The Debate Hall Social Thread

[Dre89]

I got 13/15 on the quiz, and I missed the questions about the Jewish Sabbath and the communion bread/wine becoming the body and blood of Christ.

Dre, can you explain that last one to me?

Sucumbio is pretty much correct on the transubstantiation.

Jewish Sabbath is on a Saturday, and the Catholic Sabbath is on Sunday as you probably know.

Not sure why there's a difference.

 

[Bob Jane T-Mart]

Hey guys, it looks as if an earth-like extrasolar planet has been found that exists in the habitable zone. This means that liquid water could conceivably exist on it's surface. A number of astronomers think that life could be possible there.

http://www.space.com/scienceastronomy/earth-like-exoplanet-possibly-habitable-100929.html

 

[Sucumbio]

yeah I saw a piece on Countdown about it, fascinating really.. the planet doesn't rotate so one half of it is always light, the other always dark, but because it's orbiting a red dwarf there's speculation that the radiation from the sun is really low in comparison to our sun, and so life could theoretically exist over the entire planet, though it'd mainly flourish nearer the light-equator (I've forgotten what that line is called where light/dark meet >.>)

It's also been suggested that this may lead to an iron clad refutation of intelligent design, at least the ID Theory that the Earth was created as part of a plan that intended for only 1 planet to exist with life upon it.

 

[KrazyGlue]

Hey guys, it looks as if an earth-like extrasolar planet has been found that exists in the habitable zone. This means that liquid water could conceivably exist on it's surface. A number of astronomers think that life could be possible there.

http://www.space.com/scienceastronomy/earth-like-exoplanet-possibly-habitable-100929.html

Oh yeah, read about this a few days ago. Pretty neat. Of course, they've claimed this a bunch of times, but usually they stop talking about them after a while. Whether they find a better alternative, something about the planet is wrong, or they just lose interest I don't know. It just seems like these things come and go a lot.

 

[Dragoon Fighter]

@KrazyGlue You right there is almost something wrong with the "Near earths" One is seven times the size of earth while another one is almost all water and a mix of Ice 1 and Ice 7.

 

[Bob Jane T-Mart]

yeah I saw a piece on Countdown about it, fascinating really.. the planet doesn't rotate so one half of it is always light, the other always dark, but because it's orbiting a red dwarf there's speculation that the radiation from the sun is really low in comparison to our sun, and so life could theoretically exist over the entire planet, though it'd mainly flourish nearer the light-equator (I've forgotten what that line is called where light/dark meet >.>)

It's also been suggested that this may lead to an iron clad refutation of intelligent design, at least the ID Theory that the Earth was created as part of a plan that intended for only 1 planet to exist with life upon it.

Actually, if it didn't spin, the areas of light and dark would change. It's actually tidally locked with the star it orbits. So one side always faces it's star. The star itself is a red dwarf, so it's quite dim. But to put this planet in the habitable zone, it has a very close quick orbit around it's star, once every 37 days.

Though ID can just be warped and played with to say, "Well, obviously the designer designed life on two planets."

 

[Dre89]

I love how people think ID has to exactly right on every little detail, and anticipate every future discovery, yet things like evolution, big bang, no boundary proposal etc. are allowed a margin of error and uncertainty.

 

[Bob Jane T-Mart]

I love how people think ID has to exactly right on every little detail, and anticipate every future discovery, yet things like evolution, big bang, no boundary proposal etc. are allowed a margin of error and uncertainty.

Huh? I'm just saying that ID is unfalsifiable. So you're never going to be able to disprove it. It doesn't list any predictions, so there's nothing to test it with. Scientific theories have margins of error and uncertainties because they are imperfect.

 

[Lore]

Thanks for the explanation, Sucumbio!

Sucumbio is pretty much correct on the transubstantiation.

Cool.

I really need to read more about Catholicism. A lot of the theology and philosophy stuff is great, but I always hear about something weird like transubstantiation after I think, "Hey, maybe I should be Catholic!" XD

@KrazyGlue You right there is almost something wrong with the "Near earths" One is seven times the size of earth while another one is almost all water and a mix of Ice 1 and Ice 7.

Why would it be a bad thing for the "near earth" to be seven times larger?

The only problem that I can see with this one is cat. 5 hurricane winds that are caused by the huge temperature differences on each side.

