The Foundation of the "degree" as relation of Geometric Form

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Dear WTF:

I have exhausted my convincing power. I have not been able to convince you of these facts, that ought to have been self-evident to you as well since you are a mathematician (or seem to be familiar with math):
1. The sine and tangent functions can only be taken of angle measurements;
2. The radian unit is a unit of angle, and not of length (you insist it's a length);
3. In analytic geometry dimensionality is of a primary concern, even if in calculations and notations only numbers are shown.

You opposed me in all three of these.

I have not run out of patience, but I have run out of convincing reasons to present to you. I think I have done a pretty good job at it.

At this point I think we are no longer arguing against each other... you are incapable to see reason and logic. I think at this point we have been doing what all philosophers who oppose the other's views have been doing all along history: proving to the onlooking audience who is right and who is wrong.

I am satisfied with the fact that your opinions are so important to you that you can't change them. I am more concerned now with how well either of us has been convincing in our demonstration of arguments to the onlooking members of the audience.

I shan't now partake in this quixotic battle any further. I think I said enough so others who are onlookers can form an opinion. I gave up on convincing you.

Please accept the ensuing silence not as capitulation to your ideas, which I insist are false, but as a result of realizing you are incapable of comprehending clear, logical arguments, so it is futile to keep presenting them.

[wtf]

The Sin function is DEFINED otherwise. It is defined as the farther side of a right-angle triangle devided by the longest side of a right-sided triangle. T

In high school, yeah. Not once you take some math classes at the university level. This is actually an important point, because in ancient times, sines and cosines were good for measuring Farmer Brown's land. These days sines and cosines are functions of a real or complex variable that are used via Fourier series as the basis of digital signal processing and quantum mechanics. In the modern viewpoint, the original triangle definitions are outmoded and useless.

But tell me more about sin(1) because it's a really bizarre example. Suppose for sake of discussion I grant that sine inputs an angle (whatever that means, since you don't know how to define an angle. But let that pass).

What does your sin(1) example mean? Do you not think 1 is a real number representing the radian measure of some angle? Or what?

[wtf]

you are incapable to see reason and logic.

I'll put my posting history on this forum up against that statement.

Tell me about this sin(1) thing again. You don't think 1 radian is the measure of some angle?

I don't care if you stop responding without ever having engaged with a single point I've made. For example you totally ignored the point about the Lebesgue measure of a square and a line segment, which totally refuted your silly claim that there's a "different 4" for each usage.

I would be disappointed, though, if you stopped responding without ever having explained your point about sin(1), which still makes no sense. You think there's a difference between sin(1) and sin(pi). Other than their numeric values, there is no difference in quality or type. And you won't explain why you think there is. I'd be disappointed to never find out what your mistaken idea is.

By the way I scanned your posting history here and I see no mathematical content at all. Why are you even bothering to come here and insult people based on your own lack of knowledge of the subject? Your confusion about "two different 4's," your confusion about sin(1), and all your other remarks show that you don't know anything about math. Could that be the source of your frustration?

ps -- Did you write this?

I saw no logical proof in your argument. Apparently I can't see eye-to-eye with you in your logical arguments, because you go into mathematical abstractions, every time, which I am honest enough to say I don't understand.

Looks like you admitted your mathematical ignorance. And in this thread, you demonstrated it.

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Tell me about this sin(1) thing again. You don't think 1 radian is the measure of some angle?

I never said what you asked whether I thought. The entire argument was about my trying to prove to you that 1 radian, or any amount of radians, is an angle.

If you missed that... whellll.... what else did you miss.

[-1-]

The entire argument hinged on this:

A radian is a natural measure of the circumference of a circle. Namely it's the length of the circle's radius around the circumference of the circle.

I thought a radian was an angle, not a length.

As such wouldn't a radian be the angle between two straight line segments that start from the centre of a circle and end on the perimeter of the circle, which perimeter is the same length as the radius or the length of either of these two line segments?

Been a long time since high school.

