"Veritasium" (YT) - "The Big Misconception About Electricity" ?

[snarkysparky]

https://lpsa.swarthmore.edu/BackGround/phasor/phasor.html#:~:text=A%20sinusoidal%20signal%20f(t,a%20rotational%20velocity%20of%20%CF%89.

[adx]

=SIN(3.1415926)
5.35898E-08

[adx]

And the cesium clock.

[TimFox]

=SIN(3.1415926)
5.35898E-08

In a computed-tomography application, one of our software engineers used a value of pi defined to 7 decimal places instead of the compiler's function PI().  Unfortunately, he was off in the last decimal place.  Computed tomography requires going around a circle exactly once.  It was interesting how much error this small difference in pi caused to the resulting reconstructed image.

[hamster_nz]

=SIN(3.1415926)
5.35898E-08

In a computed-tomography application, one of our software engineers used a value of pi defined to 7 decimal places instead of the compiler's function PI().  Unfortunately, he was off in the last decimal place.  Computed tomography requires going around a circle exactly once.  It was interesting how much error this small difference in pi caused to the resulting reconstructed image.

Me: Goes away and quietly adds the following to the build pipeline:

Code: [Select]

#!/bin/bash

count=`find src -type f -name '*.[ch]' | xargs grep -l -e "3[.]14" -e "1[.]57" | wc -l`
if [ "$count" != 0 ]
then
  echo "Found a PI-like constant in these files:"
  find src -type f -name '*.[ch]' | xargs grep -l -e "3[.]14" -e "1[.]57"
  exit 3
else
  exit 0
fi


[adx]

You say you're not letting it drive your thoughts... RIGHT BEFORE YOU SAY THAT YOU ARE LETTING IT DRIVE YOUR THOUGHTS.

I hoped you wouldn't pick up on that take it that way.

I  don't mean it in such an all or nothing sense. It's possible for me to be careful not to fall into traps over naming, but still appreciate and acknowledge a wider scope to the argument. You're acting as if current knowledge is complete. I'm drawing more from my own thoughts, which I intentionally avoided elaborating on, because they are not helpful at this stage. It's not as if I am starrily-eyedly fanboiing over words.

Perhaps I have been too general in my 'pondering' and skepticism, but I don't doubt for any more than a second that real numbers have more direct physical relevance than imaginary numbers do. If I am appealing to authority, which I doubt I am, it is to add a small dose of skepticism to your 'conventional' claims of absolute concrete rigour and applicability.

So things only make sense to you if they're named right?

I was saying that whatever the name there would still be a problem, and I'd still find it.

Unfortunately in trying to understand your questions here, I get more of a sense of belief and inability to see what I mean.

You don't understand my simple questions? The trouble is, as I said, is I do know what you mean. The issue is you don't like my answers or the answers of any of the references I've posted. Whatever.

Yes.

I'm not suggesting it is wrong.

Minor point but this was about your belief in mathematics, I wasn't commenting on mathematics itself.

All my objections to imaginary numbers in engineering disappear for polar notation (for obvious reason). I just prefer complex notation and operations.

You object to the imaginary numbers... but you prefer complex notation...

I didn't say I like complex notation. It just makes more sense in x y coordinates.

That's it. I'm done. You win.

By failing to learn something fundamental and useful. Except I sort of did. So I lost.

[bsfeechannel]

The entire purpose of educating students seems to ultimately get them to believe things they can go on to (never) use.

The entire purpose of educating students is to provide them with the shoulders of giants upon which they'll stand.

Especially engineering math.

For the math illiterate, everything looks like a magic number. For that ability you don't need an engineer. Any untrained person will do.

Nope and nope. I meant sqrt(-1) hardly factors (non-mathematical meaning) into complex phasors.

\$\sqrt{-1}\$ is everywhere. Since (\$\sqrt{-1}\$)² = -1, every negative number has it. And since (\$\sqrt{-1}\$)⁴ = 1, positive numbers do too.

O altar o proof, where artst [sic] thou?

In an engineering math course near you.

