1 = -1?

I posted this "proof" in the 1/0 thread yesterday. I followed it up with discussion about another strange issue in math that I thought in some abstract way might be related, and then the discussion went in the direction of the second issue. So, I am starting a new thread to discuss the "proof" below. I do not know its resolution, other than that it is a reductio ad absurdum of imaginary numbers. I am open to other perspectives.

-1 = -1

1/-1 = -1/1

sqrt (1/-1) = sqrt (-1/1)

sqrt 1 / sqrt -1 = sqrt -1 / sqrt 1

1/i = i/1

i(1/i) = i(i/1)

1 = i^2

1 = -1


What is the deal with that?

What? You've already been given a painstaking explication of exactly this. It isn't a reductio ad absurdum.

The inference from step 3 to step 4 is unfounded. The end.

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Originally Posted by Xei

What? You've already been given a painstaking explication of exactly this. It isn't a reductio ad absurdum.

The inference from step 3 to step 4 is unfounded. The end.

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With the "painstaking explication," we discussed the legitimacy of squaring the square root of a negative number and getting a negative number. Well, in this proof, the square of sqrt -1 is -1, as you believe it should be. The inference from step 3 to step 4 involves getting the square roots of negatives (not squaring them), which has not been the topic of our discussion. The inference also involves the principle that the square root of a fraction is the square root of the numerator over the square root of the denominator, an issue we have not discussed. So what is the problem in the proof? What exactly goes wrong from step 3 to step 4?

"the principle that the square root of a fraction is the square root of the numerator over the square root of the denominator"

This. It's the same issue as before, sqrt(a/c) = sqrt(a)/sqrt(c) is almost the same identity as sqrt(ab) = sqrt(a)sqrt(b), put b = 1/c.

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Originally Posted by Xei

"the principle that the square root of a fraction is the square root of the numerator over the square root of the denominator"

This. It's the same issue as before, sqrt(a/c) = sqrt(a)/sqrt(c) is almost the same identity as sqrt(ab) = sqrt(a)sqrt(b), put b = 1/c.

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I would not say it is the same identity, but it too involves an exception because an imaginary number is involved. I said in the other thread that the two principles may be connected, and I think you just helped me make the connection more concrete in my mind.

So the resolution is that getting the square root of a fraction does not involve a ratio of the square roots of the numerator and denominator when at least one is negative. That answers the question. The situation is just another exception scenario that results from assuming the reality of imaginary numbers. It is a reductio ad absurdum until "exception" is declared?

That exception is even more bizarre than the other one. Getting the square root of a fraction by putting the square root of the numerator over the square root of the denominator seems like it should always be the logical method because squaring a fraction involves getting the square of the numerator over the square of the denominator. So, going in the reverse direction, it would make sense to get the ratio of the square root of the numerator to the square root of the denominator. But what we have run into is a situation of, "Oh no, we are dealing with imaginary numbers, so we can't do that here." You answered my question, but now I am even more convinced that the "proof" is a reductio ad absurdum of imaginary numbers.

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Originally Posted by Universal Mind

It is a reductio ad absurdum until "exception" is declared?

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I don't know why you're repeating this given that I invested a lot of time resolving this misconception and you were fine with it. I'll just quote myself if you don't mind.

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This statement is not an axiom of arithmetic; it is proven from the axioms of arithmetic, and the proof only works for when a and b are non-negative reals. When a and b are not non-negative reals, the proof does not work and the identity is in fact incorrect.

Therefore, I reiterate my reiteration: there was no creation of a special case. The truth for a and b non-negative and the falsity otherwise are both deduced from the axioms of arithmetic.

There is nothing wrong with this and it is not unusual.

To help show this, here is another identity:

a^2 = a*|a| for a non-negative.

|a| means the absolute value of a, which means its size irrespective of whether it is positive or negative. You can write it as sqrt(a^2) if you want.