Huh? I'm just saying that ID is unfalsifiable. So you're never going to be able to disprove it. It doesn't list any predictions, so there's nothing to test it with. Scientific theories have margins of error and uncertainties because they are imperfect.

I think that he was just saying that even though a lot of atheists really crack down on every little fault in theological theories, they generally leave evolution, etc alone.

I could easily be wrong, but that's the way I've seen it.

 

[manhunter098]

Jewish Sabbath is on a Saturday, and the Catholic Sabbath is on Sunday as you probably know.

Not sure why there's a difference.

Jewish Sabbath begins Friday with the setting of the sun and ends Saturday with the setting of the sun. There is the difference of day, but also the difference of calendars. Jewish holidays tend to always start the "night before" our calendar date for the holiday.

 

[Sucumbio]

I thoroughly enjoyed participating in Passover, it was like, eat some, pray some, eat some more, pray some more. Good food.

 

[Budget Player Cadet_]

Whee, I placed 4th at a smashfest.

I promise I will come back to the BNW thread, it's just that I'm REALLY ****ING BUSY on many fronts and this kinda takes a back burner, sorry guys... Defending PS2 is aggravating.

 

[Amide]

I think that he was just saying that even though a lot of atheists really crack down on every little fault in theological theories, they generally leave evolution, etc alone.

I could easily be wrong, but that's the way I've seen it.

That is probably what he meant, but I'd argue that there isn't any fault in the theory of evolution- research isn't regarded as theory until lots of evidence is behind it- though there are large parts of it that we don't understand, and probably never will.

 

[Dragoon Fighter]

I read an article on programing ethics into robots, basically the idea is that ethics is computable so there working on programing it into this set of robots that help the elderly. I thought that would be an nice discussion. What do you guys think? (The source is Scientific American I can dig up the article but unless you pay it will only give you and incomplete summery. )

 

[Sucumbio]

Issac Asimov's Three Laws of Robotics:

1. Robots must never harm human beings or, through inaction, allow a human being to come to harm.
2. Robots must follow instructions from humans without violating rule 1.
3. Robots must protect themselves without violating the other rules.

Beyond this I'm not sure what ethical decisions could be programmed in without the chance for some paradox or dilemma showing up, but I might be able to come up with one or two with enough thought.

 

[#HBC | Acrostic]

Issac Asimov's Three Laws of Robotics:

1. Robots must never harm human beings or, through inaction, allow a human being to come to harm.
2. Robots must follow instructions from humans without violating rule 1.
3. Robots must protect themselves without violating the other rules.

Beyond this I'm not sure what ethical decisions could be programmed in without the chance for some paradox or dilemma showing up, but I might be able to come up with one or two with enough thought.

4. Robots may never divide by zero.
5. Robots may never take the square root of a negative number.
6. Robots may never solve for the -log of a negative number.

 

[Deleted member]

except that 5 is definitely legal. I see no reason why robots shouldn't be ably to work with complex numbers

 

[Dragoon Fighter]

I disagree with the laws of robotics as they are violated by machines used in wars all the time. I will post the incomplete artical later. Right now my computer access is limited .

 

[#HBC | Dark Horse]

What's everyone's opinion about saying the pledge of allegiance in school?

 

[KrazyGlue]

I don' think kids should be required to say it, but they should stand or sit quietly and respectfully. Basically the way it is now. If you're talking specifically about the "under God" thing, I think people make way too big of a deal of it and it doesn't really matter all that much.

 

[Lore]

except that 5 is definitely legal. I see no reason why robots shouldn't be ably to work with complex numbers

Eh? Doesn't that kind of equation involve imaginary numbers?

I disagree with the laws of robotics as they are violated by machines used in wars all the time. I will post the incomplete artical later. Right now my computer access is limited .

How are they being violated?

The "rules" are only a theoretcal set of rules for a human-like artificial intelligence. The drones used in today's wars have practically zero intelligence and are piloted by regular humans, so they do not require any ethics programming.

 

[KrazyGlue]

Eh? Doesn't that kind of equation involve imaginary numbers?

A complex number involves both imaginary and real numbers. And yeah, there's no reason why a robot shouldn't be able to work with that. Hell, I've had to work with complex numbers the last few years in math.

 

[blazedaces]

Guys, I think you misunderstand.

i = the square root of -1

When we are talking about a complex number, let's say 4 + 7 i... what we mean is 4+7 * (square root of negative 1).