[SiliconWizard]

\$\sqrt{-1} = \sqrt{e^{i.\pi}} = (e^{i.\pi})^{1/2} = e^{i.\pi/2} = i\$

Using the polar form, you can see that the square root of a complex number halves its argument.
But the root (pun?) of the "issue" is that adx doesn't see -1 as a complex number here, or the set of complex numbers as a superset of real numbers - I guess he sees them as completely separate entities.

[adx]

\$\sqrt{-1} = \sqrt{e^{i.\pi}} = (e^{i.\pi})^{1/2} = e^{i.\pi/2} = i\$

Using the polar form, you can see that the square root of a complex number halves its argument.
But the root (pun?) of the "issue" is that adx doesn't see -1 as a complex number here, or the set of complex numbers as a superset of real numbers - I guess he sees them as completely separate entities.

That's pretty much it. I see the imaginary numbers more as a parallel sequence of numbers existing in a different 'realm' than the real numbers. If they were to cross (which seems possible because of zero) then it would be at an infinitesimal angle. Or something more like a peace sign than a cross. Concepts only, not pictures! Just the idea of a different existence than the direct or obvious assignment of a physical degree of freedom implies, the sense that something no matter how small, is missing. And if so, writing a + jb is an incorrect and / or artificial construct. Rotation is an unavoidable consequence of saying (or defining) i*i = -1 (here is use j = i). Algebra isn't physical (or that's what I see) -  the universe doesn't solve, it iterates. Somewhere in there is a link I cannot make or doesn't exist. I could be wrong, but it is my mind's prerogative to not accept something that it doesn't understand.

[bsfeechannel]

But the root (pun?) of the "issue" is that adx doesn't see -1 as a complex number here, or the set of complex numbers as a superset of real numbers - I guess he sees them as completely separate entities.

It's a little worse. He also thinks 1 is not a complex number.

[adx]

I'm sure that sentiment is stretching the bounds of believability for many people here too. Sounds like you've been sniffing the chalk too long.

I can define a number 'line' based on 1 MOD 1 called moo numbers, and poke fun at anyone who suggests that 1 is not a moo number.

Yes no surprise I'd see something 'wrong' with 1's part as a complex number after "pretty much" and "peace sign", SiliconWizard's observation is closer to the mark though. And here I was thinking someone might pick up on the apparent contradiction of me talking about infinitesimal angle then saying rotation is an unavoidable consequence of i*i = -1.

Complex numbers can exist as a mathland fiction all they want, as a fundamental 'quantity' of nature if you like (I'm just not 100% convinced - it feels like a broken reality the way it has been put), but fundamental to electrical engineering? Real-valued measures of sines and cosines are not sqrt(-1), that idea is so ridiculous I shouldn't have continued arguing about it amidst the conflation with mathland fictions and quantum mechanical possibilities. That FFT I was talking about, like I said not a complex number in sight. Phasors, same.

I trolled myself.

[bsfeechannel]

I'm sure that sentiment is stretching the bounds of believability for many people here too. Sounds like you've been sniffing the chalk too long.

You get math all wrong. Math is not based on faith. Math is essentially a bunch of conveniently chosen postulates and another buch of theorems, which are deductions from those postulates, deductions which are based on another set of postules themselves.

What is a postulate? Essentially a provisory truth. Let me give you a crude example.

1. All Australians eat kangaroo meat.
2. Adx is Australian.
3. Therefore, adx eats kangaroo meat.

In the deduction above, I'm not asking you to believe in the first or the second postulates. I'm asking you to accept them as a provisory truths, i.e., if those are true, the conclusion (3) will be true.

But what happens if I eventually find out that adx is IRL a vegetarian? Well, that doesn't invalidate my reasoning, but certainly my choice of postulates doesn't help me model, describe or predict reality, does it?

So, the postulates upon which math theories are constructed have allowed these theories to have a wide range of applications and have stood the test of time. Should they be revised tomorrow because we find out that they are incomplete or that they do not cut the mustard anymore, they'll be abandoned, or updated.

