should schools stop teaching the 3 4 5 triangle?

[confission]

the most amazing thing i've ever experienced in my life is the 3 4 5 triangle

if you make a triangle with an angle that is EXACTLY 90 degrees with bases that are EXACTLY 3 and EXACTLY 4, the third side is not 4.99938347234234! the third side is not 5.0000000000078475345! the third side is not 5.0000000000000000000000000000000000000000000000000000000000010020023423! the third side is 5!!! 5.00000000000000000000000000000000000000000000000000000000000000000000000!!!!! 5!!!!!

EXACTLY 5!!!!!

this was true billions of years before any man ever existed

this is the mathematical truth of reality!

that's the most unbelievable thing i've ever heard

i ain't ever seen anything like that

the closest thing i've ever seen that even comes close to being that amazing is the 5 12 13 triangle

i should devote my entire life to the 3 4 5 triangle. i don't know exactly what that means yet, but i should really do it

check out this. if you draw a square and mark the mid-points of the sides, you can make a 3 4 5 triangle!!
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html#345

also, if you draw a perfect circle inside of the 3 4 5 triangle...................

..................

..................

the radius of that circle..........

is ONE!!!!!!!!!!!!!!!!!!!!!!!!!!!!

i kid you not!

BUT HERE'S THE PROBLEM.

i don't believe it.

i think it's just made up.

i think they're all lying.

the hypothenuse is actually 5.0000000000000000000000000000000000000106745 but nobody's measured it that well yet. that's what i think

don't believe their lies. the 3 4 5 triangle is a myth that was formulated by a collaboration of mathematicians and clergymen thousands of years ago in order to try to prove that there is some sort of divinely organized structure to reality. but we all know that the chaotic nature of this universe would never actually have the 3rd side be EXACTLY 5. that is preposterous. that is an outrage!

when will modern mathematics quit accepting the 3 4 5 triangle lie that was spread through religious doctrine and perpetuated by christian society? when will the ugly truth be uncovered that the 3rd side is actually 5.0000000000000000000000000000000000000106745? if the scientific and mathematical community is supposed to be only based on unbiased measurements and logical empirical analysis, then why is the 3 4 5 triangle myth still accepted in 2009?

if our government allows this falsehood to be taught to our children in math class, then what's next? teaching our children intelligent design in science class?

if they're going to teach it to our children, they should at least emphasize that it is only an approximation and not exactly 5. giving the impression that the hypothenuse is exactly 5 influences kids to believe that there is a divinely organized structure to reality. religion should not be taught in public schools!

[Simplicio]

Their story seems to check out (http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI47.html)

[msmith537]

the word you are looking for is "pythagorean (http://en.wikipedia.org/wiki/Pythagorean_theorem)".


And excuse my ignorance about whatever the hell you are talking about, but does A^2+B^2 no longer equal C^2?

[Jragon]

Toke jopic?

[Simplicio]

And excuse my ignorance about whatever the hell you are talking about, but does A^2+B^2 no longer equal C^2?

In a spatially curved universe. I think the OP thinks he's making some sort of point about teaching Intelligent Design in public schools though.

[Attack from the 3rd dimension]

i think it's just made up.

i think they're all lying.

the hypothenuse is actually 5.0000000000000000000000000000000000000106745 but nobody's measured it that well yet. that's what i think

don't believe their lies. the 3 4 5 triangle is a myth that was formulated by a collaboration of mathematicians and clergymen thousands of years ago in order to try to prove that there is some sort of divinely organized structure to reality. but we all know that the chaotic nature of this universe would never actually have the 3rd side be EXACTLY 5. that is preposterous. that is an outrage!

when will modern mathematics quit accepting the 3 4 5 triangle lie that was spread through religious doctrine and perpetuated by christian society? when will the ugly truth be uncovered that the 3rd side is actually 5.0000000000000000000000000000000000000106745? if the scientific and mathematical community is supposed to be only based on unbiased measurements and logical empirical analysis, then why is the 3 4 5 triangle myth still accepted in 2009?

if our government allows this falsehood to be taught to our children in math class, then what's next? teaching our children intelligent design in science class?

if they're going to teach it to our children, they should at least emphasize that it is only an approximation and not exactly 5. giving the impression that the hypothenuse is exactly 5 influences kids to believe that there is a divinely organized structure to reality. religion should not be taught in public schools!

Soooo many possible answers, but in the interest of keeping this thread in the Cartesian consensual reality, I'm gonna go with a classic:

Cite?


:)

[pancakes3]

the hypothenuse is actually 5.0000000000000000000000000000000000000106745 but nobody's measured it that well yet. that's what i think


measured what that well?

[Skylark]

This thread discusses the Pythagorean theorem, which assumes the sum of the squares of the legs of a right triangle is equal to the square of the remaining side. It is a theorem, not a fact, regarding non-collinear points on a unique plane. Because mathematicians still disagree over the details of Euclidean geometry, this material should be approached with an open mind, studied carefully, and critically considered.

[Euphonious Polemic]

hmmmmm..... Teach the controversy in math class?