You quite literally can't take the square root of a negative number. It doesn't make sense.

-blazed

 

[KrazyGlue]

I'm not sure what you mean. You can most certainly take the square root of a negative number; the only difference is that you'll end up with an imaginary number as opposed to a real number.

For instance:
sqrt(-9) = sqrt(9)*sqrt(-1) = 3i

It's part of the Fundamental Theorem of Algebra, which essentially states that any polynomial has a solution if you include complex number solutions.

 

[Budget Player Cadet_]

Once upon a time, (1/T) pretty little Polly Nomial was strolling through a field of vectors when she came to the edge of a singularly large matrix. Now Polly was convergent and her mother had made it an absolute condition that she never enter such an array without her brackets on.

Polly, however, who had changed her variables that morning and was feeling particularly badly behaved, ignored this condition on the grounds that it was insufficient and made her way in amongst the complex elements. Rows and columns enveloped her on all sides. Tangents approached her surface. She became tensor and tensor. Quite suddenly, 3 branches of a hyperbola touched het at a single point. She oscillated violently, lost all sense of directrix, and went completely divergent. As she reached a turning point, she tripped over a square root protruding from the erf and plunged headlong down a steep gradient. When she was differentiated once more, she found herself, apparently alone, in a non-Euclidean space.





She was being watched, however. That smooth operator, Curly Pi, was lurking inner product. As his eyes devoured her curvilinear coordinates, a singular expression crossed his face. Was she still convergent, he wondered. He decided to integrate improperly at once. Hearing a vulgar fraction behind her, Polly turned around and saw Curly Pi approaching with his power series extrapolated. She could see at once, by his degenerate conic and his dissipated terms, that he was up to no good.

"Eureka," she gasped. "Ho, ho," he said. "What a symmetric little polynomial you are. I can see you are bubbling over with secs." "Oh, sir," she protested. "Keep away from me. I haven't got my brackets on." "Calm yourself, my dear," said our suave operator. "Your fears are purely imaginary." "I, I," she thought, "perhaps he's homogeneous then." "What order are you?" the brute demanded. "Seventeen," replied Polly. Curly leered. "I suppose you've never been operated on yet?" he asked. "Of course not!" Polly cried indignantly. "I'm absolutelyconvergent." "Come, come," said Curly, "let's off to a decimal place I know and I'll take you to the limit." "Never," gasped Polly. "Exchlf," he swore, using the vilest oath he knew.






His patience was gone. Coshing her over the coefficient with a log until she was powerless, Curly removed her discontinuities. He stared at her significant places and began smoothing her points of inflection. Poor Polly. All was up. She felt his hand tending to her asymptotic limit. Her convergence would soon be gone forever. There was no mercy, for Curly was a heavyside operator. He integrated by parts. He integrated by partial fractions. The complex beast even went all the way around and did a counter integration. What an indignity to be multiply connected on her first integration. Curly went on operating until he was absolutely and completely orthogonal. When Polly got home that night, her mother became frightened and stated "You're traveling in a forward direction to your auntie + uncle unit in the graph of Bel Air". I whistled for a cab and when it approached, the license plane said "New" and there were dotted cubes in the reflector, if anything I could state that this cab had a lesser chance than the rest but I thought disregard that fact, if you could operator, follow the lines that lead to Bel Air! I approached the compilation of three dimensional objects about 7/12 or 2/3 and I yelled to the operator attention, smell you some other time on this planar area! Looked at my Math house, My graph had finally reached a closed point, to finalize on my algorithmically correct point as the prince of the graph known as Bel Air.

 

[GoldShadow]

BPC wins the Debate Hall.


(that story is an oldie but a goodie... also, perfect placement)

 

[blazedaces]

I'm not sure what you mean. You can most certainly take the square root of a negative number; the only difference is that you'll end up with an imaginary number as opposed to a real number.

For instance:
sqrt(-9) = sqrt(9)*sqrt(-1) = 3i

It's part of the Fundamental Theorem of Algebra, which essentially states that any polynomial has a solution if you include complex number solutions.

? You do realize you just agreed with me right? The square root of negative nine equals 3 times the square root of negative 1...

i is just a symbol we use instead of writing sqrt(-1) everywhere...

Imaginary numbers do have repercussions, but you simply can not take the square root of a negative number. Period. There are other things you can not do in math. You can't take the log of a negative number, you can't divide by 0, you can't divide infinite by infinite, etc.