Complex numbers can exist as a mathland fiction all they want, as a fundamental 'quantity' of nature if you like (I'm just not 100% convinced - it feels like a broken reality the way it has been put), but fundamental to electrical engineering?

Fundamental in the sense that you'll have to deal with them one way or another.

Real-valued measures of sines and cosines are not sqrt(-1), that idea is so ridiculous I shouldn't have continued arguing about it amidst the conflation with mathland fictions and quantum mechanical possibilities.

You shouldn't have skipped the classes on complex numbers.

$$\cos{x}=\frac{e^{ix}+e^{-ix}}{2}$$
$$\sin{x}=\frac{e^{ix}-e^{-ix}}{2i}$$

[adx]

I'm sure that sentiment is stretching the bounds of believability for many people here too. Sounds like you've been sniffing the chalk too long.

You get math all wrong. Math is not based on faith. Math is essentially a bunch of conveniently chosen postulates and another buch of theorems, which are deductions from those postulates, deductions which are based on another set of postules themselves.

I don't think the math itself is a system of belief, just what is "believed" about it.

At high school and for engineering it is approached as if it is something that must be accepted, shoulders of giants etc. If someone is unwilling or incapable of really getting into the nitty gritty of the proofs and philosophy, or simply doesn't have the time to satisfy all questions they might ever dream of, then they are taking it on faith. Being expected to trust in intellectual authorities with absolutely nil room for deviation is pretty much the definition of faith. All this talk of theorems, deductions and postulates is the setting up of a system to engender belief.

I'm not saying it is irrational, unreasonable or wrong. Just that in practical application it is faith.

What is a postulate? Essentially a provisory truth. Let me give you a crude example.

1. All Australians eat kangaroo meat.
2. Adx is Australian.
3. Therefore, adx eats kangaroo meat.

In the deduction above, I'm not asking you to believe in the first or the second postulates. I'm asking you to accept them as a provisory truths, i.e., if those are true, the conclusion (3) will be true.

That may be true, but I don't quite get it - because it is a trick to get me (or perhaps you) to accept the postulates, your logic, and the conclusion. Where is my freedom to reject any of it? That could range from calling it all "rubbish" to simply saying I am not entirely convinced. Why is the latter so completely objectionable? What if my job relies on accepting it?

Or in the real case:
1 I don't eat kangaroo meat (that I remember).
2 I'm not Australian.
3 Therefore, I don't care (apart from the concept and ethics of eating zoo animals).

Why (in principle) should I accept a provisory truth if the reasoning that is brought to bear on them is irrelevant and the conclusion is uncertain?

What if I am the only Australian? A fact I discover after accepting the conclusion on the basis of what I thought was both sound and meaningful logic?

I choose not to believe in the process or the outcome. Not necessarily because of a philosophical objection, but because I don't enjoy it (while others seem to be having the time of their lives) - if you like call it spite. Science students are told to believe and they had better enjoy it. Applied mathematics seems there to be endured and never questioned from the outset.

Also my comment was about believably of the sentiment of you saying I think 1 is not a complex number. There is more to logic than logic, as I have alluded to above.

But if someone has a firm view that complex numbers are the fundamental 'quantity' which describes the world at large, then I'm sure it would seem like any deviation from that view is the wrong one.

But what happens if I eventually find out that adx is IRL a vegetarian? Well, that doesn't invalidate my reasoning, but certainly my choice of postulates doesn't help me model, describe or predict reality, does it?

So, the postulates upon which math theories are constructed have allowed these theories to have a wide range of applications and have stood the test of time. Should they be revised tomorrow because we find out that they are incomplete or that they do not cut the mustard anymore, they'll be abandoned, or updated.

It kind of does invalidate the postulates and casts the reasoning into doubt. If it were taught that way, half the students would go away believing there is wiggle room. The alternative is to lead them into false belief in provisory truths. It's an unwinnable argument based on a sleight of hand.

I'm not against provisional belief, but unless you're a mathematics specialist, it is mostly acceptance and faith. Going against that causes friction, science suffers a similar problem but is somewhat manageable.