[Robot Arm]

i think they're all lying.

the hypothenuse is actually 5.0000000000000000000000000000000000000106745 but nobody's measured it that well yet. that's what i thinkI'll make you a deal. You give me a triangle with one side EXACTLY 3 and one side EXACTLY 4 (with an EXACTLY 90 degree angle between them) and I promise to spare no expense in measuring the third side.

[statsman1982]

Well, from a measurement-error viewpoint, the OP is right about the fact that reality isn't nice and clean like pure mathematics is. Empirically, sure, you'll never find a true 3-4-5 triangle made by humans because even the most sophisticated equipment in the world, with the most exacting tolerances ever imposed, would still have some degree of error, although probably even less than what the OP lists as the "real" length of the hypotenuse. The 3-4-5 triangle, just like the circle, square, and any other mathematical shape is a purely theoretical construct that does not exist in reality. BUT....for all intents and purposes, we can use the theorems developed from these perfect shapes to great benefit, because they are beyond "good enough."

Mathematics is only an approximation of reality, but it is a very good approximation, in fact, the best we can do. Our world is yet more complex than the most complex of predictive models, because of issues related to measurement (we can't be accurate to infinite decimals) and knowledge (every single conceivable variable cannot be put into the model, because we simply don't know what they all are and how they are related to one another). Yes, the 3-4-5 rule only applies to triangles that are axiomatically constructed, but for any real-world problem, that rule and others will work consistently, and thus predictably.

The same is true of the theory of evolution. As it stands, no, it is not 100% accurate, and it is far from complete. But it is a very good model (the best we have) for understanding the development of life, and its predictions are reasonably consistent and predictable.

[Try2B Comprehensive]

There is a 'triangle pose' in yoga. It isn't particularly hard either... to approximate. Funny though, I never seem to get quite into a right-angled triangle- it is always a little. bit. off.

Therefore the OP is correct. I have experienced it personally!

[Dissonance]

/yawn

Non-Euclidean Geometry (http://en.wikipedia.org/wiki/Non-Euclidean_geometry) - they actually teach it in public schools.

[Skylark]

Yeah, at the Boy's Preparatory Academy of R'lyeh. What vast, loathsome shapes are these?!

[runcible spoon]

Mathematics is only an approximation of reality, but it is a very good approximation, in fact, the best we can do.

Some would argue that the world is only an approximation of mathematics.

[RTFirefly]

Toke jopic?
That's my thought too.

[JohnnyMac]

Well, from a measurement-error viewpoint, the OP is right about the fact that reality isn't nice and clean like pure mathematics is. Empirically, sure, you'll never find a true 3-4-5 triangle made by humans.

I dispute this under the "infinite monkeys with infinite typewriters" theory ;).

[Grumman]

I dispute this under the "infinite monkeys with infinite typewriters" theory ;).
Problems:
There aren't an infinite humans creating 3-4-5 triangles.
There aren't an infinite yous trying to find the true 3-4-5 triangle.

[Francis Vaughan]

Euclidian geometry is pretty well sorted. I can't think of any issues that actually exist with it. But of course if you quuote Pythagoras, you need to preface that with "in a Euclidean space." If you don't, well it isn't true. And we don't live in a Euclidean world. If you get out a compas and measuring tape, and draw a nice triangle on the ground, it won't be right. Worse, the bigger you make it the worse it will be.

Not only that, even if you try to get away from the surface of the shere we live on, you won't find it easy to find any flat space. We live near gravitational bodies, and space is generally a bit warped. Even deep deep intergalactic space is not flat. The exact current theory on the nature of the universe varies a bit, but on the whole, the one thing that is pretty clear, it isn't flat.

Of course the OP's assertion that even in a Euclidian space the answer is 5.000....xxx is simply amusing. The deeper you look at the assertion, the more deep mathematics it violates. I would like to know what he has been smoking. Then I can avoid it. :D

[Marley23]

That's not so amazing. You know what's amazing? Grape Nuts cereal contains no grapes AND no nuts. Fight the power!

So, confission, I don't know if this is an I.D. parody or what, but if you want it to stay open, you should explain what you're driving at here.

[Attack from the 3rd dimension]

I just reread the OP, and I think it was written by Doctor Bronner.

[Ludovic]

The Moral 3-4-5's?

[Chief Pedant]

Schools should teach simpler concepts at lower levels and more complex concepts at higher levels.

Those levels are determined by grade level in general and mental capacity in particular.

As a parallel, I might teach a two year old very simple rules about how to speak English correctly. When he is in college he might learn all of those rules are wrong and there are no real "rules." It was still OK to teach the simple stuff.

Remove the bug from your butt about the 3-4-5 being taught in schools. If you want to defend that the concept wrong, go right ahead and explain why.

At this point the structure of your OP suggests you are a nutcase rather than someone with an actual concern. If you didn't mean it that way, please clarify.

[CalMeacham]

To the OP:

If that amazes you, consider this:


1.) Every odd number is the short side of a right triangle in which the longer side and the hypoteneuse differ by exactly one.

2.) If the short side has length 2n + 1, then the longer side has length 2n^2 + 2n (and the hypoteneuse is obviously 2n^2 + 2n +1). n = 1 gives you 3-4-5. n = 2 gives you 5-12-13, and so on.