-blazed

 

[Deleted member]

whahahaha.
Typical not mathematicians response.

We defined the square root of negative one as "i".
The rest (and by that I mean a lot of things) follows.
In the set of the Real number, yes you can't take the square root of negative numbers as the answer lies outside of the set. that is all. when your work within the set of Complex numbers everything is fine.

And you are also not perfectly right about the other things.
While I indeed know that you can't take the log of a negative number. (this just means the domain of the log function is the positive Real set, there are plenty of other function that are not defined on the entire real set. there is no such thing as "you can't do" in math, there are just things that don't have an answer like "2")

both dividing by zero and inf/inf are both fine in their limit cases btw, and you don't even look at infinity other than in limit cases.



EDIT
just had a thought.
can you write down Sqrt(2) for me without just writing Sqrt(2)? No you can't, and you could proceed to argue that irrational number don't exists either, hell people did before the time of Euler. Or how about negative numbers? The "-" is, much like i (italics is a force of habit before you ask), just a symbol.

 

[#HBC | Acrostic]

I love discussions regarding mathematical principles, especially due to the fact that the subject never sat well with me in the past and much knowledge continues to elude me. Please continue.

 

[Deleted member]

I suppose you mean Complex numbers? What is that part that doesn't sit well with you?

After checking some literature, perhaps a better definition is the following:

The imaginary unit i is defined as the solution to the equation x^2 = -1.
This gets rid of the "problem" of square roots of negative numbers, as well as the following equation:
1 = Sqrt(1) = Sqrt(-1 * -1) = Sqrt(-1) * Sqrt(-1) = i * i = -1

Now there isn't really an existential question about i as it is simply defined the way it is.
The same goes for Pi really, which is simply defined as the ratio between the diameter and circumference of a circle. It is not even possible to draw a straight line with length Pi (unlike Sqrt(2))

 

[blazedaces]

Just want to say I'm enjoying the conversation a lot, so I hope all is taken in good fun as that's how I mean it...

whahahaha.
Typical not mathematicians response.

That was a bit crude...

We defined the square root of negative one as "i".
The rest (and by that I mean a lot of things) follows.
In the set of the Real number, yes you can't take the square root of negative numbers as the answer lies outside of the set. that is all. when your work within the set of Complex numbers everything is fine.

Paprika, have you ever stopped and thought about these concepts? What does "working within a set" mean to you? Why was it "invented"? What were the original mathematicians thinking?

Every number... in existence... rational/irrational can be represented in the set of complex numbers. Why don't we work in that set all the time? Why do we strive to avoid it at all? Because it's easier to... the set we're working in is simply syntax. It doesn't mean much more than that. When you define x and you say it's contained in the reals... does that mean you CAN'T POSSIBLY represent it as a complex number?

And you are also not perfectly right about the other things.
While I indeed know that you can't take the log of a negative number. (this just means the domain of the log function is the positive Real set, there are plenty of other function that are not defined on the entire real set. there is no such thing as "you can't do" in math, there are just things that don't have an answer like "2")

I don't see the difference. You're using word salad to mix things around but the same stays true, you can't take the log of a negative number. You have to tell a machine this fact so it can spit back an error to you.

both dividing by zero and inf/inf are both fine in their limit cases btw, and you don't even look at infinity other than in limit cases.

Right... l'Hôpital's rule is completely useless... glad to hear it

And you do too look at dividing by zero and inf/inf in non-limit cases... but really there isn't much of a difference between a case with a limit and a case without a limit. The limit as x -> 0 is simply plugging in 0 for x... which is why we even talk about asymptotes, because in such cases where there is a zero on the denominator possible we take the limit as that function goes to 0 to determine what that function does...

EDIT
just had a thought.
can you write down Sqrt(2) for me without just writing Sqrt(2)? No you can't, and you could proceed to argue that irrational number don't exists either, hell people did before the time of Euler. Or how about negative numbers? The "-" is, much like i (italics is a force of habit before you ask), just a symbol.

You can approximate the sqrt(2)... can you approximate the square root of negative 1?

I want to point out the following logical truth to you:

If you say you can't "not do something" in math it's just like saying there are no rules.

Well guess what, saying so is a rule, which means it's not logically possible to imply there are no rules.