Complex numbers can exist as a mathland fiction all they want, as a fundamental 'quantity' of nature if you like (I'm just not 100% convinced - it feels like a broken reality the way it has been put), but fundamental to electrical engineering?

Fundamental in the sense that you'll have to deal with them one way or another.

Where? Because it is popular and convenient? That's not fundamental, it's a circular argument. When something is as optional as it seems to be, arguments in support are expected to collapse into various logical fallacies.

An idea I kept forgetting to suggest, is a 'cheat sheet' of example(s) to demonstrate the fundamental (necessary) applicability (whatever that is) of sqrt(-1) to engineering, then that could help multitude(s) of 'disbelievers'. I'm not suggesting you or anyone here do it, it's just an idea that might work better than pointing to nonexistent proofs for bringing more hapless victims into the fold.

Real-valued measures of sines and cosines are not sqrt(-1), that idea is so ridiculous I shouldn't have continued arguing about it amidst the conflation with mathland fictions and quantum mechanical possibilities.

You shouldn't have skipped the classes on complex numbers.

$$\cos{x}=\frac{e^{ix}+e^{-ix}}{2}$$
$$\sin{x}=\frac{e^{ix}-e^{-ix}}{2i}$$

I didn't. I might have slept through them, perhaps forgot it all or blocked it out. Who knows. Actually there is a story I'll mostly spare you where I did accidentally (due to injury) miss all the lectures of one of the maths classes of one type (I think linear algebra - had I gone I might have a better idea). I am usually pretty good at panicked cramming, but in that case it worked neither well nor at all.

But sines are not sqrt(-1). They are real-valued. I and Q representation doesn't require any sort of 'imaginary'. Despite an idea of complex frequency domain representation being supposedly embedded, it's not necessary. I'll just have to stick to my "that idea is so ridiculous I shouldn't have continued arguing about it".

Is there anything fundamentally unknowable about the quadrature signal in engineering?

[bsfeechannel]

Being expected to trust in intellectual authorities with absolutely nil room for deviation is pretty much the definition of faith. All this talk of theorems, deductions and postulates is the setting up of a system to engender belief.

There is plenty of room for "deviation". The thing is that no one has been able, as of this day, to come up with something better.

Science students are told to believe and they had better enjoy it.

What scientists are doing right now is putting all the known theories to the limit, either to confirm or to disprove them. I don't think science is a place for faith to thrive.
 

Applied mathematics seems there to be endured and never questioned from the outset.

1 + 1 = 2. You can question it, but, before that, you need to understand why it is held true that 1 + 1 = 2.

When something is as optional as it seems to be, arguments in support are expected to collapse into various logical fallacies.

You've got a point here. Cockroaches survive without math. However they're not engineers.

...Or are they?

But sines are not sqrt(-1). They are real-valued.

Alas, you slept through the class where they demonstrated that ALL real numbers are complex, too.

[SiliconWizard]

As we already said, there is no fundamental difference between irrational numbers and complex numbers (when having a non-zero imaginary part.) Both are defined by equations. Neither can *directly* be defined, so if your sense of what is "physical" and what is not is tickled here, both should tickle equally.

adx, you seem to be convinced that "real numbers" are physically real, while "complex numbers" are just a tool from human's imagination. That itself is a belief. It looks like the more accurate would be to say that you're "more comfortable" with real numbers, not that they inherently make more sense.

sqrt(2) is one solution of x^2 = 2. i is one solution of x^2 = -1. Big deal.

And, you have a problem with complex numbers because they are actually "two quantities" rather than just a single one.
But then you're OK with manipulating both sin and cos values, which are two quantities linked together.

Ultimately, I'm not sure this has really anything to do with science or reality, but mostly just with perception.

And IMHO, the universe and its physical reality does not freaking care about our qualms regarding numbers. It probably doesn't care about numbers altogether. Your perception does, and it's fine. Just maybe do not assume that you hold a "physical truth" just because it appears so in your own perception.