This is exactly analogous. The identity is not an axiom of arithmetic: it is proven from the axioms of arithmetic, and the proof only works when a is non-negative. If a is negative, the proof fails, and the correct identity is in fact

a^2 = -a*|a|

What your argument says is that we have invented a special case because the identity doesn't work for negative numbers; and therefore, negative numbers are somehow fake. As you don't believe that negative numbers have the same problem as complex numbers, hopefully you now see why your argument doesn't make sense.

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Originally Posted by Universal Mind

That exception is even more bizarre than the other one. Getting the square root of a fraction by putting the square root of the numerator over the square root of the denominator seems like it should always be the logical method because squaring a fraction involves getting the square of the numerator over the square of the denominator. So, going in the reverse direction, it would make sense to get the ratio of the square root of the numerator to the square root of the denominator. But what we have run into is a situation of, "Oh no, we are dealing with imaginary numbers, so we can't do that here." You answered my question, but now I am even more convinced that the "proof" is a reductio ad absurdum of imaginary numbers.

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I am repeating myself yet again by saying this: the identity does not apply to matrices either. If it 'disproves the existence of complex numbers' then it disproves the existence of matrices too. You repeatedly insist on referring to it as if it is some kind of ancillary axiom necessitated by consistency when it has been clearly explained to be no such thing; like all mathematics, propositions are proven from the axioms. This particular proposition can be proven for non-negative integers, but it doesn't happen to go through for other entities.

Saying 'it seems like it should make sense' is not a proof. By not delineating the actual mathematical argument properly you obfuscate the subtlety which here leads to the mistake you are making. The subtlety is that the square function is not injective. Two numbers having equal squares does not prove that they are equal.

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Originally Posted by Xei

I don't know why you're repeating this given that I invested a lot of time resolving this misconception and you were fine with it. I'll just quote myself if you don't mind.

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You are mixing up the two scenarios again. The proof for this thread is the only thing I have said is a reductio ad absurdum. The equation involving products of square roots demands an imaginary number exception, but I never said it was a reductio ad absurdum or proves anything. I have just made the point that it was an exception that had to be made when something new and fictitious was introduced to the system. Make sure you understand the difference. The topic of this thread stands out to me as a reductio ad absurdum because of the principle I explained regarding the purpose of getting the square root of the numerator and denominator. It is the logical thing to do in all situations that are rooted in reality. Multiplying square roots by getting the square root of the product of the radicands doesn't seem to follow such invincible logic, in my mind, so I have not declared the need for the exception a reductio ad absurdum. Understand?

I am going to study the logic behind the product rule before ever declaring the other scenario a reductio ad absurdum. It might be one, but I have yet to claim that it is.

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Originally Posted by Xei

I am repeating myself yet again by saying this: the identity does not apply to matrices either. If it 'disproves the existence of complex numbers' then it disproves the existence of matrices too. You repeatedly insist on referring to it as if it is some kind of ancillary axiom necessitated by consistency when it has been clearly explained to be no such thing; like all mathematics, propositions are proven from the axioms. This particular proposition can be proven for non-negative integers, but it doesn't happen to go through for other entities.

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A matrix is a man made construct. It's like a board game in that way. People made the rules for what they represent and what to do with them. I am going to repeat myself yet again too. The proposition cannot be proven for negative numbers because even roots of negative numbers are a crock of shit.

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Originally Posted by Xei

Saying 'it seems like it should make sense' is not a proof. By not delineating the actual mathematical argument properly you obfuscate the subtlety which here leads to the mistake you are making. The subtlety is that the square function is not injective. Two numbers having equal squares does not prove that they are equal.

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I was being polite by saying "seems like it should." I'll put it another way-- it is what is logical. I explained my reasoning. Your argument that two numbers having equal squares does not prove they are equal does not disprove my argument. Yes, a number has two square roots, one positive and one negative. What does that have to do with this? The radical symbol represents principal square root, which is the positive one. When more than the principal square root can be involved, such as in an equation with a variable, you fix the problem by using +/-. That is not necessary here. There is no variable, and it is principal square root involved, and that is one number.