-blazed

 

[Deleted member]

actually not every number in existence can be represented in the Complex plane, but I'll leave Quaternions out of the discussion for now.
and why don't we work within the complex set all the time? actually we do, since the real set is within the complex set.
In real life however it is enough to limit ourselves to the Real set because you don't come along Imaginary numbers in daily life.
And the set we're working in is NOT just syntax, as the structure is inherently different (cardinal and dimension vary when going through N,Z,Q,R and C).

And approximate it in what set? in the Reals? because it's not in the Real set.

Just to repeat my previous post: this isn't a matter of existence, since i is simply defined as i^2 = -1 and we work with it like that.
Is it the fact that i has no solution as such that bothers you or is it something else?

btw, are you a exact science student or something because you do know what you're talking about.

 

[Sucumbio]

I agree w/GS. Good job, BPC I actually lol'd.

wow this is some heavy stuff, it's been forever since I thought about i and square roots and complex numbers. that's what I get for changing from chemistry to english as my major

 

[blazedaces]

actually not every number in existence can be represented in the Complex plane, but I'll leave Quaternions out of the discussion for now.
and why don't we work within the complex set all the time? actually we do, since the real set is within the complex set.
In real life however it is enough to limit ourselves to the Real set because you don't come along Imaginary numbers in daily life.
And the set we're working in is NOT just syntax, as the structure is inherently different (cardinal and dimension vary when going through N,Z,Q,R and C).

Can you elaborate on why the structure is different? I can turn cardinal coordinates into polar coordinates and vice versa... they're just a syntax to make things look simpler to us.

And approximate it in what set? in the Reals? because it's not in the Real set.

I don't know what you mean. Why is sqrt(2) approximately 1.44, why does our calculator spit that out?

Just to repeat my previous post: this isn't a matter of existence, since i is simply defined as i^2 = -1 and we work with it like that.

This is my point, we just work with it like that... there's no TRUE WAY to work with anything. There's no ALWAYS RIGHT ANSWER in math. People think there's some kind of trick to math. That you're either good at math or bad at it. I think that's major BS.

There's many ways to arrive at the right answer. Half the stuff we do in math to prove things are little quarks of syntax. Like when we have a cos(x), many proofs will say cos(x) = the real part of cos(x) +i*sin(x) = the real part of e^i*theta, etc. Or when we multiply by 1 = multiply by 1/1 = multiply by sqrt(2)/sqrt(2), etc. These are all just manipulations of the same thing, but we haven't CHANGED anything, not REALLY.

Is it the fact that i has no solution as such that bothers you or is it something else?

It's not bothering me at all. Seriously, I enjoy these topics. It's more like I think a lot of the way we teach math and science is wrong in our education (and I guess I'm specifically talking about the US education as that's what I've researched and experienced mainly). Math is taught in such a way that "these are the right things to learn" and "this is the ONLY way this works" and "this is the ONLY way you can think about this problem"...

It's all nonsense. Math, in my opinion, needs to be taught in such a way where we consider why, why do we do this or that. What was the original purpose, or the purpose of the person who invented or used this method? What was going through their head at the time?

btw, are you a exact science student or something because you do know what you're talking about.

I'm a senior in electrical engineering, but even for an EE student I like math a lot and have taken quite a few extra electives in math classes.

Edit: One other thing, I was re-reading that post from earlier:

1 = Sqrt(1) = Sqrt(-1 * -1) = Sqrt(-1) * Sqrt(-1) = i * i = -1

This equation is incorrect. 1 does NOT equal sqrt(1)... sqrt(1) = +/- (plus or minus) 1. +1 is certainly NOT equal to -1.

-blazed

 

[Deleted member]

(obviously that equation isn't correct, the part where you split the root in two is illegal)

In reaction:
Can you show me the transformation from R to C then, because if they are the same thing with different syntax there should be a transformation (isomorphism) between them.

My calculator spits out Sqrt(2). It kind of depends on the calculator (would be a nice analogy between the expensiveness of your calculator and the number fields)

And yes Equations never truly change anything, that's why they're equations.
anecdote: My physics teacher once said that it is perfectly fine to introduce a pink elephant in your equations as the end result would (read: should) not change.
I don't really see the point though, I mean Sqrt(2) is also just the answer to the equation x^2 = 2.
And 1.414... doesn't really count as it is not exact. approximation is fine for real life purposes but it has no place in pure mathematics.

