Thank you for the shout out! Totally made my day! I am looking forward to working with geometry and chromogeometry over a finite field. Being able to know a quantifiable number of interacting shapes over this space/matrix/maxel(yeah, still gotta work on being more precise about terminology) and how they appear to the human mind (i.e. there should be a naturally generating order) would definitely provide the education system what is needs to begin specifying terminology when working with mathematical manipulatives, i.e. "cubes", to make learning mathematics engaging and meaningful in the context of the learners concrete situation. Hopefully... the next time I get a shout out I will take have taken "my understanding" and shown how "cubing" can be used to: 1. Engage 2. Foster 3. Validate mathematical reasoning the same way a local running race engages, fosters, and validates runners. Until then, thanks again for making my day!
Video essentially details the construction on a math model that describes complex numbers, and it does so in clear riggerous detail. For me, any individual complex number is better thought of as it's own unique math model, not 'something' called a number.
isn't that the crux of it. that many objects are being classified under the "real number" umbrella that ought to be constructed and studied as their own algebraic structures. "real numbers" even admit nonsensical objects such as non-computable "numbers", which appears to be the result of the lazy definition of them (a definition that actually prevents any constructions of its elements since they're infinitely long to identify in the general case).
It is puzzling why the notion of an additive inverse [a-bar], defined as the [unique] element for which a + a-bar = 0, was avoided. It would seem that a-bar is more fundamental than a^(-1) since it always exists for any a, and 1 + 1-bar = 0 is just a special case of this general property. It is shown below that both 0 x a = 0 and [sequentially] 1-bar x a = a-bar can be proven as a consequence of this general property: 1. 0 + 0 = 0 2. (0 + 0) x a = 0 x a 3. 0 x a + 0 x a = 0 x a 4. [0 x a + 0 x a] + (0 x a)-bar = 0 x a + (0 x a)-bar 5. 0 x a + [0 x a + (0 x a)-bar] = 0 ; based upon additive inverse 6. 0 x a + 0 = 0 ; based upon additive inverse 7. 0 x a = 0 ; which is the zero-product property Thus, as a resulting property: 1-bar x a = 1-bar x a + 0 = 1-bar x a + [a + a-bar] ; based upon additive inverse = [1-bar x a + a] + a-bar = [1-bar x a + 1 x a] + a-bar = [1-bar + 1] x a + a-bar = 0 x a + a-bar ; based upon additive inverse = 0 + a-bar ; based upon zero-product property = a-bar
Video Content 00:00 Introduction 03:46 Classical complex numbers 07:21 A field is an "arena for doing arithmetic" 10:46 A finite field 17:33 Properties of a field 22:25 The new path towards complex numbers
Norman you are a genius. No matter what anyone says, you have proven over and over again the power of restoring creativity to math to solve problems in a finite and exact and I would argue a superior way. Above all things, you are a great teacher and your series has really inspired me that sanity can be restored to math so it stops mixing math and linguistics. THANK YOU!
This was very interesting. But isn't your approach to the (usual) complex numbers equivalent to the (Hamilton's?) usual description as pairs of real numbers with co-ordinate -wise addition and multiplication defined by (a, b) (c, d) =(ab-cd, ad+bc)? Also, you are implicitly assuming that the fields are not of characteristic 2, else your three algebras wouldn't be all two-dimensional.
Prof. Wildberger, I have a question. I respect your desire to prove everything algebraically and using only finite processes, and the potential this could have in math. However, in the past we have often "stumbled" upon results before we could prove them rigorously, or perhaps we found the shortcut first. For instance, it was well after the Prime Number Theorem was first "discovered" and "proven" using complex analysis that an elementary proof was found. So my question is this: do you think that (real) analysis is still useful in the sense that it could be a shortcut to finding results about the rational/algebraic numbers, etc.? It seems that even if the real numbers are logically incoherent, we produce results using them which lead us to the rational analog (i.e., your restated and as of yet unproven analog of the FTA). Is it wrong to still pursue real analysis, if only for the practicality of this correspondence? -Ben T.
@Ben Thayer, That is a great question! For sure we should not dismiss many of the remarkable insights and results of modern analysis --- but we must honestly recognize that they are mostly not logically correct. They are often useful in an applied sense, and they encode important understandings and intuitions. But they do not stand independently in pure mathematics --- they need to be examined more deeply, rethought and then recast if possible into a more solid algebraic and computational mathematical reality. I want to emphasize how much work there is to do here! Young analysts should not be seduced into ever more sophisticated pursuits of complexity, when there is so much important and fundamental work to be done in dismantling and reconstructing what has already been done, and creating a solid and powerful analysis for the future.
So much of complex analysis was developed to understand physical processes. Those processes couldn't care less about whether humans could represent it in a finite process. OMHO, the approach in this video is not the best one to understand physical processes.
@@factChecker01 sorry but you're wrong; representing i for example as a matrix/transformation which geometrically can be interpreted as a rotation by a right angle makes far more sense wrt physical interpretations than a number whose square is -1, which is completely nonsensical
@@TheRosyCodex , The consequences of complex numbers and complex analysis are deep and profound. But your argument is not with me. Your argument is with geniuses like Gauss, Cauchy, etc., and with every person who has studied physics, math, or engineering in any depth. A good n-dimensional extension of the concepts can be studied in geometric algebra and geometric calculus. PS. Complex numbers give a much more convenient representation of the rotations in RxR than matrices do. Many mathematicians consider the equation e^(2*pi*i) = -1 to be the most important equation in mathematics.
