a _lot_ more than just ω and ε to get the surreal numbers. This can be seen just from the fact that the surreal numbers are too big to be a set, but the real numbers form a set. You would need to add a collection of elements that is too big to be a set in order to get a proper class.
It has shown up in a couple of videos this series, and I know I am not the only one bothered by this, I think it is worth noting that saying i is the square root of negative one is an improper use of the number. i is a number with the property i^2=-1. If you say i is the square root of negative one you can end up with very incorrect results, e.g, i x i = sqrt(-1) x sqrt(-1) = sqrt(-1 x -1) = sqrt(1) = 1, which is obviously wrong.
Eh. I know a lot of people get their knickers in a twist over this, but I'm not convinced. The "very incorrect results" you get come from assuming that any "square root function" defined over the complex plane must have all of the same properties as the principal square root function defined over the positive real numbers. And perhaps that's what you want! But you could just as easily define a new "square root function" over the complex plane by choosing a square root for each complex input. (For example, you could choose the one with positive real part or with positive imaginary part if the real part is 0.) Under such a definition, it could be perfectly consistent for i = sqrt(-1), because such a function would no longer be multiplicative. So there are options. It's not inherently _wrong_ to say that i = sqrt(-1). You just have to specify what you mean by the square root.
Yes, the quaternions do form a ring, so the quaternions are a ring extension of the complex numbers. :) I do want to say a little more about the quaternions. The _only_ way in which the quaternions fail to be a field is commutativity of multiplication. Structures that satisfy all of the properties of a field except commutativity of multiplication are called "division rings" (because "division" can be defined in such rings - though you have to be careful whether you're dividing on the left or dividing on the right) or "skew fields". So the quaternions are very close to a field without being one. The quaternions is the most common example of a divison ring. Wedderburn's Little Theorem implies that there are no _finite_ non-commutative division rings, so all non-commutative division rings are infinite.
I still to this day find it so ironic that one can journey up the mathematics ladder to the esteemed level of abstract algebra, then return to the lesser abstract. All the while asking yourself, "Why didn't they just teach me this from the beginning?" By analogy, the same reason that a toothless infant can't devour a steak.
Seems simple and beautiful but i know it wont take long for me to get stuck trying some problems. I would trade a lot (everything except my dogs) for an extra 10-20 IQ points. ;-)
The narration is too too too fast, especially for someone who is trying to understand it for the first time. It would serve well for the purpose of revision, i.e almost useless for comprehension purposes.
I'm "not good" at this form and format of thinking, but "having worked in the field", out in the paddock in all types of weather, I'm aware of the reasons why the Observable Universe is a different experience for people who have different memory association techniques and application strategies. The square root of two exists as an hypotenuse, in the general field of orthogonality positioning between repeat unit vectors, by projected reciprocal. You "know this intuitively" because it's "wired into" your brain, learning by doing, repeating the strategic calculations of derivation. Ie you can either remember the picture association, or the strategic map by which to "hunt it down". My wife, a former Teacher of "nebulous concepts" of society, remembers names and faces, I'm delegated to the "fix this" jobs. So by teaching us about the natural reciprocal field functions of Mathematics, Socratica demonstrates the innate processes of Evolution in the context of Actuality. Thanks to Sean Carroll and Lynne Kelly's podcast, I can refer everyone interested in Education, to her book, THE MEMORY CODE. Also, in a similar qualitative assessment of long-term memory vs short-term, the conflict of nominally male female social requirements for survival in a troupe/tribe of Baboons as revealed by Professor Sapolsky.. the males had the equivalent of an honor code, do or die, and the females had an avoidance and exclusion from society, compensation reaction. Very Educational, and mathematically binary.
I was the 38.963 viewer of this video.. I do like it 774 liker so I do 38963/774 it makes 50,3397933 and Iike a lot as number what I can not understand is the persent of likes.. 0,5 so if in some year this percent become 50,3397933 that means that at least human has a reasonable season above infinity and then we complete a mission
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A real video that makes complex things rational.
halit ince EYYYYY ;)
Legend status is yours..
this inspires me to keep learning for its own sake! thanks :)
Beautiful job explaining the trouble you have to go through to create the real numbers and not just saying "all the scalars" or "the decimals". The real number are actually quite complex (lol pun), and this is largely overlooked by a lot of professors.
please do one on finite (Galois) field
This is Great ! I like your way of explaining this stuff in brief and compact way , Thanks for your effort
Also not to mention Hausdorff and simply connected. Hey we get a trivial fundamental group. Happy day.