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Originally Posted by Universal Mind

You are mixing up the two scenarios again. The proof for this thread is the only thing I have said is a reductio ad absurdum. The equation involving products of square roots demands an imaginary number exception, but I never said it was a reductio ad absurdum or proves anything. I have just made the point that it was an exception that had to be made when something new and fictitious was introduced to the system. Make sure you understand the difference. The topic of this thread stands out to me as a reductio ad absurdum because of the principle I explained regarding the purpose of getting the square root of the numerator and denominator. It is the logical thing to do in all situations that are rooted in reality. Multiplying square roots by getting the square root of the product of the radicands doesn't seem to follow such invincible logic, in my mind, so I have not declared the need for the exception a reductio ad absurdum. Understand?

I am going to study the logic behind the product rule before ever declaring the other scenario a reductio ad absurdum. It might be one, but I have yet to claim that it is.

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The misconception is about it being an 'exception', not about it being a reductio ad absurdum... as should be clear given that that is what the quoted response is all about.

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I was being polite by saying "seems like it should." I'll put it another way-- it is what is logical. I explained my reasoning.

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If it were logical it would have taken the form of a proof. You are yet to write one. It is necessary that you write one because you are currently relying on a heuristic description which turns out to be incomplete. The issue stems from where you vaguely said 'going in the reverse direction'. This is not actually well defined, because the function isn't injective, like I just said. Principal roots have nothing to do with it.

You need to prove it properly and then you will see why the identity does not apply to all numbers, and that will fully answer the problem in the OP.

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Originally Posted by Xei

If it were logical it would have taken the form of a proof. You are yet to write one. It is necessary that you write one because you are currently relying on a heuristic description which turns out to be incomplete. The issue stems from where you vaguely said 'going in the reverse direction'. This is not actually well defined, because the function isn't injective, like I just said. Principal roots have nothing to do with it.

You need to prove it properly and then you will see why the identity does not apply to all numbers, and that will fully answer the problem in the OP.

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I made an argument. That can be done in math without the writing of a proof. I will make my point another way. The square roots of the numerator and denominator are, based on the definition of "square," the numbers that are squared to get the numerator and denominator. Right? I mentioned principle square roots because of the off target issue you brought up.

If a formal proof is all you will see, here you go.

b = 1/c (valid everywhere c =/=0)

sqrt(a)sqrt(b) = sqrt(ab)
sqrt(a)sqrt(1/c) = sqrt(a*1/c)
sqrt(a)*1/sqrt(c) = sqrt(a/c)
sqrt(a)/sqrt(c) = sqrt(a/c)

My complements to PhilosopherStoned for typing that in the other thread.

Congrats! You've proved that when sqrt(a)sqrt(b) = sqrt(ab), we also have sqrt(a)/sqrt(c) = sqrt(a/c) for c =/= 0. Can you go the other way? Remember, you'll have to take b=0 as a special case.

Also contingent upon sqrt(1/c) = 1/sqrt(c).

Just note that (1/sqrt(c))^2 = 1/(sqrt(c)^2) = 1/c by the definition of multiplication.

Has the same caveat as sqrt(a)sqrt(b) = sqrt(ab)and its proof, that's what I meant to highlight.

Xei, I am thoroughly impressed by your perseverance with this topic. It seems like some people just want you to keep repeating yourself...

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Originally Posted by Pensive Patrick

Xei, I am thoroughly impressed by your perseverance with this topic. It seems like some people just want you to keep repeating yourself...

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:rolllaugh: I'm sure you have an extremely clear understanding of what this debate is about. However, prove it. Also, I am just one person, and I have not been repeating Xei's self. I have made some of the same points more than once because they have needed to be taken into account more than once.

Is the "Pensive" in your screen name a reference to the high price of your gold teeth?

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Originally Posted by PhilosopherStoned

Congrats! You've proved that when sqrt(a)sqrt(b) = sqrt(ab), we also have sqrt(a)/sqrt(c) = sqrt(a/c) for c =/= 0. Can you go the other way? Remember, you'll have to take b=0 as a special case.