Subsequently, you kept on insisting that a radian is a length. I kept saying that that is wrong, because a radian is an angle.

The entire argument grew out of that.

I even juxtaposed two of your statements, in one of which you said radian is a length, and in the other, you said radian is an angle. I showed that to you and you ignored it.

No, I did not ignore your texts. You kept saying that in "higher" math numbers are the only things that matter; I said dimensionality is just as important, even if they are not at all stated.

I even gave you and example that showed that to any reasonable person, and you replied,

Ok anyway I'll write more later. I thought your point about the square and the line segment was the most interesting thing you've said so far. I do have a response but it will take me a while to write up, it's kind of involved in my mind.

Involved in your mind? You simply could not refute it, and you chose to become illogical (due to narcissistic rage) over admitting defeat.

Finally you said that it's not only angles that can be taken the sine or the tangent functions of. That took the cake. I knew then you were going into a narcissistic rage.


I said you were incapable to see logic. You replied you put up your entire posting history against my claim.

I admit that outside this argument you have behaved very logically each time. In this argument, however, you failed, and again, I blame your narcissism. You are capable perfectly well of logical reasonable thinking, except when it would force you to admit you were wrong.

That's it in a nutshell.

[Noax]

I hate to butt in and especially counter what wtf is saying concerning a subject about which I know far less, but every definition I can find says that the radian is a measurement of an angle, not a length measurement of any kind.

A radian is a natural measure of the circumference of a circle. Namely it's the length of the circle's radius around the circumference of the circle.

I thought a radian was an angle, not a length.

https://upload.wikimedia.org/wikipedia/ ... adians.gif

From here ... https://en.wikipedia.org/wiki/Radian

From the 2nd link, my bold:

The radian (SI symbol rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees

This says that the arc length is numerically equal to the measurement of the angle that the arc subtends, not the length of the arc that the angle subtends.
Now I'm the first person to doubt a wiki quote in a situation like this, but you posted it, and every other definition of radians I found agrees that it is an angular measure, not a measure of arc length (in units of radius).

I'm not claiming to know better, just noting that I can find no reference to what you are describing. Yes, I know that sin/cos/tangent all have mathematical applications that go well beyond direct application to actual angles, and in those cases, radians is still used, but those applications don't have arc lengths either. Yes, sin(x) is just a function of scalar x in the end. X is a scalar, but a measurement of an angle is also a scalar.

[Noax]

… Mine was awful too, a screechy old woman who I thought was saying "Piece of oh, piece of oh," when she was actually saying "P-sub-0."

Har! Har! That reminds me about when as a mature student fresh out of Poly with a Philosophy degree I blagged my way into Imperial College to do an Msc in Foundations of Advanced IT (first course I'd done where the core books were written by my lecturers!!) and this guy kept saying subaye all through the tutorial, now normally I pride myself on asking the stupid questions as I've found others wanted to ask them as well but were too scared but this time some instinct told me best not, as I was well out of my depth here amongst mathematicians, physicists and computer science grads, so after the lesson I asked them and they were aghast I'd not heard of subscript i. "I'm in trouble here I thought." as did they.

I had excellent teachers in the early days, but there was one Asian college TA that could only string so many consonants together at once, and could not say 'function' without dropping the middle 'n', which distracted the teaching since we were too busy laughing. Finally he just wrote the word on the board and pointed to it with 'THAT!' each time he needed to say it.

I had another TA that was so bad, he read the role call in the first day and not one student recognized his own name. Somebody else took attendance for him and handed back the sheet with the names checked. Thankfully that class didn't need much instruction in any verbal language.

[wtf]

I hate to butt in and especially counter what wtf is saying concerning a subject about which I know far less, but every definition I can find says that the radian is a measurement of an angle, not a length measurement of any kind.

Most definitely. The Wiki page on the sine function starts by saying that the sine is a "function of an angle." Whatever that means, but no matter. The phrasing is ubiquitous. It's taught in high school and nobody ever thinks more deeply about it. And no harm is done if you want to think of it that way.