@@factChecker01 choosing a particular interpretation of complex numbers doesn't negate their usefulness (I.e. it doesn't ignore or throw away the work of pioneers who used a more symbolic or abstract algebraic approach to expressing the same ideas and patterns). given the deep connections between mathematics and computation (e.g. type theoretic foundations, typed lambda calc and logic correspondence, etc) it would seem there exist computable interpretations of most anything we invent purely formally.
This is good stuff. It would be interesting to compare this approach with Von Staudt's introduction of complex lines and points into projective geometry.
I think the best way to arrive at complex arithmetic is via geometric algebra - complex number theory is the even-order sub-algebra of two-dimensional geometric algebra. Specifically, it's the even order sub-algebra of Cl(2,0). Starting with only scalars and bivectors, you are "trapped" with those - you can't produce vectors arithmetically. So it's a closed arithmetic system. And it's easy to show that a) there is just one unit bivector and b) that it's square is -1. So, the unit bivector represents i, and the rest just comes falling out.
Fun. Just trying to reproduce these bizarre addition and multiplication table for F7: It's basically doing computing addresses: In Octave (free version of Matlab). It uses addresses counting from 1 like good (bad) old Fortran: Define: F7 = -3:1:3 ; gives [-3 -2 -1 0 1 2 3]. To create the addition table: Initialize A=zeros(7,7); M=zeros(7,7); F7 = [-3 -2 -1 0 1 2 3]; for k=1:7, for j=1:7, S = F7(k) + F7(j); P = F7(k) * F7(j); A(k,j) = F7(mod(S+10, 7)+1); % addressing in F7 M(k,j) = F7(mod(P+10,7)+1); % endfor; endfor;
Matrix 1 is the identity and encodes dilation. Matrix j is a swap between horizontal and vertical Matrix k is the horizontal inversion (x -> -x) Matrix i is inversion and swap and is direct 90° rotation.
I would like to thank you for your contribution in making things smoother & easier to understand by providing a meaning for matrices of transformations used Obama much thankful & indebt to you , sir
Do you have any ideas on the foundations of probability without the real numbers or measure theory? In particular, how can we model probability distributions like the 'normal distribution'?
@Element118 Probability theory needs a careful re-examination. Its current set and measure theoretic foundations are highly suspect. But that is a big task. Answering about how to deal with the "normal distribution" is a lot easier; currently a lot of stats programs deal very effectively with this and many other transcendental distributions without any underlying "real number arithmetical computing". How do they do it? They just work with approximations. You just choose how many terms of the Taylor series you are going to work with for any particular application. The point is you do not need to have a quasi-religious belief that there is an "infinite object" out there beyond the universe which you are getting closer and closer to. Rather you stay firmly rooted on the ground, not in the heavens, by restricting attention to computations that can actually be run on our machines. Simple! Yes, but it means we have to be more concerned with numerical analysis issues -- what is the range of validity of our computations at any given point etc.
@@WildEggmathematicscourses So when computing partial finite representations of 'infinite objects' in general, we would need numerical analysis to figure out how much computation we need for the particular 'infinite object'? For interest sake, how will such a proof/demonstration look like?
@@element118_5 It is much better to remove the "infinite object" entirely from the discussion. If we want to consider the exponential, then for any natural number k and any number x we can define exp_k(x)=1+x+x^2/2!+... + x^k/k! Great, now for a particular problem you could use exp_3, or perhaps exp_5 or perhaps exp_100. Is it true that exp_k (x) exp_k (y) =exp_k (x+y) ? No not exactly, so you would need to keep track of the error, ie difference between left and right hand sides of this equation, which might depend not only on k, but also on x and y. More complicated, and more in line with what computer programmers have to deal with. But at no point should you ask: how close are we to the "real exp (x)" ?--- just because there is no such number.
Georg Cantor became insane when he tried to proof the continuum hypothesis. Which is still an unsolved problem. Number one on Hilberts problem list. In physics we can not measure infinity!
Suppose you try to get the mean of a tensor quantity involving a multivariate normal random variable. Since the expressions you get after integration by parts and so on are pretty manageable, they could be used as *the definition* of the expectation operator, which then operates on formal expressions. You proceed to verify all properties you want the expectation operator to have - some of them algebraic in nature, and some of them approximate.
I really enjoyed your wider view of complex numbers, particularly combining the Euclidean with Relativistic geometries, and representing i, j and k in terms of matrices, rather than the imaginary unit sqr(-1), which gives them more "realistic" meaning, albeit it is still fundamentally geometric. In the end, "numbers" are for "counting" which is intrinsically one dimensional because it is fundamentally related with "time", whereas geometry is related to "space". Therefore, we cannot legitimately call these structures "numbers" unless we unlock the relation between space and time, which will reveal the ultimate connection between geometry and algebra. For this we need the Duality of Time Theory, that is based on the Single Monad Model of the Cosmos, which explains how multiplicity is emerging from absolute oneness. I cordially would like to have a look at my work in this regard. In particular, please have a look at: TEMPORAL NUMBERS: Hypercomplex-Time Geometry and Relativistic Mathematics. ... I would also be very grateful if you have any comments or suggestions, because in the end, although I am not a mathematician, I find many of your principal ideas related to what I have discovered about dynamic creation of geometry in the inner levels of time, which will redefine many foundational mathematical concepts, such as discreteness and continuity, as well as the unity and infinity, thus affecting the structure of number sets. You can find more details on my website smonad.com .
