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PostPosted: Wed Jul 12, 2017 11:46 am 
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rainwarrior wrote:
There are possible paradoxes in all formal systems of logic

To be pedantic (that's why we're all here, right?) there are possible paradoxes in all sufficiently powerful formal systems of logic. Predicate logic and first-order logic are famously sound and complete. Second-order logic is incomplete, but certain rather esoteric fragments of it aren't.


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PostPosted: Wed Jul 12, 2017 11:48 am 
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Lots of teaching seems to be
"We told you this thing was impossible because it would have been a distraction when we were working on basics. Now we need you to unlearn a little so that we can fill in that gap".

Not just with math but lots of engineering fields too.


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PostPosted: Wed Jul 12, 2017 2:46 pm 
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Here's a brief article about one way that complex numbers naturally arose in the 16th century from trying to solve cubic equations:
https://www2.clarku.edu/~djoyce/complex/cubic.html

One mathematician came up with a generic formula solution for solving cubic equations (like the quadratic formula does for quadratics).

From quadratic problems they were already used to "impossible" solutions where there's a square root of a negative number, but it was noticed that in some cases of the cubic formulae you could get a pair of such square roots that cancel each other out, leaving a valid solution!

Down some lines of inquiry, the idea of the "imaginary" number i2=1 is just kind of staring you in the face. This wasn't some arbitrary hack just tacked on there on a whim, it was a thing that kept coming up in what they were doing. It seems a bit bizarre at first, but it very "naturally" comes out of the study of roots. This is also probably hard to see if you're taught about i before you've ever seen a place where it springs out of.


Actually, maybe that's a good way to show one of their uses... you know that a square root has 2 solutions, negative and positive. Why does a cube root have only 1 solution and not 3? Why does a 4th root have 2 solutions and not 4? Shouldn't a 5th root have 5 solutions?

Well, with complex numbers it does! On the complex plane, these roots are evenly spaced along a circle:
Attachment:
fifth_root_of_1.png
fifth_root_of_1.png [ 11.38 KiB | Viewed 429 times ]

Anyhow, I'm not really trying to explain complex exponents here, I'm just trying to show how having i as a tool might begin to shed light on these things in a useful way.


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PostPosted: Wed Jul 12, 2017 4:18 pm 
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A deep feeling of depression comes over me when I think about all the stuff I don't know. And with all the new stuff that I learn, I end up forgetting an equal amount of the old stuff.

I barely remember calculus, for example. I couldn't solve an integral if you asked me.


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PostPosted: Wed Jul 12, 2017 4:32 pm 
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rainwarrior wrote:
Here's a brief article about one way that complex numbers naturally arose in the 16th century from trying to solve cubic equations:
https://www2.clarku.edu/~djoyce/complex/cubic.html

One mathematician came up with a generic formula solution for solving cubic equations (like the quadratic formula does for quadratics).

From quadratic problems they were already used to "impossible" solutions where there's a square root of a negative number, but it was noticed that in some cases of the cubic formulae you could get a pair of such square roots that cancel each other out, leaving a valid solution!

Down some lines of inquiry, the idea of the "imaginary" number i2=1 is just kind of staring you in the face. This wasn't some arbitrary hack just tacked on there on a whim, it was a thing that kept coming up in what they were doing. It seems a bit bizarre at first, but it very "naturally" comes out of the study of roots. This is also probably hard to see if you're taught about i before you've ever seen a place where it springs out of.


Actually, maybe that's a good way to show one of their uses... you know that a square root has 2 solutions, negative and positive. Why does a cube root have only 1 solution and not 3? Why does a 4th root have 2 solutions and not 4? Shouldn't a 5th root have 5 solutions?

Well, with complex numbers it does! On the complex plane, these roots are evenly spaced along a circle:
Attachment:
fifth_root_of_1.png

Anyhow, I'm not really trying to explain complex exponents here, I'm just trying to show how having i as a tool might begin to shed light on these things in a useful way.


Cool, thanks for that...makes me even more interested in the subject.

If by arbitrary hack you were referring to my complaint about the name "imaginary," I didn't mean to say I thought it was an arbitrary invention, just simply to say that it is unfortunate to call something imaginary and define something which appears to be a falsehood without context---it's hard to swallow for many students I think. But information like what you just shared, I wish could be more seamlessly integrated into math courses. Instead its like an assembly line. Learn your formulas! Got 'em? Do a test! Failed? Oh well you have a good job 11 years later anyway


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PostPosted: Thu Jul 13, 2017 12:30 am 
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Quote:
So... okay maybe it doesn't immediately sound useful for a quadratic, but as soon as you want to solve cubic (x3) or quartic (x4) equations, you'll discover that the "equivalent" to the quadratic formula for them is maddeningly complicated. Complex analysis becomes a light leading out of the tunnel for this.

Oh I remember having studied this, and had to use the cubic formula, it was insanely complicated. Basically if I remember well solving arbitrary equation of 3rd and 4th order is possible but insanely complicated, and 5th order and upper is basically impossible without resorting to numerical approximations. Or you can be smart and factor the equation away by "guessing" solutions, but it's not always possible.

Quote:
That's what I thought. Who the hell says "-5 meters above sea level"?

Well, the sign simply shows the direcition. The same could be said about latitude and longitude positioning, the sign indicate the direction whether it's north/south or east/west. Basically it carries extra information in the numebr itself. Complex numbers is the same concept, they carry information for a number and an angle at the same time.
Quote:
It is not the square root of -1, but rather it is the "imaginary" number that when squared will equal -1. i2 = -1

Actually every (positive real) number has two "square roots" (i.e. two reals that multiplied by themselves gives this result) but the "square root" function only returns the positive one. For example both 3 and -3 will give 9 when multiplied with themselves, but only 3 is the square root of 9, not -3.

Similarly, both i and -i gives -1 when multipplied by themselves. Since neither is "positive" nor "negative" (they are something else entierely !) we can't decide which one is the "square root" of -1. We could decide that we keep the one in the first hemicycle (angle is smaller than 180°), this is simple when having the square root of negative numbers (the two "roots" will always be conjugated), but when taking the square root of an already complex number there is no simple solution (the two "roots" can be anything if I'm not mistaken).


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PostPosted: Thu Jul 13, 2017 7:18 pm 
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Location: Texas, USA
DoNotWant wrote:
Who the hell says "-5 meters above sea level"?

HEIGHT REACHED.....-240m.

I think it's kind of wierd though. In that game, the levels during game play are called DEPTH xxx m when you're under ground and will be called HEIGHT xxx m when you're above ground in the full game. I think it's kind of weird it switches from DEPTH x during game play to HEIGHT -x on the game over screen, but I guess it makes it easier to compare progress with other people at a glance. (The instructions encourage you to take a picture of this progress screen to share with others.)


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