[snarkysparky]

The fundamental theorem of algebra says

Any polynomial of degree N will have N roots or solutions.

You can plot Y = x^2 +1 

It never crosses the x axis.  Yet the FTA says it has two roots.

The FTA has many proofs as I gather from some googling. 

See if you can invalidate one of the proofs.

https://mathbitsnotebook.com/Algebra2/Polynomials/POfundamentalThm.html

Here is a proof to get you started.

https://www.math.ucdavis.edu/~anne/WQ2007/mat67-Ld-FTA.pdf

[penfold]

Being expected to trust in intellectual authorities with absolutely nil room for deviation is pretty much the definition of faith. All this talk of theorems, deductions and postulates is the setting up of a system to engender belief.

There is plenty of room for "deviation". The thing is that no one has been able, as of this day, to come up with something better.
[...]

But... whilst it is faith and trust, the faith and trust should be in the rationalism and logical framework in which maths exists, as with science where it is the scientific method in which we must trust and believe - it is just a necesary contradiction that we must trust prior work to be valid, though proof and review processes contribute to rationalising that assumption. So, surely the argument there is that the maner in which maths is presented to engineering students, in the non-rigorous sense (i.e. very different to maths-degree maths), the student is expected to assume what is presented as true... but must trust the logic from which it is derived.

The fundamental theorem of algebra says

Any polynomial of degree N will have N roots or solutions.
[...]

But... that is a theorem of algebra, the complex number does not arrive until one starts to pose questions. Starting with natural numbers, all positive, whole numbered, countable, possesable etc quantities, we seek the answer to a+b=1 which is not defined for all both "a" and "b" in the set of natural numbers, enter the integer, the rational, the irrational and complex as we seek more or less general solutions to problems involving numbers in each set. But that's all find and dandy, but it isn't a general property of all sets of numbers and any relationship with reality depends on the formulation of the problem and it is a later attribution of significance which gives the numbers and significance or relationship to reality.

[SandyCox]

Being expected to trust in intellectual authorities with absolutely nil room for deviation is pretty much the definition of faith. All this talk of theorems, deductions and postulates is the setting up of a system to engender belief.

There is plenty of room for "deviation". The thing is that no one has been able, as of this day, to come up with something better.
[...]

But... whilst it is faith and trust, the faith and trust should be in the rationalism and logical framework in which maths exists, as with science where it is the scientific method in which we must trust and believe - it is just a necesary contradiction that we must trust prior work to be valid, though proof and review processes contribute to rationalising that assumption. So, surely the argument there is that the maner in which maths is presented to engineering students, in the non-rigorous sense (i.e. very different to maths-degree maths), the student is expected to assume what is presented as true... but must trust the logic from which it is derived.

The fundamental theorem of algebra says

Any polynomial of degree N will have N roots or solutions.
[...]

But... that is a theorem of algebra, the complex number does not arrive until one starts to pose questions. Starting with natural numbers, all positive, whole numbered, countable, possesable etc quantities, we seek the answer to a+b=1 which is not defined for all both "a" and "b" in the set of natural numbers, enter the integer, the rational, the irrational and complex as we seek more or less general solutions to problems involving numbers in each set. But that's all find and dandy, but it isn't a general property of all sets of numbers and any relationship with reality depends on the formulation of the problem and it is a later attribution of significance which gives the numbers and significance or relationship to reality.

You should read more carefully.

The fundamental theorem of algebra states that any polynomial of degree N will have N roots over the Complex numbers.

[penfold]

[...]

The fundamental theorem of algebra says

Any polynomial of degree N will have N roots or solutions.
[...]

But... that is a theorem of algebra, the complex number does not arrive until one starts to pose questions. Starting with natural numbers, all positive, whole numbered, countable, possesable etc quantities, we seek the answer to a+b=1 which is not defined for all both "a" and "b" in the set of natural numbers, enter the integer, the rational, the irrational and complex as we seek more or less general solutions to problems involving numbers in each set. But that's all find and dandy, but it isn't a general property of all sets of numbers and any relationship with reality depends on the formulation of the problem and it is a later attribution of significance which gives the numbers and significance or relationship to reality.