3.) You can inscribe a circle in each of these triangles, and the radius of the circle will be ---- n ! (Not n! -- I'm expressing surprise, not factorial)

4.) Furthermore, where the inscribed circle touches each side divides that side into integral lengths, as well.



By the way, the Egyptians knew about the 3-4-5 right triangle long before Pythagoras.

[Keweenaw]

the word you are looking for is "pythagorean (http://en.wikipedia.org/wiki/Pythagorean_theorem)".


And excuse my ignorance about whatever the hell you are talking about, but does A^2+B^2 no longer equal C^2?

Actually I think the word he's looking for is Pythagoreanism (http://en.wikipedia.org/wiki/Pythagoreanism) or possibly Platonic idealism (http://en.wikipedia.org/wiki/Platonic_idealism)

Or for fans of Neal Stephenson's Anathem the Hylaean Theoric World (http://anathem.wikia.com/wiki/Hylaean_Theoric_World)

[DanBlather]

My guess, is that the OP is a form of witnessing. Isn't the 3 4 5 triangle amazing. Something like that could not occur naturally, just like life. If we teach the 3 4 5 triangle as "gospel" why can't we acknowledge the role of God in the design of man.

Know what else is amazing? Cancer.

[Bryan Ekers]

Well, if one wants to create a strawman rebuttal, the OP is wrong: The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.

[SentientMeat]

confission, are you saying that the square root of (32 + 42 = 25) is not precisely 5?

Also, when you say that Pythagoras' Theorem (actually Baudhayama's centuries beforehand) was "true for billions of years", what precisely do you mean?
As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality.
In order to do useful maths, you must accept certain axioms (http://en.wikipedia.org/wiki/Peano_axioms), and one cannot prove those axioms without appealing to others just as arbitrary.

[Attack from the 3rd dimension]

On rerereading, I think the OP is trying to make some sort of point about schools and teaching, but I can't figure out if its against ID, against Evolution or against paragraphs.

[Lemur866]

The OP is arguing that the features of the 3-4-5 right triangle are so amazing it proves that God exists. Therefore, the liberals should be scared to teach the truth about it in schools, just like they are scared of creationism, because what if Timmy learns the amazing features of this triangle, and therefore realizes that God exists?

I guess it's kind of amazing that 9+16=25. Makes you think, doesn't it? Also, what are the odds that the square root of 64 would be exactly 8? Not 8.0000000001 or 7.999999999, but EXACTLY 8. Pretty far fetched if you ask me.

The alternate explanation is that this is a case of Poe's law, and the OP is writing a parody of how he thinks creationists think.

[Euphonious Polemic]

Or, he's been drinking the bong water again.

[Gyrate]

Yawn. Come back when you want to discuss eiπ + 1 = 0.


And then go away again, because I sure don't understand that shit.

[tomndebb]

I just reread the OP, and I think it was written by Doctor Bronner.Come on! Dr. Bronner was smart enough to develop a pretty decent soap. There is no way that he could be accused of having written the OP--even if you base your judgment on the sides of his soap bottles.


I'm with Marley, however, that, barring a return post explaining his intention, this thread is about to die a swift death.

[Attack from the 3rd dimension]

Come on! Dr. Bronner was smart enough to develop a pretty decent soap. There is no way that he could be accused of having written the OP--even if you base your judgment on the sides of his soap bottles.


I'm with Marley, however, that, barring a return post explaining his intention, this thread is about to die a swift death.

I have nothing but respect for Dr. Bronner, his soap, his story and his writing style. In fact, I am currently redolent of peppermint thanks to him.

[ElvisL1ves]

Yawn. Come back when you want to discuss eiπ + 1 = 0.


And then go away again, because I sure don't understand that shit.

It's just a special case of vector algebra in the complex plane. R ei θ = R (cos θ+ i sin θ) where R = 1 and θ = π .

[Stealth Potato]

Dear Og, I had my suspicions all along, but this just proves it: confission is the Timecube guy! :eek:

[Darwin's Finch]

I guess it's kind of amazing that 9+16=25. Makes you think, doesn't it?

My mind? BLOWN!

[Kobal2]

Dear Og, I had my suspicions all along, but this just proves it: confission is the Timecube guy! :eek:

Yeah, I also thought the OP read something like Timecube Does Pythagore!. Mind you, as stoner entertainment it has the value it lacks in the math department; just like the original.

[Thudlow Boink]

At this point the structure of your OP suggests you are a nutcase rather than someone with an actual concern. If you didn't mean it that way, please clarify."P.S. I am not a crank."

[Thudlow Boink]

Pythagoras' Theorem (actually Baudhayama's centuries beforehand)Cite?