Man I’m really having a field day with these excellent videos!
Is your day algebraicly closed?
@@Grassmpl Let’s just say that if I was a surgeon, I’d have to do two operations!
Truly appreciate the hard work and passion behind these videos. They are the complex numbers cardinality worthier than producing a bunch of discouraging low quality content.
The best teacher ever😊
i have a degree in astrophysics and absolutely love watching math videos...the only problem i have is understanding how the math presented is used in physics...
i guess i have to see 'what is the point' of the topic to understand the process better...i remember taking a statistical mechanics course that showed how the math can be used to represent the flow of electricity as an 'electron gas'
that freaking blew me away...
fyi, you do a fantastic job of presenting the topic....
You are amazing, but everything was a bit too fast. Please continue doing such high quality videos!
Another great explanation. Thanks. Keep up the good work.
شكرا جزيلاً لك أيتها المعلمة الفاضلة
Amazing presentation! Thank you.
Nice explanation and background music is very good
Do a video on number fields and their algebraic integers. Include some prime ideal factorizations.
Your speech is just awesome. I also want some classes about remain integral and sequence of functions. Please 🙏🏻🙏🏻🙏🏻🙏🏻🙏🏻
What a good video. Thank you. I prefer you to make videos. I understand better.
Outstanding video lecture.
I just got my BA in math, I failed group theory once and barely made it out alive my second time around, thanks to this channel :)
Grade: 50.000001% congratulations.
This stuff is mindblowing, is this when mathematicians could prove some infinities are greater or smaller than others?
Ok. Blow it up. Resolve your mind singularities. Hopefully they are all nodal. Cusps are a pain.
Great concept
Superb mam....this video is very useful to us ....waiting for next video ...pls teach about extension fields...
You are really best
Hi Socratica. Many thank fo your videos. I have aquestion that has been with me for years. What is the difference between Complex numbers and 2 dimensional vectors. Could there be a way in which 2x1 vectors behave in the same way than complex numbers? I would love to listen to your answer. I hope you can see this comment
thank you madam.......
@1:18 I'm not sure why this is phrased as "adding rational numbers to the group" instead of as "adding the division operation to the group." Should I take this as simply being your choice of phrasing and division is implicitly included like subtraction is, or is there more significance to having not included the operator?
I also find simply saying that "adding a variable is how you extend the complex numbers" is not very illuminating without defining "variable" in the context of set members. Is this like adding root(2) to the member set without defining it?
great video
Mam can you explain in details with solutions of problems
No doubt your vedioes are best
Oh thank fuck, I finally understand what makes a field a field. I was getting really worried. Thanks. Big help.
The set of all insects, denoted by double-stroke B (for Beetles), is a STRAWBERRY FIELD. 🍓
Please create videos on other topics like trigonometry, linear algebra, calculus etc
Trigonometry is trivial
What We must realize is that a field not only contains the possibility of any given field of any given type of number, yet, it additionally contains the very notion of possibility itself. What We are speaking of primarily is the very possibility of Field. This Field takes place in an abstract space. However, We must not assume what We mean by an an abstract Space. It has been my encounter with the Study of Mathematics that this "Abstract Space" that "contains" any given field may in fact takes place in the Mind. However, We must further investigate the very notion of Mind, for what We mean by "Mind" is quite a complex undertaking. It has been the Story of Western Philo-Sophical thought that Mind along with Being is either Identical or Different. This issue which disturbs every authentic Mathematician concerns the very distinction between Plato and Aristotle: Are Ideas Real? Nevertheless, What We mean by Field, and it would be quite limited to say, that It is merely a collection of elements of such and such. I understand We are speaking in terms of the System known as Abstract Algebra, which truly gets to the Heart of Math as a Whole: What is the the relation between any given possibility of what We refer to as Numbers?
Thank You for Your Love of Math,
Kelly Phoenix Rose
Outstanding.
Thanks
Would you mind doing videos in euclidean rings maximal ideals?? It would be more helpful 👍
Thank you so much.
At 4:03, I don't think so. What about quaternions? We definitely can extend complex numbers to quaternions and further to even higher dimension numbers.
We have to be very careful about what we mean by "extend" here. Her answer to her question is correct - you cannot extend the complex numbers by topological closure or algebraic closure. The field of complex numbers is Cauchy complete (meaning all Cauchy sequences converge) and algebraically complete (meaning all nonconstant polynomials have a root).
