pi
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Ryan Binns wrote:
I see most people say that i = sqrt(-1), which is NOT correct - it implies that i2 = 1
Uh, how? If you say
i = SQRT(-1)
squaring both sides will give you
i * i = -1
How does i = sqrt(-1) imply i2 = 1 ? Cheers, Vikram.
I don't know and you don't either. Militant Agnostic
assume i = sqrt(-1) i * i = sqrt(-1) * sqrt(-1) i * i = sqrt(-1 * -1) i * i = sqrt(1) i * i = 1
Ryan
"Punctuality is only a virtue for those who aren't smart enough to think of good excuses for being late" John Nichol "Point Of Impact"
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it's a pure definition problem then? I think the issue here is the same as vegetable and fruit. Some people would say tomatoes is a fruit, some people would say it is not. And then they each refer to their own definition. Doens't matter too much. Anyway I will stick to my definition which means that PI is part of super set of the real (hence it is not a real number). And that Math teacher don't bother make the difference explicit until you are in advanced math studies....
Super Lloyd wrote:
Anyway I will stick to my definition which means that PI is part of super set of the real (hence it is not a real number).
And you will be wrong. Pi is an element of the set of Real numbers. If you'd like to hold to an errant belief that Pi is outside the set of real numbers, please keep it to yourself, lest you confuse those, like Jeremy, who ask questions regarding mathematics.
They dress you up in white satin, And give you your very own pair of wings In August and Everything After
I'm after everything
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:) Simple, that would be 4. Next please. Jeremy Falcon
Jeremy Falcon wrote:
that would be 4
:omg: 4!? Really? I thought it was 5 :~
Ryan
"Punctuality is only a virtue for those who aren't smart enough to think of good excuses for being late" John Nichol "Point Of Impact"
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assume i = sqrt(-1) i * i = sqrt(-1) * sqrt(-1) i * i = sqrt(-1 * -1) i * i = sqrt(1) i * i = 1
Ryan
"Punctuality is only a virtue for those who aren't smart enough to think of good excuses for being late" John Nichol "Point Of Impact"
Ryan Binns wrote:
i * i = sqrt(-1) * sqrt(-1) i * i = sqrt(-1 * -1)
The property that sqrt(a) * sqrt(b) = sqrt(a * b) only applies to real x >= 0. So you really can't do that.
They dress you up in white satin, And give you your very own pair of wings In August and Everything After
I'm after everything
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Observe: A Simple Proof that 22/7 exceeds Pi[^]. 22/7 is merely a convenient Diophantine approximation that people are taught in basic math so that they can have some frame of reference for Pi.
They dress you up in white satin, And give you your very own pair of wings In August and Everything After
I'm after everything
David Stone wrote:
Observe: A Simple Proof that 22/7 exceeds Pi[^]. 22/7 is merely a convenient Diophantine approximation that people are taught in basic math so that they can have some frame of reference for Pi.
Good heavens! I am not even gonna click that link! You math types are weirdos! Regards, Nish
Nish’s thoughts on MFC, C++/CLI and .NET (my blog)
The Ultimate Grid - The #1 MFC grid out there! -
David Stone wrote:
Observe: A Simple Proof that 22/7 exceeds Pi[^]. 22/7 is merely a convenient Diophantine approximation that people are taught in basic math so that they can have some frame of reference for Pi.
Good heavens! I am not even gonna click that link! You math types are weirdos! Regards, Nish
Nish’s thoughts on MFC, C++/CLI and .NET (my blog)
The Ultimate Grid - The #1 MFC grid out there!Oh c'mon Nish, you know you want to find out what a Diophantine Approximation[^] is. ;P
They dress you up in white satin, And give you your very own pair of wings In August and Everything After
I'm after everything
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Super Lloyd wrote:
do you mean never repeat?