I looked up "Baudhayama" (which referred me to "Baudhayana"), and found:

A Wikipedia page (http://en.wikipedia.org/wiki/Baudhayana) that claimed "The now known Pythagorean theorem is believed to have been invented [:dubious:] by Baudhayana. This theorem is used to calculate the sides of a right angle triangle. There is evidence to this fact exists all over India.[citation needed]"

A page (http://www.gap-system.org/~history/Biographies/Baudhayana.html) which claims he was "Born: about 800 BC in India / Died: about 800 BC in India" (which doesn't seem to leave much time for doing mathematics), and

A page (http://www.experiencefestival.com/a/Baudhayana/id/1928990) which says He is the author of one of the earliest and most important Sulbasutras — appendices to the Vedas giving rules for the construction of altars — called the Baudhayana Sulbasutra, which contained several important mathematical results.

The Baudhayana Sulbasutra contains one of the earliest references to what is known today as the Pythagorean theorem.

The rope which is stretched across the diagonal of a square produces an area double the size of the original square.
This is a special case of the Pythagorean theorem for a 45° right triangle.Stating (without proof) a special case of the Pythagorean theorem hardly seems to justify getting the theorem named after you. But then, in mathematics, it's not uncommon for theorems, etc., to be commonly known by names other than those of the person who's really responsible for them.

[adhay]

Some would argue that the world is only an approximation of mathematics.

Or mathematics is the idealization of the world.

[begbert2]

I seriously can't tell how the OP intends us to match up the categories (3-4-5, 3-4-5 is WRONG) with the analogized categories (evolution, creationism). I think the confusion comes from the fact that his analogy is presented from the position of a complete moron who doesn't comprehend math at all. Is he arguing that we should "teach the controversy"? Or is he arguing that only a moron would want to do so?

I'm so confused...

[garygnu]

What about the 5-12-13 triangle?

[Nancarrow]

What about the 5-12-13 triangle?

Splitter.

[Salix]

Maybe the OP is advocating that schools recognize answers that are calculated using floating point operations on an old Pentium processor from 1994 (http://en.wikipedia.org/wiki/Pentium_FDIV_bug). I imagine another issue up for debate is whether the current year is 2009 or 1909.

[toodlepip]

Well, I never trusted triangles anyway. So aggressive, all lines and pointy bits. And all that empty space in the middle. I just tried it out, I got my crayons out and drew a triangle, and guess what: The sides weren't 3, 4 or 5 at all! And there were no right angles. In fact, one of the lines wasn't even straight! The OP is right, it's all a big lie!!!

There must be some way of protecting the children from all this brainwashing, maybe they could wear tin foil helmets in school or something.

[Thudlow Boink]

Triangle Man hates Person Man.

[Lumpy]

To take the OP perhaps more seriously than needed: (:))

Obviously 3 squared (9) plus four squared (16) equals five squard (25) by any definition of arithmatic worth discussing. So more properly speaking, the question would be whether a triangle with those sides would really have an *exactly* 90-degree angle, not infinitesmally greater or lesser than 90 degrees. Actually, more broadly* the question is whether all right triangles meet the pattern of (side1)squared plus (side2)squared equal (side3)squared, whatever the actual side length.

The answer, proven long ago, is "yes". Given some axiomatic presumptions about plane geometry and about what constitutes logical proof, it can be shown that by definition of the terms used than the theorem must be true.

To my mind, what's even more interesting than 3²+4²=5² are the limitless series of triangles in which the two short sides differ by 1, coming ever closer to being almost equal in length. I spent quite a bit of time rediscovering how these calculated. The next in the series is 20²+21²=29², the one after that is 119²+120²=169², and so on. I once worked out a table of these to examples where two astronomically long sides would only differ by the width of a hydrogen atom.

*more broadly still, any triangle's sides can be related by a formula which only reduces to exact squares for the special case where the angle opposite the longest side is 90 degrees. This forms the basis of trigonometry.

[confission]

Well, from a measurement-error viewpoint, the OP is right about the fact that reality isn't nice and clean like pure mathematics is. Empirically, sure, you'll never find a true 3-4-5 triangle made by humans because even the most sophisticated equipment in the world, with the most exacting tolerances ever imposed, would still have some degree of error, although probably even less than what the OP lists as the "real" length of the hypotenuse. The 3-4-5 triangle, just like the circle, square, and any other mathematical shape is a purely theoretical construct that does not exist in reality. BUT....for all intents and purposes, we can use the theorems developed from these perfect shapes to great benefit, because they are beyond "good enough."

Mathematics is only an approximation of reality, but it is a very good approximation, in fact, the best we can do. Our world is yet more complex than the most complex of predictive models, because of issues related to measurement (we can't be accurate to infinite decimals) and knowledge (every single conceivable variable cannot be put into the model, because we simply don't know what they all are and how they are related to one another). Yes, the 3-4-5 rule only applies to triangles that are axiomatically constructed, but for any real-world problem, that rule and others will work consistently, and thus predictably.

The same is true of the theory of evolution. As it stands, no, it is not 100% accurate, and it is far from complete. But it is a very good model (the best we have) for understanding the development of life, and its predictions are reasonably consistent and predictable.

but what's the difference between reality and mathematics?