Of course, you can extend the complex numbers to other rings and fields, but it's not in the same ways we extended things before - we are not forcing sequences to converge or forcing polynomials to have roots.
The quaternions form a ring which has the complex numbers as a subring, but the quaternions are also not a field, so that's something to keep in mind here too - we are mostly concerned with extending to fields. There is no finite-degree field extension of the complex numbers.
can you please make videos on lie algebra and representation theory ; thank you
Hi Thanks for these videos.
Is it possible to make videos on Functions, Graphs, Recurrences?
Please upload some videos of young mathematician biography
¿Eres de Quito Ecuador? I’ve been over 20 times for work. A wonderful country. May I ask, you have a natural born American accent. How did you acquire it.
How does this relate to the surreal numbers? They boast being the "largest" field possible. Are the surreal numbers an extension of the reals when you add omega and epsilon?
Well, by the strict definition of a field, the surreal numbers don't *technically* form a field. But it's only because the collection of surreal numbers is too big to be a set. Rather, the collection of surreal numbers is what's called a "proper class". But one can generalize the notions of algebra to proper classes, and in this generalization, the surreal numbers form a field.
The surreal numbers aren't the largest field - they are the largest _ordered_ field. A field F is called an "ordered field" if there exists an ordering ≤ on the elements of F so that the ordering is a total ordering (meaning that given any two elements x and y of F, x ≤ y or y ≤ x must be true) which is compatible with the field operations. The compatibility with the field operations can be stated in the form of the following two axioms:
1. If x ≤ y, then x+z ≤ y+z for all z in F.
2. If 0 ≤ x and 0 ≤ y, then 0 ≤ xy.
The rational numbers and real numbers are examples of ordered fields. The complex numbers is an example of a field that is _not_ an ordered field.
One of the interesting things about fields is that they can _only_ be related to each other in terms of one being a subfield of the other. In other words, given a ring homomorphism f : F → E between two fields F and E, either f is the zero map (that is, f(x) = 0 for all x in F), or f is one-to-one/injective. In the case that the map is the zero map, we say that F and E have no relation to each other. If f is one-to-one, we can identify F with a subfield of E, since F is isomorphic to the image of f, which is a subfield of E.
When we say that the surreal numbers S are the "largest" ordered field, what we mean is that, given any ordered field F, there exists a ring homomorphism f : F → S which is one-to-one and compatible with the ordering (in other words, whenever x ≤ y, f(x) ≤ f(y)). So every ordered field is isomorphic with a subfield of the surreal numbers.
So, in some sense, the surreal numbers are an extension of the real numbers. But you would need to add a _lot_ more than just ω and ε to get the surreal numbers. This can be seen just from the fact that the surreal numbers are too big to be a set, but the real numbers form a set. You would need to add a collection of elements that is too big to be a set in order to get a proper class.
how to find determinant of 6 x 6 matrix ( taking 3 columns at time ) ?
it was hepful thnx
It has shown up in a couple of videos this series, and I know I am not the only one bothered by this, I think it is worth noting that saying i is the square root of negative one is an improper use of the number. i is a number with the property i^2=-1. If you say i is the square root of negative one you can end up with very incorrect results, e.g, i x i = sqrt(-1) x sqrt(-1) = sqrt(-1 x -1) = sqrt(1) = 1, which is obviously wrong.
Eh. I know a lot of people get their knickers in a twist over this, but I'm not convinced. The "very incorrect results" you get come from assuming that any "square root function" defined over the complex plane must have all of the same properties as the principal square root function defined over the positive real numbers. And perhaps that's what you want!
But you could just as easily define a new "square root function" over the complex plane by choosing a square root for each complex input. (For example, you could choose the one with positive real part or with positive imaginary part if the real part is 0.) Under such a definition, it could be perfectly consistent for i = sqrt(-1), because such a function would no longer be multiplicative.
So there are options. It's not inherently _wrong_ to say that i = sqrt(-1). You just have to specify what you mean by the square root.
2:31
Every element in ℚ(√2)can be witten as : 𝛼+𝛽√2 because :
• 𝛼=a/b | a,b ∈ ℤ and 𝛽=c/d | c,d ∈ℤ ; c=0⇒ (𝛼+𝛽√2) ∈ ℚ
• 𝛼=a/b | a,b ∈ ℤ and 𝛽=c/d | c,d ∈ℤ* ⇒ (𝛼+𝛽√2) ∈ ℝ
awesome
When you talk about finite fields are you going to include the transcendental extensions, ie infinite fields of non-zero characteristic.