You should read more carefully.

The fundamental theorem of algebra states that any polynomial of degree N will have N roots over the Complex numbers.

Had I stated otherwise? Or did you pick up on my deviation to set theory? In either case, the relationship between, say, the width of a square field and the number of square cars I can tessellate within it does not intrinsically result in any complex numbers until I pose the question of what width would I need to hold a negative number of cars... my point remaining that the definition of the problem and how the problem is abstracted is important to prevent the descartian absurdity of "imaginary" numbers which the abstract nature of modern maths avoids by removing that very intrinsic link between numbers on the page and measureable quantities in reality.

[bsfeechannel]

But... whilst it is faith and trust, the faith and trust should be in the rationalism and logical framework in which maths exists, as with science where it is the scientific method in which we must trust and believe - it is just a necesary contradiction that we must trust prior work to be valid, though proof and review processes contribute to rationalising that assumption.

We don't trust science. Science is just a method for accumulating knowledge based exactly on distrusting current hypotheses.

So, surely the argument there is that the maner in which maths is presented to engineering students, in the non-rigorous sense (i.e. very different to maths-degree maths), the student is expected to assume what is presented as true... but must trust the logic from which it is derived.

I don't know where you had your engineering math courses, and I don't care, but where I learned about math, still in high school, they taught us that math has axioms, or postulates, that are provisional truths, subject to denial if convenient.

It is the case for instance of the so called parallel postulate: true in euclidean geometry; false in, you guessed it, non-euclidean geometry (that one Einstein used for the GTR). 

When we arrived in college, for our engineering degree, we all had this concept in mind. Postulates were accepted as ad hoc truths, we had to prove the deductions from these postulates and then test their application in the lab.

No one told us to trust or believe anything.

If the experience you had with math in your engineering degree is the one you described, I feel bad for you.

[penfold]

But... whilst it is faith and trust, the faith and trust should be in the rationalism and logical framework in which maths exists, as with science where it is the scientific method in which we must trust and believe - it is just a necesary contradiction that we must trust prior work to be valid, though proof and review processes contribute to rationalising that assumption.

We don't trust science. Science is just a method for accumulating knowledge based exactly on distrusting current hypotheses.
[...]

I think you're adding a bit more weight to "trust and belief" than I intended, I was using it in the sense that to pick up any one "eastablished" topic from "science", whilst it is possible to go through all the proofs from first principals and review the entirity of research and experiments, one must accept/trust/believe that the scientific processs has both been followed and is itself correct. So to clarify, it is not that I am suggesting we must have any trust in 'science the findings/theories/hypotheses", but in "science the method". We must also trust that rationalism is itself correct and capable of producing the answers we are looking for, it has worked so far, but we do not yet know whether we are just chasing our tails. Don't forget that practioners of science make up a very small percentage of the global population, the remaining majority still contains a large number of people who find it easier to accept religious teachings: the easy response would be to question their inteligence but; perhaps it is evident of a set of necesary beliefs that are just part of the human existance but not everybody recognises belief as belief when they believe something.

[...]
I don't know where you had your engineering math courses, and I don't care, but where I learned about math, still in high school, they taught us that math has axioms, or postulates, that are provisional truths, subject to denial if convenient.

It is the case for instance of the so called parallel postulate: true in euclidean geometry; false in, you guessed it, non-euclidean geometry (that one Einstein used for the GTR). 

When we arrived in college, for our engineering degree, we all had this concept in mind. Postulates were accepted as ad hoc truths, we had to prove the deductions from these postulates and then test their application in the lab.

No one told us to trust or believe anything.

If the experience you had with math in your engineering degree is the one you described, I feel bad for you.

I'm not really sure why you should feel bad for me, not only because I've not really described my maths education but because I've said nothing to describe how that method of teaching has affected my career or any general metric of success etc. It clearly hasn't done too well to help me justify a philosophical argument on an internet forum, but that is an incredibly small part of my life and one that there's no need for you to feel bad.