It's all nonsense. Math, in my opinion, needs to be taught in such a way where we consider why, why do we do this or that. What was the original purpose, or the purpose of the person who invented or used this method? What was going through their head at the time?

Could you elaborate on that? It sounds like an interesting PoV but i don't entirely grasp it.


I also came across this:
http://www.math.toronto.edu/mathnet/answers/imaginary.html

an interesting read in general, and also shows an argument why fractional numbers exist and then proceeds to use the same argument for complex numbers.

 

[blazedaces]

(obviously that equation isn't correct, the part where you split the root in two is illegal)

Actually no, splitting a root is not illegal. Think of the rules of powers:

(a*b)^x = a^x*b^x. This applies the same no matter what number you put in for x, such as 1/2, so (a*b)^1/2 = a^1/2 * b^1/2. This is the same as writing sqrt(a*b)=sqrt(a)*sqrt(b).

This can be done at any time. Let me ask you this, why is sqrt(27)=3*sqrt(3)? Why can we reduce a square root in this way? Simple, because sqrt(27) = sqrt(9*3) = sqrt(9)*sqrt(3) = 3 * sqrt(3).

In reaction:
Can you show me the transformation from R to C then, because if they are the same thing with different syntax there should be a transformation (isomorphism) between them.

Certainly:

A real number is a any number contained in the reals. An imaginary number is any number constant b * i where i = sqrt(-1). A complex number is the set of all numbers a+b*i where a and b are any constants.

Therefore, for all a in the reals, there exists a complex number z = a+0*i which also equals our original a. Similarly, all imaginary numbers b*i also equal a complex number 0+b*i.

My calculator spits out Sqrt(2). It kind of depends on the calculator (would be a nice analogy between the expensiveness of your calculator and the number fields)

Smart ***

You know full well what I'm talking about though.

And yes Equations never truly change anything, that's why they're equations.
anecdote: My physics teacher once said that it is perfectly fine to introduce a pink elephant in your equations as the end result would (read: should) not change.
I don't really see the point though, I mean Sqrt(2) is also just the answer to the equation x^2 = 2.
And 1.414... doesn't really count as it is not exact. approximation is fine for real life purposes but it has no place in pure mathematics.

Mathematics only exists to apply to the real world (imo). What's wrong with discrete sets, or approximating? What's wrong with linearizations? What's wrong with Taylor Series Expansions? They make life easier. If you can't apply something to the real world in some way it's just speculation for the sake of speculation. Its purpose becomes lost.

Could you elaborate on that? It sounds like an interesting PoV but i don't entirely grasp it.

It's hard to explain, but I'll give it my best shot. In general, I just think the way most people choose to teach math is wrong. First, there is this arrogance in math, about how it's for some people, and NOT for others. I've tutored a lot of children in math. Granted, I'm sure you might call my viewpoint biased, but many of the children I taught were gifted in other aspects of their life (often writing), so their math teachers always called them failures. Granted it's the viewpoint of the child that the teacher called them such a thing, but yet again, I'm digressing...

The point I'm trying to get to is that I was able to teach these kids math on a level I don't believe their teachers foresaw possible. And I don't think it's very hard to do. It's not about who you are, or how good a teacher you are... it's just about HOW you teach.

First, the concept of ONE RIGHT ANSWER AND METHOD needs to be removed. We need to explain clearly that in math there's almost always more than one way to skin a cat. This brings me to the next concept, and it comes from a book called "outliers". A good read and I definitely recommend it. He refers to a national test (the one everyone takes at 8th grade in the US I believe, but it's been a while since I read the book) where people answer questions on all of the subjects offered in school. There's one extra, optional portion of the test, which consists of about 25/50 questions (I don't remember, whatever) about you, your school, your personal information, how your teachers tended to teach a subject, etc. Most people skip it. But guess what, lo and behold, those who were good at math also filled in the most of those optional questions.

Being good at math is just about trying more things (according to the author of the book). Now, I don't think this is 100% true, but I do believe it's true to a very large extent. The biggest concept that succeeds for me in tutoring children is telling them this story, and then asking them to try again, because that's all it takes. Most people just stop trying in math, they say "I'm just not good at it". This usually causes them to doubt themselves further.