"reality" is what i see and measure outside of myself? and mathematics is at the core of where who i am actually touches reality head on?

mathematical logic is in my being?

mathematical logic is what i know for sure, but "reality" is what i can only make approximations of?

if the point in which whatever i truly am touches reality is not itself part of reality, then what is it?

what do you make of the electrical light energy that makes up logic in a human's head? is there any reality to it, or is it all complete fiction?

where does our idea of the number 1 come from? what is the explanation of our idea of the number 1?

what does the number 1 mean, on its own, without any units put to it? what does it mean that we can use the number 1 without ever assigning any units to it? if the number 1 does not have any real existence in the "real world" without units, where does the number 1 come from? does it not come from reality? is it not real? how can we talk about it if it's not a real part of reality?

how do you explain the mapping of the 3 4 5 triangle on a real sheet of paper using real ink (even if it's not exact, it's close enough that it appears amazing, because it maps to something real in our heads)

are the light particles that are bouncing in our heads not real? what are they? and how do you explain them? how on earth would the real world.. the sun and the earth, go about putting light particles in your head so that you can think of a 3 4 5 triangle?

why would the universe create a human through evolution and place the thought of a 3 4 5 triangle in its head?

isn't your head a consequence of real life?

how do 3 4 5 triangles exist in real life in your head?

how do we know what a 3 4 5 triangle is if it doesn't exist, at least in our head. and if it exists in our head, then can it be said that what is in our head is a real 3 4 5 triangle. we hold light particles in our head in the shape of a 3 4 5 triangle?

how can we form light particles in our head in the shape of an exact 3 4 5 triangle when we don't even know the exact angles of the 2 non-90 sides?

how can a real 3 4 5 triangle exist in my head (i know what it is) and not exist in my head (i don't know the exact angles) at the same time?

and if the real 3 4 5 triangle does not exist in my head, then shouldn't it be said that there is no truth at all to a 3 4 5 triangle? and nobody should even mention them or use them, because they are a myth?

if what's happening inside my head is not real life, then what is it? why would evolution go through millions of years of struggling just to create something that's not real? why is the brain not real?

[pmwgreen]

What about the 20-21-29 triangle? (On edit, Lumpy beat me to it.)

[Kobal2]

why is the brain not real?

I think most of us are beginning to doubt the reality of yours, so you've got a point there.

[statsman1982]

...
where does our idea of the number 1 come from? what is the explanation of our idea of the number 1?

(snip)

and if the real 3 4 5 triangle does not exist in my head, then shouldn't it be said that there is no truth at all to a 3 4 5 triangle? and nobody should even mention them or use them, because they are a myth?

if what's happening inside my head is not real life, then what is it? why would evolution go through millions of years of struggling just to create something that's not real? why is the brain not real?

I forgot I was dealing with the Sperm Guy :).

As to your first question (what is "1"), well, good question. We probably (IANA mathematical historian) got the idea for it by making notches sticks to keep track of animals. Adding up "1s" gives you every other number. But then, what is a number? It is also a mental construct, eminently useful to survival but still theoretical. You can show me nine of something, but you cannot show me plain ol' nine without making up a symbol for it. But once we have the symbolism, we can do all sorts of things. In fact, one of the most important concepts of modern mathematics is that of notation. It's visual shorthand that everyone agrees on to communicate concepts; if you want once instance demonstrating how useful notation is, look up Liebniz notation (http://en.wikipedia.org/wiki/Leibniz_notation). It has made understanding derivatives easier for students for centuries!

On your second point, I thought I covered this in my last post, but here goes. We use the idea of 3-4-5 triangles (and perfect circles, squares, lines, etc.) because they are useful. If you build me something that conforms to, within the smallest tolerance of any measurement device on earth, a triangular structure with sides 3-4-5, all the properties derived from the axiomatic definitions will apply, but only insofar as we can measure them. We can't measure to infinite decimals, so there is no practical way of verifying the exactness of the 3-4-5 properties other than to note that they logically follow from the axioms of geometry. So far, all triangular structures following the 3-4-5 design obey (up to our ability to measure them) the properties predicted by the axioms, so we dub the concept useful and teach it.

Lastly, there is nothing that says that what goes on in my brain has to be "real life." Heck, a good part of the entertainment business is built on the idea that people like to see things that can't happen in reality. It's called imagination. The mechanics of how we imagine lie in the chemical reactions that take place between neurons; IANA neuroscientist, so I don't have the specifics. But to claim that what goes on in one's head has to be real because the brain is real doesn't follow.

[Indistinguishable]

Yawn. Come back when you want to discuss eiπ + 1 = 0.


And then go away again, because I sure don't understand that shit.
Keep in mind some basic definitions/properties:

ert: the amount a quantity multiplies by after t time-units, if its rate of change at any moment is always r (per time-unit) * its current value [that this depends only on the product rt is apparent with a little thought about the arbitrariness of the time-unit]
π: the distance halfway around a circle of radius 1
i: square root of -1

Now, let's find another way to look at i: Suppose you were to take some vector and rotate it 90° (a quarter revolution). Now rotate it 90° again. What've you accomplished in total? You've rotated a full 180°, a half-revolution; you've ended up with the opposite vector from what you started with. That is, (rotate 90°) * (rotate 90°) * v amounts to the same thing as -v; i.e., in this sense, (rotate 90°)^2 = -1. That is to say, "i" essentially means "90° rotation", nothing more esoteric than that. This is the fundamental insight.