I suppose it is only a subtle difference in teaching approaches, as you were taught "...math has axioms, or postulates, that are provisional truths, subject to denial if convenient. ", I don't recall anybody suggesting "denial if convenient", more along the lines of alternative geometries can be constructed from alternative axioms (to varying degrees of validity and applicability to physical things), but ofcourse then require their own treatment... just a pushing vs pulling difference in phrasing I guess.

Nobody in my education has ever asked me to trust or believe anything either, but I accept that in order to actually get along and do some engineering, I won't always be deriving things from first principals and I must often trust what is written on paper and believe it to be true (a more appropriate term may be "to have confidence in").

[adx]

Being expected to trust in intellectual authorities with absolutely nil room for deviation is pretty much the definition of faith. All this talk of theorems, deductions and postulates is the setting up of a system to engender belief.

There is plenty of room for "deviation". The thing is that no one has been able, as of this day, to come up with something better.

Good point. But that is where the problem arises. People can only be expected to 'buy in' up to their own level of understanding (or gullibility), to feel like they are being led on a journey to the "truth". A kind of 'multilevel subjective experience', part of that unwinnable argument. If instead students were told to remain skeptical, not believe a thing, deviate, and revel in their own failure - well, that's not what most people would call an education. If no one believes deviation is possible, they won't try. Postgrad might be the first opportunity the unwashed masses have to think 'deviantly' (or critically).

... I don't think science is a place for faith to thrive.

Nor do I. But thrive it does.

It can only get worse as the expectation for human knowledge grows.

Applied mathematics seems there to be endured and never questioned from the outset.

1 + 1 = 2. You can question it, but, before that, you need to understand why it is held true that 1 + 1 = 2.

I often wonder if 1+1=1.999...

Applied mathematics sweeps right over number theory. (I can only assume that, but there is a nonzero chance that I had a dream that exactly replicated the lesson while I napped, so that monster that was chasing me was really an amusing anecdote given by the lecturer of having a dream of being chased by pi, while neglecting number theory as I slept through it all.)

That all reminds me of a post I drafted but didn't make, about my objection over pi.

When something is as optional as it seems to be, arguments in support are expected to collapse into various logical fallacies.

You've got a point here. Cockroaches survive without math. However they're not engineers.

...Or are they?

Well I never needed math, not complex numbers anyway. There is a gap. Pure math and physics is different from engineering, that was part of my argument that engineers would get by to some potentially large extent. The fact that they do and have got by seems to have caused you some anguish. It (hopefully not your anguish, because it does seem to be laid on a bit thick) can only get worse as the expectation for human knowledge grows.

But sines are not sqrt(-1). They are real-valued.

Alas, you slept through the class where they demonstrated that ALL real numbers are complex, too.

If you keep steering the argument to loop back around to that same point, then by the 5th or 6th time I might notice, leading me to reject it on purely contrarian grounds.

Valid though that may be, your point works both ways; if all real numbers can be considered complex, then any complex can be broken down into 2 'reals' with a by definition redundant (some might say nonexistent) imaginary part. Complex is a construction on top of reals, in a similar way to negative numbers are made from positive reals. I don't have to believe that complex numbers are "better" than real numbers, when I can twist my provisional faith to believe that real numbers are more fundamentally "numbery".

[adx]

As we already said, there is no fundamental difference between irrational numbers and complex numbers (when having a non-zero imaginary part.) Both are defined by equations. Neither can *directly* be defined, so if your sense of what is "physical" and what is not is tickled here, both should tickle equally.

Doesn't work for me. Irrational numbers fit into a number line, ordered. Complex numbers are not a quantity.

Exact irrational numbers? Then yes (except as by above), but I can define pi as 314/100 and keep going as far as I think I need. "Equations" are a very mathematics thing to say, pi = 314/100 is an equation, 1+1 = 2 is too. They all tickle my sense of what is "physical", but some are less equal than others.