Then there's the way we teach math as a "things are true JUST BECAUSE I SAID SO" viewpoint. How many people do you know who can prove the Pythagorean theorem? How many people do you know who can prove ANYTHING in math? Even understand what that is asking? Why don't we teach proofs? Whenever I prove concepts to the kids I tutor they understand the concept 10 times better. Things just start to click. It's how it's always been with me.

We're so busy teaching the "what" part of math... but we don't bother about the "why"...

Does this make more sense? Now to suggest a curriculum with these concepts in mind is another story...

I also came across this:
http://www.math.toronto.edu/mathnet/answers/imaginary.html

an interesting read in general, and also shows an argument why fractional numbers exist and then proceeds to use the same argument for complex numbers.

Ah, but this author is taking on a different viewpoint as well, let me quote him below:

Since numbers are just abstract concepts anyway

I guess if one were to say "all numbers are abstract concepts" then to say imaginary numbers are just another abstract concept wouldn't be wrong. But honestly man, how can you not read his explanation and just say "yes the syntax works"? It's just like a programming language. You're defining that a number system just needs to satisfy certain parameters. It has nothing to do with "existence" in the way you and I think of the word.

-blazed

Edit: Just so you know, the book outliers explains that concept A LOT better than I did...

 

[Deleted member]

I know very well how root splitting works, but it isn't legal for negative roots.

That transformation in not what I meant, and is certainly not an isomorphism.

well imo, that imo is kind of misplaced. for far in my study we have barely focussed on real world applications outside of calculus.
Certainly in the real world approximations and such are very useful, but in theoretical mathematics they aren't used much/at all. As I mentioned earlier, imaginary numbers aren't very useful in real life, but they are in purely theoretical mathematics.

certainly an interesting story, I might read that book at some point.
from it I guess you're talking about high school teaching, which I can't really say much about as I never really saw that attitude in my teachers (I was one of the "good" people though).
debating the curriculum would be interesting.

actually that is how I think of numbers
with even numbers changing meaning in classes I kind of have to.

(what concept?)

 

[Reaver197]

Came across this story, and thought it particularly poignant and telling of current state of things in America.

http://www.reddit.com/r/WTF/comments/dmzok/censorship_at_its_finest_one_teachers_story_on/

 

[blazedaces]

I know very well how root splitting works, but it isn't legal for negative roots.

Why not? sqrt(-49) = sqrt(49)*sqrt(-1)=7*sqrt(-1)=7i

That transformation in not what I meant, and is certainly not an isomorphism.

Sorry, I didn't pay attention to the word isomorphism in the parenthesis. But all you're really saying is that not all complex numbers can be represented as real numbers. I was only implying that all real numbers can be represented by complex numbers. Like squares and rectangles, all squares are rectangles, not all rectangles are squares.

well imo, that imo is kind of misplaced. for far in my study we have barely focussed on real world applications outside of calculus.

You'd be hard pressed to find ANYTHING that involves math that can't be applied to the real world in some way. I know of a few things, but I'm betting most of the stuff you're studying does in some way of which you're not aware.

Certainly in the real world approximations and such are very useful, but in theoretical mathematics they aren't used much/at all.

In theoretical mathematics you can do either. It is not bound by anything. Why do we even distinguish the two?

As I mentioned earlier, imaginary numbers aren't very useful in real life, but they are in purely theoretical mathematics.

That's not entirely true. They are plenty useful in many applications, specifically in my experience with anything involving periodic functions/signals. Whenever you talk about laplace transforms, fourier series (and in extension most electrical and mechanical systems) you're going to involve some math that has to do deal with imaginary numbers. But because like you said, we don't want them in that form, we manipulate things so that we don't deal with them and see only their real components. Sin(x) and Cos(x) are both exponential functions with imaginary components, but we see them as real results, which implies that quite a lot of what goes on in the real domain is affected indirectly by what happens in the imaginary domain. Another words, both domains are not entirely independent from one another.

(what concept?)

I just meant what I tried to explain that being good at math is equivalent to not giving up or being willing to try more things (again according to the author). I was just saying he articulates it better than I do...

-blazed

 

[blazedaces]

Came across this story, and thought it particularly poignant and telling of current state of things in America.

http://www.reddit.com/r/WTF/comments/dmzok/censorship_at_its_finest_one_teachers_story_on/

I can't see the original story. The link works, but all I see are the comments. When I click on it a blog page appears that says something along the lines of "this blog doesn't exist".

-blazed