The only problem is that if you try to drill it down, in the end the sine is a function of a real number. Which you can think of as an angle if it makes you happy. But what does that mean? Not much. In the end it's semantics.

Consider another Wiki page, this one on Taylor series. https://en.wikipedia.org/wiki/Taylor_se ... _functions

Consider the Taylor series for the sine function, which is taught in freshman calculus.

sin(x) = x - x^3/3! + x^5/5! - ...

This formula is proved in calculus to be valid for all real numbers x. In complex analysis, the formula is shown to converge for all complex numbers x, and is taken as the definition of the complex sine function.

By the way, we see that sin(1) = 1 - 1/6 + 1/120 - ..., yet another way to regard sin(1).

Now, what kind of sense would it make for x to be considered an "angle" here? If by that you mean that by sin(1) we really mean sin(1 radian), then the sine must be an infinite sum of 1 radian - 1 radian^3/6 + 1 radian^5/120 ...

Then how could we add up all those powers of radians? In other words if you attach a unit like "radian" to the real number argument of the sine function, the sine function no longer makes any sense.

Now, did the author of the Taylor series article call up the author of the sine function article and explain this? Of course not. One person writes about the sine function in high school math terms, another person writes about the sine function's Taylor series, and the logical contradiction between these two pages goes unresolved. Yet a third person writes the article about Fourier series and quantum mechanics, and the sine is the imaginary part of the complex exponential function. It's really not a big deal. It's a change in viewpoint through increasing levels of mathematical sophistication.

The reason the trig functions have to be regarded ultimately as functions of real or complex numbers is because that is how they're used in the modern world. As I've noted, in ancient times sines and cosines helped people compute the area of their wheat fields. So the triangle definitions were primary. In modern times, sines and cosines are used as trigonometric series as in Fourier series, and are applied in digital signal processing (ie the entire Internet) and quantum mechanics. In the modern viewpoint the triangles are gone and what's left are pure functions of real or complex numbers.

Again, it makes no difference to anyone. If you're in high school or you don't care about higher math, you can think of the sine as inputting an angle. Once you start studying the sine function from a modern viewpoint, it has to be a function of a real or complex variable in order to make sense and to be conveniently used to run the Internet and serve as the mathematical foundation for quantum physics.

BUT! As I've pointed out, all this was already hinted at in high school. They showed you the right-angle definition of adjacent/hypotenuse, then the next day they showed you how to graph the sine function, and they put the entire real number line on the x-axis. If you could transport your adult brain back to that day in high school, you'd say to yourself, "Hey, they told us sine is a function of an angle, but those are actually the REAL NUMBERS there on the x-axis, and what they are calling an "angle" is really nothing more than a real number!".

From the 2nd link, my bold:
The radian (SI symbol rad) is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics.

Right. Note that a radian is a MEASURE of an angle, not the angle itself.

What is the radian measure of a right angle? A right angle subtends an arc of length pi/2 (no units) on the unit circle. So we define the radian measure of a right angle to be pi/2 radians. How else would you define it?

The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees

Yes, but that's a little circular. The measurement in radians is DEFINED as the length of that circular arc. For example what is the radian measure of a right angle? It's pi/2. What is the length of 1/4 of the way around the unit circle? Again it's pi/2. Coincidence? I think not!

This says that the arc length is numerically equal to the measurement of the angle that the arc subtends, not the length of the arc that the angle subtends.

But the measurement of the angle is defined as that arc length. So this definition is circular -- hey that's a pun, because of course trigonometry is about the unit circle, not about triangles. But the def really is circular. The measure of the angle is defined as the arc length the angle subtends.

Now I'm the first person to doubt a wiki quote in a situation like this,

Good instinct. One could spend their life fixing all the errors on Wiki.

but you posted it, and every other definition of radians I found agrees that it is an angular measure, not a measure of arc length (in units of radius).