Thank you, your notation fixes everything as far as rational square roots. For instance, the square root of two is (3j-i)/2 (or (3j+i)/2 and the negations of these).
Unfortunately (?) there are many more square roots of 2 than those. For example j+k, j-k, k-j and -j-k are also square roots of 2. And i+k is actually a square root of zero! But I assume our host will say you are not meant to mix and match 'geometries' like this.
@@russellsharpe288 I do not see that as unfortunate, but a more complete set of solutions. And why should we not mix and match geometries to create coquaternions?
@@whig01 I don't know what you mean by 'coquaternions'. Unlike for Hamilton's quaternions, it doesn't seem to be Wildberger's intention that one should adjoin all of i, j, k at once, since he specifically says that one has to "choose a geometry", which means: EITHER i, OR j, OR k combine with the unit matrix to generate each of his three analogues of the complex numbers. As I mentioned, if you *do* combine them all, things go badly wrong, and not just in the way that the quaternions are noncommutative. You end up with the entire algebra of 2x2 matrices over the underlying field (This is obvious if you consider 1+k, 1-k, j+i, j-i). Not only do all field elements (ie the isomorphic copy of the field generated by the unit matrix 1) now have an *infinite* (I suppose Wildberger would want to say 'unbounded') number of 'square roots', but there are also an *infinite* (unbounded) number of nontrivial divisors of zero too. This means they are not even a division algebra: that is, there are nonzero elements which have no multiplicative inverse, so division cannot be defined. Nor can a norm. (But as I say, this is not a problem for Wildberger, since he seems not to have had any intention of combining them in this way).
@@whig01 Thanks! I see there is a wikipedia article about them under the name "Split-quaternion" which confirms the (real 2x2) matrix representation. It also suggests the determinant can be taken as a norm; this puzzled me since I had assumed that a norm of a nonzero element could not be zero, but the definition (of norm) is apparently weaker than I thought. (Obviously many nonzero matrices have a determinant of zero) However a *metric* certainly cannot allow the distance between distinct elements to be zero, so this determinant norm cannot be converted into a metric in the standard way. That leaves me wondering whether you meant to say "isomorphic" rather than "isometric" here. If you did mean "isometric", which metric on the matrices do you have in mind?
I'm only just now finding my way into this channel, but it's definitely interesting. There are things in this video that are reminiscent both of quaternions and of the Dirac equation. I know that the Dirac equation itself was derived with some knowledge of this kind of theory, thereby producing something like the square root of a 4-dimensional Laplacian operator under a Minkowski metric. I wonder if the series is going to get into this kind of stuff? But yeah, the idea of a multiplicative inverse on a finite field was pretty cool. As was the idea of going more abstract so you can define different kinds of complex numbers for all sorts of fields, no matter how abstract.
1,0,T are not distinct in the binary case, because T=1. what's intersting is that putting the symbols this way you can still make a number system, but the number of digits is different. Normal base 3 goes, 1,2,10,11,12,100,101 ... etc, but with 1,0,T as the only symbols it goes, 1,1T,10,11,1TT,1T0,1T1,... and so on. numbers of the form 3^n are still 10... (of the form one follwed by zeros) but unlike the standard base three system they are not the first number with that many digits, but the middle number with that many digits. So 9 = 100, but there are 4 three digit numbers before and 4 three digit numbers after. Unlike normal base 3, there is no carry associated with multiplication. There is a carry of 1 for addition 1+1 and there is carry of T for T+T. Also if you know a number is positive integer, then the first digit will always be a 1, so is essentially redundant.
wow ,........I'm just 3 minutes into this lecture and i am hooked. i thought i was the only one that saw something odd about complex numbers and questioned it.
Dear Mr. Wlidberger, thank you for your excellent video. At the moment I am reading the book Matrix Gateway to Geometric Algebra, Spacetime and Spinors by Garret Sobczyk which is very close to the topic of you lesson. I am personaly interessted in the relation between geometric algebra and matrices. Some sketches of this topic can be found on my Geometric Algebra and Matrices page on Wacker Art. Your lesson is a great inspiration! Hermann
One correspondence to his 2 by 2 matrices could be (in a 4 dimensional geometric algebra): 1 as the scalar 1, i as e123, j as e4, and k as e1234 (the 4 dimensional pseudoscalar).
@Gennady Arshad Notowidigdo There is another book of Sobczyk called "New Foundations in Mathematics: The Geometric Concept of Number" which I don't own. The table of contens and the first pages are online there he is taking about modular numbers. By the way the first group Norman Wildberger introduces in Wild Linear Algebra is isomorphic to the geometric algebra G2. This algebra contains the same matrices as Normans chromogeometry. (see my geometric matrices page that I have updated yesterday.)
Nice! So, _there are three possible interpretations of the imaginary number!_ The first is that it represents a rotation of 90 degrees in the negative direction in a plane. (Euclidean) The second is that it represents a reflection along the line x = y. (Lorentzian) The third is that it represents a reflection in the y-axis. What _that_ has to do with special relativity is beyond me! Still, I have a critique. Why not choosing the first as a rotation in the positive direction in a plane? _Don't be more revolutionary than necessary!_
@@hermannwacker1902 Ye, I know. I have studied several books of Geometric Algebra. A very nice introduction is the following book of John Vince. (Illegal? Download) www.pdfdrive.com/geometric-algebra-an-algebraic-system-for-computer-games-and-animation-e32878980.html I must warn you, though, that the first edition contains a huge mistake in one of its derivations. In my comment on Amazon of this book you can find where this mistake is, and an alternative derivation that does not contain the mistake. I think my derivation is a beautiful demonstration of the power of Geometric Algebra. If you want to download via the link above, and your anti-virus program complains, switch it temporarily off. The download is safe.