Ok, so what? So now let's think about what happens if you were to keep rotating a vector at a constant rate. Well, as the endpoint of that vector traces out a circle, its velocity at any moment will be perpendicular to the vector itself; that is, it will be aligned with the vector rotated 90 degrees. And, for convenience, let's pick units of distance and time such that the vector itself has length 1, and its endpoint is always moving at a speed of 1. Thus, we will have that the endpoint's velocity (i.e., the vector's rate of change) at any moment is always equal to the vector itself rotated 90°.

Thus, by the defining property of e above, after t time-units of such rotation, the vector will have multiplied by e(rotate 90°)*t, which is to say, by eit. And since the endpoint is moving at speed 1 around a circle of radius 1, it will take precisely π time-units for this to go halfway around the circle (this being the defining property of π above). That is, eiπ = 180° rotation = -1.

That's all there is to it. When you boil down past all the defining properties, that "eiπ = -1" simply means "If a quantity whose rate of change, per time-unit, is always its value rotated 90°, then after the number of time-units equal to the ratio between half the distance round a circle's circumference and its radius, that quantity will have reached the negation of its original value", just cast in slightly different words. And that this is true follows immediately from and essentially amounts to nothing more than the simple geometric fact that if I keep facing you while moving to my right, then I will trace out a circle around you.

[Lemur866]

Thank you Indistinguishable. That's the first time I've ever understood how that relation could possibly make sense.

[pancakes3]

this thread is an exemplary example of a little learning being a dangerous thing. i still can't figure out if confission is just being contrary for contrary's sake or if he actually believes what he writes.

the pythagorean theorum is well proven.
http://www.cut-the-knot.org/pythagoras/

pythagorean triples are just integer values that fit into the pythagorean theorum nicely. 3-4-5 is just the most common example.

numbers exist just as much as the units they keep track of. if you don't believe in the number 1, what's to stop you from losing faith in a mile, a degree, or even discrete objects like books or sheep? this conversation went from mathematical to philosophical without ever deviating from moronic.

[statsman1982]

this thread is an exemplary example of a little learning being a dangerous thing. i still can't figure out if confission is just being contrary for contrary's sake or if he actually believes what he writes.

the pythagorean theorum is well proven.
http://www.cut-the-knot.org/pythagoras/

pythagorean triples are just integer values that fit into the pythagorean theorum nicely. 3-4-5 is just the most common example.

numbers exist just as much as the units they keep track of. if you don't believe in the number 1, what's to stop you from losing faith in a mile, a degree, or even discrete objects like books or sheep? this conversation went from mathematical to philosophical without ever deviating from moronic.

I don't think confission is disputing the fact that the Pythagorean Theorem can be easily proven. I think his questions here, like those he wrote for a thread on what a sperm feels, are intended to be philosophical. Heck, the Sperm thread (http://boards.straightdope.com/sdmb/showthread.php?t=539219&highlight=sperm+feel) managed to generate 83 posts out of a pretty simple question.

The point I am trying to make, and I am far from the first one to make it, is that mathematics is an abstraction and simplification of reality. It's based on axioms and theorems that logically follow from them. When stepping into the real world, all one ever finds are very accurate--so accurate that the deviation doesn't matter--approximations of the purely theoretical ideas of "triangle", "circle", "square", etc. Sure, for all intents and purposes, we can treat these objects as if they were "perfect" in the axiomatic sense, and we do. But the OP seemed to be wanting a very literal discussion of what is real and what isn't. Another example more dear to my heart is the idea that data we observe can come from a normal distribution. The theoretical normal has infinite tails, and I can't think of anything real that is "infinitely small" or "infinitely large". But chosen properly, the normal can model real data quite well, since the activity in the tails becomes arbitrarily close to 0 the further one goes out.

[Bryan Ekers]

In high school, I wrote a program to find Pythagorean triples and eventually discovered this pattern:

[2n(n+1)]2 + (2n+1)2 = [2n(n+1)+1]2, for all n = N0

This generates (where a2 + b2 = c2):

0 1 1
4 3 5
12 5 13
24 7 25 etc.

Every odd number is a valid b-term, and a and c differ by 1. I only found out much later that other forms of this equation exist to generate other triples.

[Keweenaw]

I seriously can't tell how the OP intends us to match up the categories (3-4-5, 3-4-5 is WRONG) with the analogized categories (evolution, creationism). I think the confusion comes from the fact that his analogy is presented from the position of a complete moron who doesn't comprehend math at all. Is he arguing that we should "teach the controversy"? Or is he arguing that only a moron would want to do so?

I'm so confused...

The op's mathematical premise is not a unique one. It's essentially Intuitionism (http://en.wikipedia.org/wiki/Intuitionism). It could certainly be considered that L. E. J. Brouwer (http://en.wikipedia.org/wiki/Luitzen_Egbertus_Jan_Brouwer)was misguided, but probably not a complete moron.

Is there is the connection between consideration of the abstract 3 4 5 triangle, which does not exist in the physical world, and the consideration of supernatural to explain the source of goodness?

Don't both atheism and intuitionism have a some connection in the form of anti-realism?