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Why do I need to go the other way? I proved my point Xei challenged me to prove. To be a little more to the point, the next step in your proof that I cut and pasted could be wrtitten as sqrt(a/c) = sqrt(a)/sqrt(c). (Symmetric Property of Equality) I know that sqrt(a)sqrt(b) = sqrt(ab) doesn't apply to imaginary numbers, fantasy numbers (I just added that rule.), or the Flying Spaghetti Monster.

UM, this reminds me of that clip that one of us posted one time for a creationist. Was it Noogah? Or maybe Ne-Yo. This time, you're the Black Night and Xei is Arthur. Enjoy!

http://www.youtube.com/watch?v=zKhEw7nD9C4

Ironically, your nonresponsiveness to a very detailed and on-point post suggests that it is you who has lost his arms and legs. I used your proof to show that the square root of a fraction is the square root of the numerator over the square root of the denominator and agreed that it cannot apply to imaginary numbers. We have different takes on why that is, but I would have to say something I have already said to explain why that is, and I would hate to make 'pensivePatrick think that reveals a flaw in my argument because he doesn't know enough about this stuff to get what is actually happening.

However, I will be glad to talk about movies if you have had enough of this. My reaction to the Black Knight scene when I saw that movie for first time was literally one of the three best laughs of my life. Have you seen Kentucky Fried Movie? Man, it's up there with Holy Grail and Life of Brian, but it was written by the Zucker Brothers and Jim Abrahms. They wrote Airplane and the Naked Gun movies. Great stuff.

No. There was no irony in my response. The irony here is that you claiming to have made an on-point post is equivalent to the Black Night claiming that his arm being off is only a flesh wound.

Specifically, Xei has been asking you to prove that the square root is multiplicative.

You posted my proof that the square root being multiplicative on a domain implies that it distributes over division on that domain as well. Of course this would work for any multiplicative function. So it already assumed what you were asked to prove.

So your post was not just not on-point but it was not even wrong. You were asked to prove one thing and proved another.

What time is it? Fifty kilograms.

Oh, sorry. Here we go.

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Originally Posted by Universal Mind

-1 = -1

1/-1 = -1/1

sqrt (1/-1) = sqrt (-1/1)

sqrt 1 / sqrt -1 = sqrt -1 / sqrt 1

1/i = i/1

i(1/i) = i(i/1)

1 = i^2

1 = -1


What is the deal with that?

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Originally Posted by Xei

The inference from step 3 to step 4 is unfounded. The end.

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Quote:

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Originally Posted by Universal Mind

we discussed the legitimacy of squaring the square root of a negative number and getting a negative number. Well, in this proof, the square of sqrt -1 is -1, as you believe it should be. The inference from step 3 to step 4 involves getting the square roots of negatives (not squaring them), which has not been the topic of our discussion. The inference also involves the principle that the square root of a fraction is the square root of the numerator over the square root of the denominator, an issue we have not discussed. So what is the problem in the proof? What exactly goes wrong from step 3 to step 4?

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Quote:

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Originally Posted by Universal Mind

Getting the square root of a fraction by putting the square root of the numerator over the square root of the denominator seems like it should always be the logical method because squaring a fraction involves getting the square of the numerator over the square of the denominator. So, going in the reverse direction, it would make sense to get the ratio of the square root of the numerator to the square root of the denominator. But what we have run into is a situation of, "Oh no, we are dealing with imaginary numbers, so we can't do that here." You answered my question, but now I am even more convinced that the "proof" is a reductio ad absurdum of imaginary numbers.

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Originally Posted by Xei

Saying 'it seems like it should make sense' is not a proof.

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Originally Posted by Xei

You need to prove it properly and then you will see why the identity does not apply to all numbers, and that will fully answer the problem in the OP.

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Originally Posted by Universal Mind

b = 1/c (valid everywhere c =/=0)

sqrt(a)sqrt(b) = sqrt(ab)
sqrt(a)sqrt(1/c) = sqrt(a*1/c)
sqrt(a)*1/sqrt(c) = sqrt(a/c)
sqrt(a)/sqrt(c) = sqrt(a/c)

My complements to PhilosopherStoned for typing that in the other thread.