@@konradswart4069 i have the book Geometric Algebra for Computer Graphics from John Vince in my bookshelf. But had started with geometric algebra by the thesis of Alan Miller and a discussion on geometric algebra on the old fractal forum. What Norman is doing in this lesson is close to the book of Garret Sobczyk on Geometric Algebra and Matrices. You can also visit my Geometric Algebra page on Wacker Art.
You are right that the properties are not independent, but Prof. Wilderberger does not claim that. Some may perhaps argue that 0xa=0 is a more fundamental property than the ones you would use to deduce 0xa=0.
After thinking about it I do not see how 0xa=0 is deduced from the other properties. If 1xa=a and 0xa=0 and 1+T=0 then the distributive law implies that a+Txa=1xa+Txa=(1+T)xa=0xa=0 implying that Txa is the additive inverse of a. Distributivity implies that 0xa=(0+0)xa=0xa+0xa so if 0xa has an additive inverse then 0xa=0, but I do not se how should be deduced without additive inverses. In a sense the existence of additive inverses is a more fundamental property than 1+T=0 in the sense that this is an important property in that it make (F,+) into a group even if F has no multiplication.
Allow me to suggest the word 'anti' instead of 'minus'. Antarctica is not called minusarctica. The antiproton is not called a minusproton. Let -1 be denoted '-' . Subtraction should be written a+-b or a+b- . Say: 'a plus anti b' or 'a plus b anti' . a-b is the product a(-)b = - a b = a b - . Say: 'a anti b equals anti a b equals a b anti'. 1+- = -+1 = 0 . One plus anti is equal to anti plus one, which is equal to zero. -- = 1 . Anti anti is one. -1 = 1- = - . Anti one is equal to one anti, which is equal to anti. -+- = 2- = -2 . Anti plus anti is two anti, or anti two. Complex numbers may be: 5 + i3 = 5 + 3i , - 5 + 3i- = -5 + 3-i = -5 + -3i . (Your letter t is indistiguishable from a plus sign , and x is indistiguishable from a multiplication sign, and a bar in the second row of a matrix is indistinguishable from an underscore in the first row). Thank you! Bo.
Normally I enjoy watching your content despite my heavy disagreements (you can offer additional insights or ways of thinking that can be useful or intuitive). However, this just overcomplicates complex numbers in a way that doesn't seem very helpful or intuitive, compared to the usual approach that Cauchy introduced. Being frank this is just unapproachable. Just speaking from the perspective of a physicist, the real number approach to complex fields (I took courses in complex analysis and complex variables years back). When doing field theory in physics the more conventional approach to complex fields is far more elegant and simpler to use. This is especially true when working in Quantum Fields and Electromagnetic Fields. The traditional approach just simplifies things a lot more than this modular approach. Additionally, it allows us to use the Fourier and Laplace transforms in ways that give us power over some complicated geometry that one might find in a crystal. This isn't to condemn the modular approach to this arithmetic because it does have its uses (miller indices and Pauli spin matrices are good examples), but This can't replace the Approach proposed by Cauchy. It isn't approachable or understandable and it certainly isn't as usable for practical purposes either.
@CrownedFalcon I am trying to develop a good pure approach to complex numbers, that extends the understanding also to relativistic analogs, as well as others also to come. I am sure if you watch the remaining 3 videos in this little series you may more clearly see the advantages. I am not proposing however that every physicist you uses complex numbers has to frame everything in terms of Dihedrons. By all means, work with a + bi etc, but now you have a better answer for what this actually is, and why it naturally connects with two dimensional geometry and motion. That is an understanding that is quite useful to a physicists I believe.
@@njwildberger Thanks for your reply. I will watch the other videos. Please forgive me if I approach them with a great deal of skepticism though. I just don't see the utility in this approach to complex numbers yet.
@@CrownedFalcon00 This stuff connects with Pauli matrices, the Dirac equation, quaternions, and lots of other super useful stuff. It's all related, and seeing the abstract connections among them can give you insights you wouldn't otherwise have. It's like introducing a metric tensor into special relativity, and distinguishing covariant and contravariant tensors when you're only using rectilinear orthogonal coordinates. It's not obvious why you'd make it more complicated by introducing this stuff. But when you get into differential geometry and general relativity, all the insight you built up by working with the seemingly-needlessly-complicated formal abstract theory suddenly pays off.
![Imaginary Numbers Are Real [Part 1: Introduction]](/img/n.gif)
Thank you for the shout out! Totally made my day!
I am looking forward to working with geometry and chromogeometry over a finite field. Being able to know a quantifiable number of interacting shapes over this space/matrix/maxel(yeah, still gotta work on being more precise about terminology) and how they appear to the human mind (i.e. there should be a naturally generating order) would definitely provide the education system what is needs to begin specifying terminology when working with mathematical manipulatives, i.e. "cubes", to make learning mathematics engaging and meaningful in the context of the learners concrete situation.
Hopefully... the next time I get a shout out I will take have taken "my understanding" and shown how "cubing" can be used to:
1. Engage
2. Foster
3. Validate
mathematical reasoning the same way a local running race engages, fosters, and validates runners.
Until then, thanks again for making my day!