How do you measure an angle? The measure of an angle is DEFINED as the arc length.

I'm not claiming to know better, just noting that I can find no reference to what you are describing.

If I sent you off to consult a text on complex analysis where they showed how to properly define the trig functions and the notion of angle, would you be happy? Probably not. If you want to think of the trig functions as inputting angles, that's fine with me, but how do you define an angle? The only sensible way to define an angle is that it's a real number, or perhaps a real number mod 2pi. And that the measure of an angle is the length of the arc it subtends. Think of the example of an angle of pi/2 subtending a quarter of the unit circle, the length of which is pi/2. When we called that angle pi/2, where did we get pi/2 from? From the length of the arc subtended by the angle.

Yes, I know that sin/cos/tangent all have mathematical applications that go well beyond direct application to actual angles, and in those cases, radians is still used, but those applications don't have arc lengths either.

That's right. The proper definitions are the infinite series, which input real or complex numbers.

Yes, sin(x) is just a function of scalar x in the end.

Now you are agreeing with me! What is another word for a scalar? It's just a real (or complex) number. Right?

X is a scalar, but a measurement of an angle is also a scalar.

A scalar is a real number. You agree or no?

Well I'm sure this will satisfy no one.

[Eodnhoj7]

To get back on track, however I am on "1"'s side (for once) on this one.

What we observe in measurement (as evidenced through the degree) is fundamentally "relation" where the foundation is a 1 line relative to another line which in itself is 1 line, with both lines being actual and the potential space which forms that relation between them as potential actual lines (considering if we invert two parallel lines through the space between them we get one line).

In simpler terms, a part is both composed of parts and composed further parts and in these respects all measurement through relations observes "1" as a means with the line as "1 directional" observing "1" and direction being inseperable in the respect relation is dependent upon directive qualities as one part is directed towards another part.

The degree, or radian for that matter, is founded under a relation of parts with all parts being composed of and composing "1" through "1".

[Noax]

I hate to butt in and especially counter what wtf is saying concerning a subject about which I know far less, but every definition I can find says that the radian is a measurement of an angle, not a length measurement of any kind.

Most definitely. The Wiki page on the sine function starts by saying that the sine is a "function of an angle." Whatever that means, but no matter.

Well, that's already wrong since it is possibly a function of the measure of an angle, but not of the angle itself. You've pointed out this distinction above.

The phrasing is ubiquitous. It's taught in high school and nobody ever thinks more deeply about it. And no harm is done if you want to think of it that way.

I admitted in my post that sine function is a function of real number (I called it a scalar), which need not be considered an angle. My cutting in on the discussion was concerning the definition of a radian, not the definition of the argument type of the sine function. It was here that you seemed to be quite adamant about a definition contrary to all the published ones I can find. Yes, the real-number measure of an angle is numerically identical to the real number ration of the arc length (assuming there is an arc somewhere) and the radius of that arc. But the word is applicable to only one of those quantities.

There are applications where the input is indeed typically presumed to be an angle, say in the modified sine function (found on my calculator) that takes degrees (or grads!) as an argument instead of radians. The function is a different function then, and doesn't have nice clean properties like being identical to its own 4th derivative.

Then how could we add up all those powers of radians? In other words if you attach a unit like "radian" to the real number argument of the sine function, the sine function no longer makes any sense.

Agree.

What is the radian measure of a right angle? A right angle subtends an arc of length pi/2 (no units) on the unit circle. So we define the radian measure of a right angle to be pi/2 radians. How else would you define it?

In posts to -1-, you seem to have been insisting on defining the radian as the measure of the arc which subtends a right angle, not a measure of the angle itself. Same real number result, but different definition. This is why I posted, since it seemed contrary to every definition I could find.

The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees

Yes, but that's a little circular. The measurement in radians is DEFINED as the length of that circular arc.

No, the above doesn't say that. It says it is numerically equal to the arc length, but not that it is defined as the arc length.