[CurtC]

Don't both atheism and intuitionism have a some connection in the form of anti-realism?

???

[Bryan Ekers]

Don't both atheism and intuitionism have a some connection in the form of anti-realism?

Only if you can show them both recapitulating phylogeny, preferably caught in the act in a seedy motel room.

[gonzomax]

Indian chief has 3 pregnant women.
He puts 1 in a teepee with a deer skin
He puts the 2nd in a teepee with an elk skin
He looks about and all that he has left is a hippopatamus skin for the 3rd one
Next day they all gave birth. The last one had twins others had singles
The chief observed
The squaw on the hippopatamus hide is equal the the sum of the squaws on the other two hides.

[Marley23]

I think most of us are beginning to doubt the reality of yours, so you've got a point there.
Personal insults aren't allowed in this forum.

[begbert2]

The op's mathematical premise is not a unique one. It's essentially Intuitionism (http://en.wikipedia.org/wiki/Intuitionism). It could certainly be considered that L. E. J. Brouwer (http://en.wikipedia.org/wiki/Luitzen_Egbertus_Jan_Brouwer)was misguided, but probably not a complete moron.No, it's denial of intuitionism. It's a statement that the math that is correct based on proofs and internal logic is wrong. It's a ratinalle-less rejection of mathematics itself.

It's kind of like what it takes to get creationism into a science class, actually.

Is there is the connection between consideration of the abstract 3 4 5 triangle, which does not exist in the physical world, and the consideration of supernatural to explain the source of goodness?No. Triangles exist in the abstract model called "mathematics/geometry" which does not exist in reality, but is useful because when reality is found to match the model to some degree, the model can be used to infer additional useful data that is accurate in proportion to the degree that the chunk of reality you are observing matches the model.

The supernatural is also an abstact model that does not exist in reality, but unlike math it is useless - you cannot match the supernatural elements with reality directly, since we can't find anything resembling actual ghosts and gods running around, and attempts to use observed "results" to infer the presence of the supernatural consistently fail to yeild better predictive power than chance, once you correct for observer bias.

Using the supernatural to "infer" the souce of goodness is like "inferring" that ((334 * 999 + 1) * 37 - 12345678) happened just because you run across the number 1. In theory that might have been the math that led to the answer you see, but there's no real reason to think so.

Don't both atheism and intuitionism have a some connection in the form of anti-realism?I looked up anti-realism, and atheism doens't qualify. Quoting wikipedia (http://en.wikipedia.org/wiki/Anti-realism), "In philosophy of science, anti-realism applies chiefly to claims about the non-reality of "unobservable" entities such as electrons or DNA, which are not detectable with human senses." Which is to say, it disregards all secondary evidence - no matter how consistent or compelling the secondary evidence is. This is not something atheists generally do - atheists not only almost universally believe in electrons and DNA, they also would accept sufficently compelling secondary evidence for God. The problem is, of course, that all the so-called evidence for god is the crudest level of misinterpretable personal anecdotes - and it's wildly inconsistent across varying populations besides.

Though this is just the crap argument that skepticism isn't justifiable repeated again, I will say that this is one of the most intellectually complex phrasings of it I've ever seen. Kudos.

[begbert2]

I'm going to give you better answers than your rambling questions deserve.

but what's the difference between reality and mathematics?Reality is real. Mathematics is an abstract model - it's all in your head. However, if you find things in reality that approxomate things in mathematics, the math can tell you additional facts about the real things, to the degree that reality matches the math.

"reality" is what i see and measure outside of myself? and mathematics is at the core of where who i am actually touches reality head on?Yes to the reality part. The math stuff here is gibberish.

mathematical logic is in my being?Apparently not - and even those of us who have it, had to learn it; it's not inherent, not any more than the Klingon language is.

mathematical logic is what i know for sure, but "reality" is what i can only make approximations of?Mathematical knowledge can be known for sure, though you might not know it. However, the only thing that math is sure about is other math. When you start to talk about the bits of reality that resemble the things in the abstract math, then errors will exist between the abstract math results and the actual reality, because the reality doesn't precisely match the model.

That said, reality can be known directly, to the limits of your sensory apparatus. Those aren't perfectly accurate, but that's not math's fault.

if the point in which whatever i truly am touches reality is not itself part of reality, then what is it?Math isn't where you "touch reality" (with me, that's my skin), so this is gibberish.

what do you make of the electrical light energy that makes up logic in a human's head? is there any reality to it, or is it all complete fiction?Electricity isn't fictional. (Dunno about "light" energy.) There is as much reality to your mind as there is to your computer browser; both are things that exist due to and created by the organized behavior of electrons operating within and interacting with a material substrate.

where does our idea of the number 1 come from? what is the explanation of our idea of the number 1?Most humans learn very early on how to distinguish between separate objects - the awareness that distinct objects exists gives rise to the notion of the number 1 - each distinct object that you can recognize is 1 object.