Video essentially details the construction on a math model that describes complex numbers, and it does so in clear riggerous detail. For me, any individual complex number is better thought of as it's own unique math model, not 'something' called a number.
isn't that the crux of it. that many objects are being classified under the "real number" umbrella that ought to be constructed and studied as their own algebraic structures. "real numbers" even admit nonsensical objects such as non-computable "numbers", which appears to be the result of the lazy definition of them (a definition that actually prevents any constructions of its elements since they're infinitely long to identify in the general case).
It is puzzling why the notion of an additive inverse [a-bar], defined as the [unique] element for which a + a-bar = 0, was avoided. It would seem that a-bar is more fundamental than a^(-1) since it always exists for any a, and 1 + 1-bar = 0 is just a special case of this general property. It is shown below that both 0 x a = 0 and [sequentially] 1-bar x a = a-bar can be proven as a consequence of this general property:
1. 0 + 0 = 0
2. (0 + 0) x a = 0 x a
3. 0 x a + 0 x a = 0 x a
4. [0 x a + 0 x a] + (0 x a)-bar = 0 x a + (0 x a)-bar
5. 0 x a + [0 x a + (0 x a)-bar] = 0 ; based upon additive inverse
6. 0 x a + 0 = 0 ; based upon additive inverse
7. 0 x a = 0 ; which is the zero-product property
Thus, as a resulting property:
1-bar x a
= 1-bar x a + 0
= 1-bar x a + [a + a-bar] ; based upon additive inverse
= [1-bar x a + a] + a-bar
= [1-bar x a + 1 x a] + a-bar
= [1-bar + 1] x a + a-bar
= 0 x a + a-bar ; based upon additive inverse
= 0 + a-bar ; based upon zero-product property
= a-bar
I'll be your student till the end
I have just realized how much wisdom is contained in the comments in this channel, nevermind the videos...
mind blown.
Video Content
00:00 Introduction
03:46 Classical complex numbers
07:21 A field is an "arena for doing arithmetic"
10:46 A finite field
17:33 Properties of a field
22:25 The new path towards complex numbers
Norman you are a genius. No matter what anyone says, you have proven over and over again the power of restoring creativity to math to solve problems in a finite and exact and I would argue a superior way. Above all things, you are a great teacher and your series has really inspired me that sanity can be restored to math so it stops mixing math and linguistics. THANK YOU!
This was very interesting. But isn't your approach to the (usual) complex numbers equivalent to the (Hamilton's?) usual description as pairs of real numbers with co-ordinate -wise addition and multiplication defined by (a, b) (c, d) =(ab-cd, ad+bc)? Also, you are implicitly assuming that the fields are not of characteristic 2, else your three algebras wouldn't be all two-dimensional.
Prof. Wildberger,
I have a question. I respect your desire to prove everything algebraically and using only finite processes, and the potential this could have in math. However, in the past we have often "stumbled" upon results before we could prove them rigorously, or perhaps we found the shortcut first. For instance, it was well after the Prime Number Theorem was first "discovered" and "proven" using complex analysis that an elementary proof was found. So my question is this: do you think that (real) analysis is still useful in the sense that it could be a shortcut to finding results about the rational/algebraic numbers, etc.? It seems that even if the real numbers are logically incoherent, we produce results using them which lead us to the rational analog (i.e., your restated and as of yet unproven analog of the FTA). Is it wrong to still pursue real analysis, if only for the practicality of this correspondence?
-Ben T.
@Ben Thayer, That is a great question! For sure we should not dismiss many of the remarkable insights and results of modern analysis --- but we must honestly recognize that they are mostly not logically correct. They are often useful in an applied sense, and they encode important understandings and intuitions. But they do not stand independently in pure mathematics --- they need to be examined more deeply, rethought and then recast if possible into a more solid algebraic and computational mathematical reality.
I want to emphasize how much work there is to do here! Young analysts should not be seduced into ever more sophisticated pursuits of complexity, when there is so much important and fundamental work to be done in dismantling and reconstructing what has already been done, and creating a solid and powerful analysis for the future.
So much of complex analysis was developed to understand physical processes. Those processes couldn't care less about whether humans could represent it in a finite process. OMHO, the approach in this video is not the best one to understand physical processes.
@@factChecker01 sorry but you're wrong; representing i for example as a matrix/transformation which geometrically can be interpreted as a rotation by a right angle makes far more sense wrt physical interpretations than a number whose square is -1, which is completely nonsensical
@@TheRosyCodex , The consequences of complex numbers and complex analysis are deep and profound. But your argument is not with me. Your argument is with geniuses like Gauss, Cauchy, etc., and with every person who has studied physics, math, or engineering in any depth. A good n-dimensional extension of the concepts can be studied in geometric algebra and geometric calculus.
PS. Complex numbers give a much more convenient representation of the rotations in RxR than matrices do. Many mathematicians consider the equation e^(2*pi*i) = -1 to be the most important equation in mathematics.
@@factChecker01 choosing a particular interpretation of complex numbers doesn't negate their usefulness (I.e. it doesn't ignore or throw away the work of pioneers who used a more symbolic or abstract algebraic approach to expressing the same ideas and patterns). given the deep connections between mathematics and computation (e.g. type theoretic foundations, typed lambda calc and logic correspondence, etc) it would seem there exist computable interpretations of most anything we invent purely formally.
So interesting! I hope you'll comment about quaternions, octernions etc., as well as Pauli matrices.
This is good stuff. It would be interesting to compare this approach with Von Staudt's introduction of complex lines and points into projective geometry.