Yeah that's exactly when separability and perfection may fail.
what if we add the other quaternions to the Complex numbers?, that would be an extension, right?
Not exactly. The issue is that the quaternions are not technically a field - multiplication isn't commutative in the quaternions.
then it would be a ring, right?
Yes, the quaternions do form a ring, so the quaternions are a ring extension of the complex numbers. :)
I do want to say a little more about the quaternions. The _only_ way in which the quaternions fail to be a field is commutativity of multiplication. Structures that satisfy all of the properties of a field except commutativity of multiplication are called "division rings" (because "division" can be defined in such rings - though you have to be careful whether you're dividing on the left or dividing on the right) or "skew fields". So the quaternions are very close to a field without being one.
The quaternions is the most common example of a divison ring. Wedderburn's Little Theorem implies that there are no _finite_ non-commutative division rings, so all non-commutative division rings are infinite.
What about the set of irrational numbers? Are they a field, a ring or a group?
No. Two irrational numbers can be added to form a rational number, thus they do not form a group.
@@thomaspeck4537 Makes sense. Thank you!
Here's a trivial explanation. Zero is not in this set, so it can't even begin to form an additive group.
great
Great
What is the differences between finite fields and infinite fields?
An infinite field may have many extensions of any degree, but a finite field has only one.
whether complex number is field ?
I still to this day find it so ironic that one can journey up the mathematics ladder to the esteemed level of abstract algebra, then return to the lesser abstract. All the while asking yourself, "Why didn't they just teach me this from the beginning?" By analogy, the same reason that a toothless infant can't devour a steak.
is finite field video already available?
Find every irreducible polynomial over every prime order field. Now you can construct every finite field.
Seems simple and beautiful but i know it wont take long for me to get stuck trying some problems. I would trade a lot (everything except my dogs) for an extra 10-20 IQ points. ;-)
Set of positive integers is a field?
Taqi Ahmed Nope. Missing the additive inverses.
And multiplicative inverses.
The integers don't contain any subring that's a field.
The narration is too too too fast, especially for someone who is trying to understand it for the first time. It would serve well for the purpose of revision, i.e almost useless for comprehension purposes.
I'm "not good" at this form and format of thinking, but "having worked in the field", out in the paddock in all types of weather, I'm aware of the reasons why the Observable Universe is a different experience for people who have different memory association techniques and application strategies.
The square root of two exists as an hypotenuse, in the general field of orthogonality positioning between repeat unit vectors, by projected reciprocal. You "know this intuitively" because it's "wired into" your brain, learning by doing, repeating the strategic calculations of derivation. Ie you can either remember the picture association, or the strategic map by which to "hunt it down".
My wife, a former Teacher of "nebulous concepts" of society, remembers names and faces, I'm delegated to the "fix this" jobs.
So by teaching us about the natural reciprocal field functions of Mathematics, Socratica demonstrates the innate processes of Evolution in the context of Actuality.
Thanks to Sean Carroll and Lynne Kelly's podcast, I can refer everyone interested in Education, to her book, THE MEMORY CODE.
Also, in a similar qualitative assessment of long-term memory vs short-term, the conflict of nominally male female social requirements for survival in a troupe/tribe of Baboons as revealed by Professor Sapolsky.. the males had the equivalent of an honor code, do or die, and the females had an avoidance and exclusion from society, compensation reaction. Very Educational, and mathematically binary.
I was the 38.963 viewer of this video.. I do like it 774 liker so I do 38963/774 it makes 50,3397933 and Iike a lot as number what I can not understand is the persent of likes.. 0,5 so if in some year this percent become 50,3397933 that means that at least human has a reasonable season above infinity and then we complete a mission
Nice - Strawberry fields forever
What about quaternions?
DarthQuantum Thanks for your reply! :D
Quaternions are not a field, because they are not commutative with multiplication.
For example i * j = k.
But j * i = - k.
They are only division rings
Hmmm… maybe that’s why quantum mechanics has solutions in the complex field… because it seems to be the most complete field there is?… 🤔
Mashallah
R is a Field and then the real analysis starts
The definition of reals presented here seems incomplete enough as to be unhelpful.
First!