what does the number 1 mean, on its own, without any units put to it? what does it mean that we can use the number 1 without ever assigning any units to it? if the number 1 does not have any real existence in the "real world" without units, where does the number 1 come from? does it not come from reality? is it not real? how can we talk about it if it's not a real part of reality?The number 1 without units is an abstraction from individual objects 'with' units - which is to say, you go from "an orange" to "1 orange" to "1". The abstraction exists because there are common properties of things that all can be describes as "1": for example, when you have two of them together, you get "2". Between when you abstract it (take the units off) and when you later use it as a model of some part of reality (put the units on), it does not exist in reality; it's just part of the mathematical model. You can do things with it, but all the things you do are still part of the abstract mathematical model.

how do you explain the mapping of the 3 4 5 triangle on a real sheet of paper using real ink (even if it's not exact, it's close enough that it appears amazing, because it maps to something real in our heads)Err, it's ink on paper. That's my explanation. It's usually done to help other people learn things about the abstract thing called "math", the same way that books exist to help people learnd about the abstract thing called "Klingon".

And it's mostly cool because integers unexpectedly popping up is cool in math like this. It's not cool because it maps to something "inside year head" (in the abstract model called "math") - there's lots of boring things that are equally mathematical. 13 + 2 = 15, for example.


are the light particles that are bouncing in our heads not real? what are they? and how do you explain them? how on earth would the real world.. the sun and the earth, go about putting light particles in your head so that you can think of a 3 4 5 triangle?I dunno about you, but I don't have a big hole in my head to let light in. And electrons are everywhere, and the beasties that stumbled on putting them in clever patterns outcompeted enough other beasties to survive. That's all there is to it.

why would the universe create a human through evolution and place the thought of a 3 4 5 triangle in its head?The universe didn't put 3 4 5 in my head. Some human figured it out, based on math which humans made up from their ability to tell one thing from another, and then eventually some teacher tought it to me. That's how it got in my head, and the universe couldn't care less.

isn't your head a consequence of real life?Yes.

how do 3 4 5 triangles exist in real life in your head?The same way the res of my thoughs do - they're arrangements of chemicals and electrons that are meaningful when interpreted according to rules encoded in the electrochemical arrangements in the other parts of my head. Your computer's memory works exactly the same way.

how do we know what a 3 4 5 triangle is if it doesn't exist, at least in our head. and if it exists in our head, then can it be said that what is in our head is a real 3 4 5 triangle. we hold light particles in our head in the shape of a 3 4 5 triangle?Look at your screen, here: R. See that letter R? It's held in your computer's memory, but not in an arragment of "light particles":rolleyes: in the shape of a letter R. It's encoded. Just like our thoughts are.

how can we form light particles in our head in the shape of an exact 3 4 5 triangle when we don't even know the exact angles of the 2 non-90 sides?We can't, but we don't have to, because our thoughts are encoded. And further, all we encode are the things we actually know, which is untroubled by the fact we don't know the other angles (or at least, we're not usually taught them when we learn the 3-4-5 triangle thing.)


how can a real 3 4 5 triangle exist in my head (i know what it is) and not exist in my head (i don't know the exact angles) at the same time?No "real" 3 4 5 triangle exists in your head.

and if the real 3 4 5 triangle does not exist in my head, then shouldn't it be said that there is no truth at all to a 3 4 5 triangle? and nobody should even mention them or use them, because they are a myth?Nonsense. The 3 4 5 triangle is an emergent behavior of the abstract system called mathematics/trigonometry. This system is useful because it gives us approxomately accurate information when applied to things in real life that approxomately line up with it. Because it is useful, it should be mentioned and used when it is useful to do so.

if what's happening inside my head is not real life, then what is it? why would evolution go through millions of years of struggling just to create something that's not real? why is the brain not real?
Evolution doesn't struggle, any more than 1 and 1 struggle to make 2. Evolution is just the way things happen in certain circumstances - just like in the circumstances when you have 1 and 1, they make 2. And math didn't evolve anyway; it was deduced by humans independent of their evolution. And as for what's going on in your head, it's as real as what's going on in your computer case. There are no browsers floating around in there, but you can get on the internet just the same.

[Indistinguishable]

A weird coincidence: I had been planning myself to use the abstract concept of the letter 'R' as an example to illustrate a point, and then begbert2 went and did it independently, same letter and everything.

[ToeJam]

Pffft... the 3-4-5 Triangle?

I always preferred the 30-60-90 Triangle. And who could forget it's unloved stepchild, but infinitely more likable 45-45-90 Triangle.

Screw the evolutionary distance, man, it's all about the angels in the angles!

*edit* whoops, just noticed this was by the sperm guy. I'll quietly slink away and come back in a few days to see where this train of discussion actually goes.

[BunnyTVS]

if what's happening inside my head is not real life, then what is it?

From many minutes of rigorous masturb... calculation, I have calculated 2(two) equally proble results.

1: Frickin' Frightning

2: What happens when you smoke a full 1/4oz of prime grade skunk

Strangely enough Frickin' Frightning2 + What happens when you smoke a full 1/4g of prime grade skunk2 = confission2

[tomndebb]

If confission wants to post this, again, in MPSIMS or a Joke forum on another board, that is fine, butr it is not funny enough to be left here and it is certainly nothing resembling a legitimate debate.

Closed.

[ /Moderating ]