Oh hey it's Richard Southwell! Love your videos
@@accountname1047 Good of you to say :-)
I think the best way to arrive at complex arithmetic is via geometric algebra - complex number theory is the even-order sub-algebra of two-dimensional geometric algebra. Specifically, it's the even order sub-algebra of Cl(2,0). Starting with only scalars and bivectors, you are "trapped" with those - you can't produce vectors arithmetically. So it's a closed arithmetic system. And it's easy to show that a) there is just one unit bivector and b) that it's square is -1. So, the unit bivector represents i, and the rest just comes falling out.
This is a good looking Finite Field from down under!
Fun. Just trying to reproduce these bizarre addition and multiplication table for F7: It's basically doing computing addresses:
In Octave (free version of Matlab). It uses addresses counting from 1 like good (bad) old Fortran:
Define: F7 = -3:1:3 ; gives [-3 -2 -1 0 1 2 3]. To create the addition table: Initialize
A=zeros(7,7);
M=zeros(7,7);
F7 = [-3 -2 -1 0 1 2 3];
for k=1:7,
for j=1:7,
S = F7(k) + F7(j);
P = F7(k) * F7(j);
A(k,j) = F7(mod(S+10, 7)+1); % addressing in F7
M(k,j) = F7(mod(P+10,7)+1); %
endfor;
endfor;
Hm, not totally new to me ;) But I really love the linear algebra way to represent the complex numbers ...
Matrix 1 is the identity and encodes dilation.
Matrix j is a swap between horizontal and vertical
Matrix k is the horizontal inversion (x -> -x)
Matrix i is inversion and swap and is direct 90° rotation.
I would like to thank you for your contribution in making things smoother & easier to understand by providing a meaning for matrices of transformations used
Obama much thankful & indebt to you , sir
I look forward to the next video.
Great Job, Norman.
C A Berg
Do you have any ideas on the foundations of probability without the real numbers or measure theory? In particular, how can we model probability distributions like the 'normal distribution'?
@Element118 Probability theory needs a careful re-examination. Its current set and measure theoretic foundations are highly suspect. But that is a big task. Answering about how to deal with the "normal distribution" is a lot easier; currently a lot of stats programs deal very effectively with this and many other transcendental distributions without any underlying "real number arithmetical computing". How do they do it? They just work with approximations. You just choose how many terms of the Taylor series you are going to work with for any particular application. The point is you do not need to have a quasi-religious belief that there is an "infinite object" out there beyond the universe which you are getting closer and closer to. Rather you stay firmly rooted on the ground, not in the heavens, by restricting attention to computations that can actually be run on our machines. Simple! Yes, but it means we have to be more concerned with numerical analysis issues -- what is the range of validity of our computations at any given point etc.
@@WildEggmathematicscourses So when computing partial finite representations of 'infinite objects' in general, we would need numerical analysis to figure out how much computation we need for the particular 'infinite object'? For interest sake, how will such a proof/demonstration look like?
@@element118_5 It is much better to remove the "infinite object" entirely from the discussion. If we want to consider the exponential, then for any natural number k and any number x we can define
exp_k(x)=1+x+x^2/2!+... + x^k/k!
Great, now for a particular problem you could use exp_3, or perhaps exp_5 or perhaps exp_100. Is it true that exp_k (x) exp_k (y) =exp_k (x+y) ? No not exactly, so you would need to keep track of the error, ie difference between left and right hand sides of this equation, which might depend not only on k, but also on x and y. More complicated, and more in line with what computer programmers have to deal with. But at no point should you ask: how close are we to the "real exp (x)" ?--- just because there is no such number.
Georg Cantor became insane when he tried to proof the continuum hypothesis. Which is still an unsolved problem. Number one on Hilberts problem list. In physics we can not measure infinity!
Suppose you try to get the mean of a tensor quantity involving a multivariate normal random variable. Since the expressions you get after integration by parts and so on are pretty manageable, they could be used as *the definition* of the expectation operator, which then operates on formal expressions. You proceed to verify all properties you want the expectation operator to have - some of them algebraic in nature, and some of them approximate.
I really enjoyed your wider view of complex numbers, particularly combining the Euclidean with Relativistic geometries, and representing i, j and k in terms of matrices, rather than the imaginary unit sqr(-1), which gives them more "realistic" meaning, albeit it is still fundamentally geometric. In the end, "numbers" are for "counting" which is intrinsically one dimensional because it is fundamentally related with "time", whereas geometry is related to "space". Therefore, we cannot legitimately call these structures "numbers" unless we unlock the relation between space and time, which will reveal the ultimate connection between geometry and algebra.
For this we need the Duality of Time Theory, that is based on the Single Monad Model of the Cosmos, which explains how multiplicity is emerging from absolute oneness. I cordially would like to have a look at my work in this regard. In particular, please have a look at: TEMPORAL NUMBERS: Hypercomplex-Time Geometry and Relativistic Mathematics. ... I would also be very grateful if you have any comments or suggestions, because in the end, although I am not a mathematician, I find many of your principal ideas related to what I have discovered about dynamic creation of geometry in the inner levels of time, which will redefine many foundational mathematical concepts, such as discreteness and continuity, as well as the unity and infinity, thus affecting the structure of number sets. You can find more details on my website smonad.com .
Finally I understood what complex number is . Thank you .
Thank you, your notation fixes everything as far as rational square roots.
For instance, the square root of two is (3j-i)/2 (or (3j+i)/2 and the negations of these).
Unfortunately (?) there are many more square roots of 2 than those. For example j+k, j-k, k-j and -j-k are also square roots of 2. And i+k is actually a square root of zero! But I assume our host will say you are not meant to mix and match 'geometries' like this.