That sense is driven more by what something is than how it is defined. That might sound like a nonsense, invalid to some, but to me I am more interested in the nature or existence of sqrt(-1) than the difference between pi and 3.1415926535897932384626433832795 (well that might be a bad example due to my objection over pi, but you get what I mean).

Reminds me of some scones I made a few days ago (after this post came in) that were such a twisted mess that it wasn't possible to tell where one started or even if they were connected (it was in a rush and experimental). Sizes ranged from full scone to microscopic crumbs (I ran out of liquid, couldn't be bothered adding or mixing more so dumped it out like that, made futile attempts to pound it into homogeneity with scone compressions, then proceeded to tear it into groups of congealment). I remain more alive than dead despite eating it all (except a piece that looks remarkably like a dead baby bird which remains - bad word choice). The question though is this: How many scones were there? Casts the existence (or place in the hierarchy) of whole numbers or integers into doubt.

adx, you seem to be convinced that "real numbers" are physically real, while "complex numbers" are just a tool from human's imagination. That itself is a belief. It looks like the more accurate would be to say that you're "more comfortable" with real numbers, not that they inherently make more sense.

My attempts to explain have possibly made my views seem more certain than they are. I do think real numbers inherently make more sense in some situations, based on observation, and resulting suspicion then having not yet found something to explain it (away). If you want to build a definition of belief that includes skepticism as a form of belief then you wouldn't be the first, and is fine given my recalcitrance in the face of consensus. But you could hardly call my description here "faith". In any event it is consistent with my argument that mathematics pits belief against belief - like these ones.

Real numbers have a certain inevitability to them in a field of work like engineering. Applications would fail if some numbers went missing (say voltages >12 or numbers with an odd integer part, or negatives even), or at least be severely impacted. If complex numbers lose a whole axis, you just use a vector. No need to concoct from an impossible equation to have a new dimension imagine itself into being so you can say that has "phsycial relevance". It seems so synthetic. Hard for me to believe it is anything else.

sqrt(2) is one solution of x^2 = 2. i is one solution of x^2 = -1. Big deal.

Sqrt(2) can be approximated onto the same number line as 2. i (or really 1i) must remain orthogonal to -1. I think that is a big deal.

And, you have a problem with complex numbers because they are actually "two quantities" rather than just a single one.
But then you're OK with manipulating both sin and cos values, which are two quantities linked together.

My problem is with people saying complex numbers are a quantity for seemingly no reason beyond what was drilled into them. Sin and cos are linked together by the sides of a triangle. Real and imaginary are linked together by insanity.

It's not really an argument anyway. I have no more a problem with complex numbers being two quantities, as I have with vectors. Unless you mean the word "number", which I will have to learn to take with a grain of salt.

Ultimately, I'm not sure this has really anything to do with science or reality, but mostly just with perception.

Maybe, but my argument is that perception has driven complex numbers to this place, people seem more concerned with these perceptions than what the realities might be.

And IMHO, the universe and its physical reality does not freaking care about our qualms regarding numbers. It probably doesn't care about numbers altogether. Your perception does, and it's fine. Just maybe do not assume that you hold a "physical truth" just because it appears so in your own perception.

I don't know how you can say that (actually I do but I'm just saying it for effect). The universe is full of "thermometers" of continuously-varying stuff - maybe not numbers as we know them (we can't define a scale for something until we know what "1" is), but with quanta, it's as close to unavoidable as I think possible. So I have no real option other than to assume "physical truth" for potentially any mathematical object. That's not to say I "believe" or have "faith".

[TimFox]

"Sin and cos are linked together by the sides of a triangle. Real and imaginary are linked together by insanity."
The second sentence in that quotation is both absurd and offensive to those with mathematical education.
Just because you find something to be icky does not render it insane.

[HuronKing]

"Sin and cos are linked together by the sides of a triangle. Real and imaginary are linked together by insanity."
The second sentence in that quotation is both absurd and offensive to those with mathematical education.
Just because you find something to be icky does not render it insane.

I tried to tell you guys. He can't stop worshipping at the altar of Rene Descartes. That's actually the religion here.