For example what is the radian measure of a right angle? It's pi/2. What is the length of 1/4 of the way around the unit circle? Again it's pi/2. Coincidence? I think not!

Of course not. It says they're numerically equal, but not syntactically equal. A wheel does not spin in terms of meters per second (a unit of speed, not angular velocity), but does spin at some angular velocity of radians per second. That both (rad/sec and meters/sec linear speed of an abstract point 1M from the axis) yield the same real number value doesn't mean that the units (of angular velocity and linear speed) are interchangeable.

My apologies for dropping out of mathematics and using the physical wheel example. I lack knowledge of terminology for temporal geometry: spinning circles and such. Perhaps seconds are still allowed in geometry despite the physical nature of a second.

This says that the arc length is numerically equal to the measurement of the angle that the arc subtends, not the length of the arc that the angle subtends.

But the measurement of the angle is defined as that arc length.

The value of the angle measurement is defined to be numerically equal to value of that arc length measurement, but the radian (which isn't mentioned in your definition above) is a unit of the former, not the latter. The choice of how wide of an angle constitutes one radian was determined by that arc length, but the definition is still that a radian is an angular measurement, not a length measurement.
The definitions I find (I quit after half a dozen) all say that a radian is the measure of the angle subtended by the unit arc length, not the measurement of the arc subtended by the angle.

Yes, sin(x) is just a function of scalar x in the end.

Now you are agreeing with me! What is another word for a scalar? It's just a real (or complex) number. Right?

Correct. I was talking about what kind of unit a radian is, not the interpretation of the scalar argument to a trig function or any other function for that matter.

X is a scalar, but a measurement of an angle is also a scalar.

A scalar is a real number. You agree or no?

I don't think either of us, nor -1- propose that the measure of an angle, or of the length of an arc yields anything other than a real number. The contention is about the name of the units assigned to those different concepts.

[wtf]

My cutting in on the discussion was concerning the definition of a radian, not the definition of the argument type of the sine function. It was here that you seemed to be quite adamant about a definition contrary to all the published ones I can find.

I'm mindful that I seem to be in disagreement with all published definitions. I have thesis. I will not invoke higher math, I'll only walk through Wiki articles. I'm organizing my thoughts to minimize the word count. Another day or two maybe. Just wanted to let you know I take your point to heart and am giving this some thought.

[Eodnhoj7]

My cutting in on the discussion was concerning the definition of a radian, not the definition of the argument type of the sine function. It was here that you seemed to be quite adamant about a definition contrary to all the published ones I can find.

I'm mindful that I seem to be in disagreement with all published definitions. I have thesis. I will not invoke higher math, I'll only walk through Wiki articles. I'm organizing my thoughts to minimize the word count. Another day or two maybe. Just wanted to let you know I take your point to heart and am giving this some thought.

"Degree" and the "Radian" (which exists through the degree) are both premised on the relation of lines as the foundation of their premises and in these respects are inherently relativistic.

[Eodnhoj7]

My cutting in on the discussion was concerning the definition of a radian, not the definition of the argument type of the sine function. It was here that you seemed to be quite adamant about a definition contrary to all the published ones I can find.

I'm mindful that I seem to be in disagreement with all published definitions. I have thesis. I will not invoke higher math, I'll only walk through Wiki articles. I'm organizing my thoughts to minimize the word count. Another day or two maybe. Just wanted to let you know I take your point to heart and am giving this some thought.

"Degree" and the "Radian" (which exists through the degree) are both premised on the relation of lines as the foundation of their premises and in these respects are inherently relativistic with the line existing as the foundation of relativism in the respect it exists as a part which is both composed of and composes further lines as parts.

In these respects the folding of the line, as both composed of and composing lines, observes an inherent element of alternation with in the nature of relativism.

I may have to elaborate on these points.

[Dalek Prime]

Personally, I prefer radians. 2(pi). Now that's easy.

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Personally, I prefer radians. 2(pi). Now that's easy.

2pi or not 2pi. That is the angle. (... just radiant of wisdom.)