@@russellsharpe288 I do not see that as unfortunate, but a more complete set of solutions.
And why should we not mix and match geometries to create coquaternions?
@@whig01 I don't know what you mean by 'coquaternions'. Unlike for Hamilton's quaternions, it doesn't seem to be Wildberger's intention that one should adjoin all of i, j, k at once, since he specifically says that one has to "choose a geometry", which means: EITHER i, OR j, OR k combine with the unit matrix to generate each of his three analogues of the complex numbers.
As I mentioned, if you *do* combine them all, things go badly wrong, and not just in the way that the quaternions are noncommutative. You end up with the entire algebra of 2x2 matrices over the underlying field (This is obvious if you consider 1+k, 1-k, j+i, j-i). Not only do all field elements (ie the isomorphic copy of the field generated by the unit matrix 1) now have an *infinite* (I suppose Wildberger would want to say 'unbounded') number of 'square roots', but there are also an *infinite* (unbounded) number of nontrivial divisors of zero too. This means they are not even a division algebra: that is, there are nonzero elements which have no multiplicative inverse, so division cannot be defined. Nor can a norm. (But as I say, this is not a problem for Wildberger, since he seems not to have had any intention of combining them in this way).
@@russellsharpe288 Coquaternions are isomorphic to 2x2 matrices.
@@whig01 Thanks! I see there is a wikipedia article about them under the name "Split-quaternion" which confirms the (real 2x2) matrix representation. It also suggests the determinant can be taken as a norm; this puzzled me since I had assumed that a norm of a nonzero element could not be zero, but the definition (of norm) is apparently weaker than I thought. (Obviously many nonzero matrices have a determinant of zero) However a *metric* certainly cannot allow the distance between distinct elements to be zero, so this determinant norm cannot be converted into a metric in the standard way. That leaves me wondering whether you meant to say "isomorphic" rather than "isometric" here. If you did mean "isometric", which metric on the matrices do you have in mind?
I'm only just now finding my way into this channel, but it's definitely interesting. There are things in this video that are reminiscent both of quaternions and of the Dirac equation. I know that the Dirac equation itself was derived with some knowledge of this kind of theory, thereby producing something like the square root of a 4-dimensional Laplacian operator under a Minkowski metric. I wonder if the series is going to get into this kind of stuff?
But yeah, the idea of a multiplicative inverse on a finite field was pretty cool. As was the idea of going more abstract so you can define different kinds of complex numbers for all sorts of fields, no matter how abstract.
Excellent wild.
Very compelling. Thank you.
1,0,T are not distinct in the binary case, because T=1. what's intersting is that putting the symbols this way you can still make a number system, but the number of digits is different. Normal base 3 goes, 1,2,10,11,12,100,101 ... etc, but with 1,0,T as the only symbols it goes, 1,1T,10,11,1TT,1T0,1T1,... and so on. numbers of the form 3^n are still 10... (of the form one follwed by zeros) but unlike the standard base three system they are not the first number with that many digits, but the middle number with that many digits. So 9 = 100, but there are 4 three digit numbers before and 4 three digit numbers after. Unlike normal base 3, there is no carry associated with multiplication. There is a carry of 1 for addition 1+1 and there is carry of T for T+T. Also if you know a number is positive integer, then the first digit will always be a 1, so is essentially redundant.
Boolean logic does not form a group, if I'm not mistaken. So it would be to bark up the wrong tree, so to speak.
wow ,........I'm just 3 minutes into this lecture and i am hooked. i thought i was the only one that saw something odd about complex numbers and questioned it.
like it. TNX. AGAIN
i really really love you so much sir . i hate my physics major now. im a pure mathematics student now.
By using Ramanujan's Aproximation for circumference of an ellipse, applied to a circle, one might demonstrate that x/0=0.
Eg 0/ ( 2r)²
Dear Mr. Wlidberger, thank you for your excellent video.
At the moment I am reading the book Matrix Gateway to Geometric Algebra, Spacetime and Spinors by Garret Sobczyk which
is very close to the topic of you lesson. I am personaly interessted in the relation between geometric algebra and matrices.
Some sketches of this topic can be found on my Geometric Algebra and Matrices page on Wacker Art.
Your lesson is a great inspiration!
Hermann
One correspondence to his 2 by 2 matrices could be (in a 4 dimensional geometric algebra): 1 as the scalar 1, i as e123, j as e4, and k as e1234 (the 4 dimensional pseudoscalar).
@@jdp9994 I have some other posibilities on this page: www.wackerart.de/mathematik/geometric_matrices.html
@Gennady Arshad Notowidigdo There is another book of Sobczyk called
"New Foundations in Mathematics: The Geometric Concept of Number" which I don't own. The table of contens and the first pages are online there he is taking about modular numbers. By the way the first group Norman Wildberger introduces in Wild Linear Algebra is isomorphic to the geometric algebra G2. This algebra contains the same matrices as Normans chromogeometry. (see my geometric matrices page that I have updated yesterday.)
thx...
Wonderful. Thank you.
Interesting
Woow 👌
Nice!
So, _there are three possible interpretations of the imaginary number!_
The first is that it represents a rotation of 90 degrees in the negative direction in a plane. (Euclidean)
The second is that it represents a reflection along the line x = y. (Lorentzian)
The third is that it represents a reflection in the y-axis. What _that_ has to do with special relativity is beyond me!
Still, I have a critique.
Why not choosing the first as a rotation in the positive direction in a plane?