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Originally Posted by Universal Mind

To be a little more to the point, the next step in your proof that I cut and pasted could be wrtitten as sqrt(a/c) = sqrt(a)/sqrt(c). (Symmetric Property of Equality) I know that sqrt(a)sqrt(b) = sqrt(ab) doesn't apply to imaginary numbers, fantasy numbers (I just added that rule.), or the Flying Spaghetti Monster.

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Easy Rider is another really cool movie. The philosophy and drama of it are great enough, but the sound track and cinematography add to that and make it one of the best films of all time. The Graduate is right up there with it, for the same reasons. The comedy in the Graduate makes it a different sort of movie, though.

You proved that sqrt(a/c) = sqrt(a)/sqrt(c) whenever sqrt(ab) = sqrt(a)sqrt(b) and sqrt(1/c) = 1/sqrt(c). So like I said at the start, this is the same issue as before. It depends on the proof for both of these things, and the proof of these things does not work for all numbers.

Hey, did Xei just repeat himself again?

Hey Universal Mind, the 'Pensive' in my username is actually referring to my fifteen year old opinion of what sounded like a good username... Thanks for asking!

Here is my response to your insult:

http://www.youtube.com/watch?v=qCTLCNmnlKU&feature=youtube_gdata_player

Holy crap patrick! You were 15 when you signed up? That means you are gonna be 22 this year...and Im gonna be 24 >< I cant believe Ive known about this place for so long. Its incredible this community remains so bright and vibrant after so long. It didnt hit me how long Ive known about this place until you just said that.

Back on topic...this is getting quite circular. UM...Xei is saying that the proof you got from philosopher is not true for every number, just every positive number. So I guess I should ask, do you believe negative numbers in general are crap? Or just roots of negatives? Negative numbers are very useful and show the removal of real physical objects, just like positives show the adding of, or the state of those physical objects. I suppose you could say from the point of view of "being" negative numbers dont make sense. But they do make sense from a relativistic point of view (ex: motion). Which I think is all they are ever used for anyways arnt they?

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Originally Posted by Xei

You proved that sqrt(a/c) = sqrt(a)/sqrt(c) whenever sqrt(ab) = sqrt(a)sqrt(b) and sqrt(1/c) = 1/sqrt(c). So like I said at the start, this is the same issue as before. It depends on the proof for both of these things, and the proof of these things does not work for all numbers.

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It hit me earlier today what the miscommunication has been, and tkdyo's post spelled it out. I was not quite clear enough about my perspective on what I meant by an exception needing to be conveniently introduced with the introduction of imaginary numbers. When I said it was true for all numbers and then an exception had to be made when imaginary numbers were introduced, I meant that the principle was deemed for centuries to be true for all numbers that give the equations any real meaning. Back in the day, negative numbers were not even up for consideration as square root radicands. With the introduction of imaginary numbers, an exception had to be made to the rule in terms of numbers that are actually viable in the equations in the first place. It dawned on me that you were talking about excluded values for a and b. I was talking about the radical expression values. For a long time, the square root of a negative number was not even considered a number. That is why I said an exception had to suddenly be made. What had not been considered a number became considered a number. The rule works for all values a and b except those that give radical expressions imaginary values. We of course agree on that because we both agree that the equations do not work with negative radicands. We just disagree on the nature of negative radicands.

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Originally Posted by Pensive Patrick

Hey, did Xei just repeat himself again?

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Hey, I have done that too. Don't ban me from your tree house just yet. Feel free to make arguments about the actual topic. Being a mere sideline heckler is kind of weak.

The video guy's insults majorly lack creativity, but that makes the video hilarious.


tkdyo, you figured out the miscommunication. I explained somewhere in the sea of posts in the other thread why I do think negative numbers are legitimate and non-fictitious. If I owe you 5 dollars, you owe me -5 dollars. If I am 4 feet in front of a line, I am -4 feet behind it, etc. Negative numbers denote values below zero, which I think makes perfect sense.

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Originally Posted by Universal Mind

tkdyo, you figured out the miscommunication. I explained somewhere in the sea of posts in the other thread why I do think negative numbers are legitimate and non-fictitious. If I owe you 5 dollars, you owe me -5 dollars. If I am 4 feet in front of a line, I am -4 feet behind it, etc. Negative numbers denote values below zero, which I think makes perfect sense.

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Even the positive integers didn't make sense to many people for the whole history of human culture before their introduction. There are people today that I've encountered in real life that don't accept even zero let alone the negative integers. Of course, as somebody that understands them, you think that that's stupid. They normally come up on some travesty of mathematical reasoning that depends upon taking a trivial statement out of context and then call it a proof. :cackle:

Most of them are fairly smart outside of mathematics and formal logic. They must be to care.

I once saw a man, within the same hour no less, deny the "existence" of negative numbers and then point out that we could use peanut butter that we had at hand as a sealant for an impromptu bong.

That's pretty smart.

When I get around to it, I'll go back to the other thread and point out how your assumptions about the "fantasy" unit, together with the assumption that they satisfy field axioms, imply that the fantasy unit is a square root for -1. I'll then leave it for you as an exercise to show that the square root isn't multiplicative on the fantasy numbers.

We will prove that the fantasy numbers are isomorphic to the complex numbers.

That is to say that however you take the algebraic completion of the reals, one gets the complex numbers. They're determined by the question "How do I factor every polynomial into linear roots?" to the same degree that the reals are determined by the question "How do I take the limit of any convergent series of rationals?".

The reals are the analytic completion of the rationals and the complex numbers are the algebraic completion of the reals. Even if one just wants the algebraic completion of the rationals, one is still looking at a subset of the complex numbers that isn't contained in the reals, i.e. one has to factor the equation x² + 1. The way that I originally learned to construct the complex numbers (and this is kinda backward, one would normally first encounter it in terms of R²) is as the quotient ring R[x]/(x² + 1). It can also be embedded in matrix rings as Xei has pointed out. It represents rotations and dilations. It can also be mapped onto a sphere where it represents rotations in three space. C is a very real thing. Or at least as real as N, Z, Q, and R. Pfft. These can only represent rotations and dilations in one space and only rotations in two space.

It would be nice if we had something that satisfied field axioms on general n space. A lot of mathematics (related to matrices and tensors in general, i.e. grassman algebras and differential forms) can be regarded as attempting to build some useful approximation of this.

It would be nice though it's been proven impossible. We only get numbers containing the reals and using archimedian norms on one and two dimensions. :(

Plus they're apparently even useful for practical things on occasion and that's always nice.

ur my hero

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It would be nice if we had something that satisfied field axioms on general n space. A lot of mathematics (related to matrices and tensors in general, i.e. grassman algebras and differential forms) can be regarded as attempting to build some useful approximation of this.

It would be nice though it's been proven impossible. We only get numbers containing the reals and using archimedian norms on one and two dimensions.

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I don't know about this, want to talk about it some more?

Also out of curiosity where and when do you get all of this learning?

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Originally Posted by PhilosopherStoned

There are people today that I've encountered in real life that don't accept even zero let alone the negative integers. Of course, as somebody that understands them, you think that that's stupid. They normally come up on some travesty of mathematical reasoning that depends upon taking a trivial statement out of context and then call it a proof. :cackle:

Most of them are fairly smart outside of mathematics and formal logic. They must be to care.

I once saw a man, within the same hour no less, deny the "existence" of negative numbers and then point out that we could use peanut butter that we had at hand as a sealant for an impromptu bong.

That's pretty smart. Plus they're apparently even useful for practical things on occasion and that's always nice.

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Are those travesties proofs that you yourself posted? It is another case of an argument I would love to see. It would be pretty easy to show them zero and negative numbers in the real world.

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Originally Posted by PhilosopherStoned

When I get around to it, I'll go back to the other thread and point out how your assumptions about the "fantasy" unit, together with the assumption that they satisfy field axioms, imply that the fantasy unit is a square root for -1. I'll then leave it for you as an exercise to show that the square root isn't multiplicative on the fantasy numbers.