_Don't be more revolutionary than necessary!_
True
You can go even further with the help of geometric algebra!
@@hermannwacker1902 Ye, I know.
I have studied several books of Geometric Algebra.
A very nice introduction is the following book of John Vince.
(Illegal? Download)
www.pdfdrive.com/geometric-algebra-an-algebraic-system-for-computer-games-and-animation-e32878980.html
I must warn you, though, that the first edition contains a huge mistake in one of its derivations.
In my comment on Amazon of this book you can find where this mistake is, and an alternative derivation that does not contain the mistake.
I think my derivation is a beautiful demonstration of the power of Geometric Algebra.
If you want to download via the link above, and your anti-virus program complains, switch it temporarily off.
The download is safe.
@@konradswart4069 i have the book Geometric Algebra for Computer Graphics from John Vince in my bookshelf. But had started with geometric algebra by the thesis of Alan Miller and a discussion on geometric algebra on the old fractal forum. What Norman is doing in this lesson is close to the book of Garret Sobczyk on Geometric Algebra and Matrices. You can also visit my Geometric Algebra page on Wacker Art.
@@hermannwacker1902 Thanks for your response.
Wacker Art?
-Could you give a link to your page?-
_Found it!_
_Thanks!_
A wonderful presentation from a math guru. Always your disciple
0 x a = 0 is not a property but a consequence that can be deducted from all other properties.
You are right that the properties are not independent, but Prof. Wilderberger does not claim that. Some may perhaps argue that 0xa=0 is a more fundamental property than the ones you would use to deduce 0xa=0.
0a=0 is rightly seen as the nullary case of distributivity.
After thinking about it I do not see how 0xa=0 is deduced from the other properties. If 1xa=a and 0xa=0 and 1+T=0 then the distributive law implies that a+Txa=1xa+Txa=(1+T)xa=0xa=0 implying that Txa is the additive inverse of a. Distributivity implies that 0xa=(0+0)xa=0xa+0xa so if 0xa has an additive inverse then 0xa=0, but I do not se how should be deduced without additive inverses. In a sense the existence of additive inverses is a more fundamental property than 1+T=0 in the sense that this is an important property in that it make (F,+) into a group even if F has no multiplication.
@@PeterHarremoes this is actually a really good remark, I took additive inverse for granted.
Allow me to suggest the word 'anti' instead of 'minus'.
Antarctica is not called minusarctica. The antiproton is not called a minusproton.
Let -1 be denoted '-' .
Subtraction should be written a+-b or a+b- . Say: 'a plus anti b' or 'a plus b anti' .
a-b is the product a(-)b = - a b = a b - . Say: 'a anti b equals anti a b equals a b anti'.
1+- = -+1 = 0 . One plus anti is equal to anti plus one, which is equal to zero.
-- = 1 . Anti anti is one.
-1 = 1- = - . Anti one is equal to one anti, which is equal to anti.
-+- = 2- = -2 . Anti plus anti is two anti, or anti two.
Complex numbers may be: 5 + i3 = 5 + 3i , - 5 + 3i- = -5 + 3-i = -5 + -3i .
(Your letter t is indistiguishable from a plus sign , and x is indistiguishable from a multiplication sign, and a bar in the second row of a matrix is indistinguishable from an underscore in the first row).
Thank you!
Bo.
Normally I enjoy watching your content despite my heavy disagreements (you can offer additional insights or ways of thinking that can be useful or intuitive). However, this just overcomplicates complex numbers in a way that doesn't seem very helpful or intuitive, compared to the usual approach that Cauchy introduced. Being frank this is just unapproachable. Just speaking from the perspective of a physicist, the real number approach to complex fields (I took courses in complex analysis and complex variables years back). When doing field theory in physics the more conventional approach to complex fields is far more elegant and simpler to use. This is especially true when working in Quantum Fields and Electromagnetic Fields. The traditional approach just simplifies things a lot more than this modular approach. Additionally, it allows us to use the Fourier and Laplace transforms in ways that give us power over some complicated geometry that one might find in a crystal. This isn't to condemn the modular approach to this arithmetic because it does have its uses (miller indices and Pauli spin matrices are good examples), but This can't replace the Approach proposed by Cauchy. It isn't approachable or understandable and it certainly isn't as usable for practical purposes either.
@CrownedFalcon I am trying to develop a good pure approach to complex numbers, that extends the understanding also to relativistic analogs, as well as others also to come. I am sure if you watch the remaining 3 videos in this little series you may more clearly see the advantages. I am not proposing however that every physicist you uses complex numbers has to frame everything in terms of Dihedrons. By all means, work with a + bi etc, but now you have a better answer for what this actually is, and why it naturally connects with two dimensional geometry and motion. That is an understanding that is quite useful to a physicists I believe.
@@njwildberger Thanks for your reply. I will watch the other videos. Please forgive me if I approach them with a great deal of skepticism though. I just don't see the utility in this approach to complex numbers yet.
@@CrownedFalcon00 This stuff connects with Pauli matrices, the Dirac equation, quaternions, and lots of other super useful stuff. It's all related, and seeing the abstract connections among them can give you insights you wouldn't otherwise have.
It's like introducing a metric tensor into special relativity, and distinguishing covariant and contravariant tensors when you're only using rectilinear orthogonal coordinates. It's not obvious why you'd make it more complicated by introducing this stuff. But when you get into differential geometry and general relativity, all the insight you built up by working with the seemingly-needlessly-complicated formal abstract theory suddenly pays off.