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If you can do that, we are having a purely semantic argument. I made up fantasy numbers for satiricical purposes. Could you do anything useful and logical with a number that is equal to both 2 and 7 but not equal to 2? If so, our argument is ultra-semantic.

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Originally Posted by PhilosopherStoned

That is to say that however you take the algebraic completion of the reals, one gets the complex numbers. They're determined by the question "How do I factor every polynomial into linear roots?" to the same degree that the reals are determined by the question "How do I take the limit of any convergent series of rationals?".

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I agree that it works. However, remember what I have said about the hypothetical nature. The FBI makes up characters who satisfy the elements of a given suspect situation based on profiling techniques. That doesn't make the characters real. I think this paragraph addresses most of the rest of your post.

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Originally Posted by PhilosopherStoned

It would be nice though it's been proven impossible. We only get numbers containing the reals and using archimedian norms on one and two dimensions. :(

Plus they're apparently even useful for practical things on occasion and that's always nice.

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I think that raises some issues concerning our discussion. "Usefulness" vs. "true existence."

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Originally Posted by Xei

I don't know about this, want to talk about it some more?

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Sure. Like I said I'm rusty and this is all very informal, but think about it like this. Take the most basic linear equation in one dimension w = cz. We are here interpreting number in two ways, as a function and as an element in the domain/range. Of the two numbers c is playing the role of a function and z is playing the role of an element.

So when we go to generalize to higher dimensions (and solve more equations) we have two distinct roles to fulfill. One gets fulfilled with vectors and the other with matrices. So we end up with something like w=Cz.

The analogy is however still striking. Take matrices. if M is a matrix ring over R then we can define maps M -> M. Take the map A |-> A². Differentiate it. The derivative at A will be a linear function on H in M where H is interpreted as the displacement. The derivative is H |-> AH + HA! Consider that if A and H commute than this is just 2AH. But in one dimension, x |-> x² has a derivative at x which is a linear function on h real of the form h |-> 2xh.

So the analogy can be very tight at times. It's of course important to realize that matrices are different from numbers in that everything doesn't carry over by a long shot but there is some intuition to be had there.

As far as grassman algebras and differential forms go, they're ways to generalize the purpose of numbers in measuring various volumes. They're pretty deep for me but I could probably bullshit my way through an explanation.

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Also out of curiosity where and when do you get all of this learning?

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Books. Ending mostly about seven years ago or so. Though I do try to at least keep sharp with my basic algebra and analysis. I wasn't above dropping in on a math department and asking random people if I got stuck with something. I'm actually very slow and stupid so it's not like I'm a genius or anything. At least not at math ;) Nobody was impressed with anything other than that I work very hard.

Another example is positive and negative matrices. For the reals, the invertible elements are in an open set with two disconnected components, N = (-inf, 0) and P = (0, inf). Numbers in P multiplied by each other end up in P so it's closed under multiplication. Numbers in N multiplied together end up in P. Multiplying a number in P and a number in N ends up in N. This is just the negative times a negative is a positive rule and its variants.

Now consider the invertible elements of some matrix ring over the reals. These will be the matrices with non-zero determinants. Now N is the set of matrices with Det(A) < 0 and P will have Det(A) > 0. Note that Det(AB) = Det(A)Det(B) so matrix multiplications satisfies all the same negative-times-a-negative-is-a-positive type rules.

Also note that it is an open set with two disconnected components. To see that it's open, just note that the determinant is a polynomial. Because polynomials are continuous, given an invertible matrix A, we can put a sufficiently small open ball around it so that the determinate is non-zero on that too.

To show that it's disconnected, note that a continuous function that switches signs must take on a value of zero. Hence any path s |-> A(s) that we can draw from a positive matrix to a negative matrix or vice-versa will have Det(A(s)) = 0 for some s. Here the s are just real numbers.

So again, it's very much like numbers.

no wai.