I was under the impression it was infinite, just as 1/3 would also be. Jeremy Falcon
You're getting confused here because math people don't like to hear that a number is infinite. The term you're looking for is infinitely repeating. :)
They dress you up in white satin, And give you your very own pair of wings In August and Everything After
I'm after everything
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assume i = sqrt(-1) i * i = sqrt(-1) * sqrt(-1) i * i = sqrt(-1 * -1) i * i = sqrt(1) i * i = 1
Ryan
"Punctuality is only a virtue for those who aren't smart enough to think of good excuses for being late" John Nichol "Point Of Impact"
Hmm, it's like those 1 = 2 proofs. :-D I know SQRT(a) * SQRT(b) = SQRT(a * b), but does it hold for imaginary numbers as well? I'm uncomfortable with
SQRT(-1) * SQRT(-1) = SQRT(-1 * -1). It looks dubious to me. :suss: Cheers, Vikram.
I don't know and you don't either. Militant Agnostic
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Hmm, it's like those 1 = 2 proofs. :-D I know SQRT(a) * SQRT(b) = SQRT(a * b), but does it hold for imaginary numbers as well? I'm uncomfortable with
SQRT(-1) * SQRT(-1) = SQRT(-1 * -1). It looks dubious to me. :suss: Cheers, Vikram.
I don't know and you don't either. Militant Agnostic
Vikram A Punathambekar wrote:
I know SQRT(a) * SQRT(b) = SQRT(a * b), but does it hold for imaginary numbers as well?
Nope. That property only holds true for real numbers greater than or equal to zero.
They dress you up in white satin, And give you your very own pair of wings In August and Everything After
I'm after everything
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Ryan Binns wrote:
i * i = sqrt(-1) * sqrt(-1) i * i = sqrt(-1 * -1)
The property that sqrt(a) * sqrt(b) = sqrt(a * b) only applies to real x >= 0. So you really can't do that.
They dress you up in white satin, And give you your very own pair of wings In August and Everything After
I'm after everything
Ah, just what I'd suspected. Thanks, David. :) Cheers, Vikram.
I don't know and you don't either. Militant Agnostic
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it's a pure definition problem then? I think the issue here is the same as vegetable and fruit. Some people would say tomatoes is a fruit, some people would say it is not. And then they each refer to their own definition. Doens't matter too much. Anyway I will stick to my definition which means that PI is part of super set of the real (hence it is not a real number). And that Math teacher don't bother make the difference explicit until you are in advanced math studies....
Super Lloyd wrote:
it's a pure definition problem then?
NO, dude, it's not. Can you please tell me how you got the idea that PI is not a real number? Like I said, if a number is not real, it's square is negative. Is PI's square negative? Like I said, if a number is real, it's square is not negative. Is PI's square negative or not?
Super Lloyd wrote:
Some people would say tomatoes is a fruit, some people would say it is not.And then they each refer to their own definition.
Your analogy has nothing to do with the discussion.
Super Lloyd wrote:
Anyway I will stick to my definition
Please do. But since it's blatantly false, don't preach it to others, esp those who don't know and are trying to learn. Cheers, Vikram.
I don't know and you don't either. Militant Agnostic
-- modified at 1:18 Friday 17th March, 2006
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Super Lloyd wrote:
Anyway I will stick to my definition which means that PI is part of super set of the real (hence it is not a real number).
And you will be wrong. Pi is an element of the set of Real numbers. If you'd like to hold to an errant belief that Pi is outside the set of real numbers, please keep it to yourself, lest you confuse those, like Jeremy, who ask questions regarding mathematics.
They dress you up in white satin, And give you your very own pair of wings In August and Everything After
I'm after everything
Trying to help you out of this error and reading again this[^] memory refresher I realized I confound real numbers with their subset, the algebric numbers. Pi is part of the transcendental numbers set, which is included in real numbers set as well. Hoho.... Anyway I doubt it has confused anyone as these are just obscure definition problem regarding obscure property only (mathematical) scholar could have any interest about.
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Trying to help you out of this error and reading again this[^] memory refresher I realized I confound real numbers with their subset, the algebric numbers. Pi is part of the transcendental numbers set, which is included in real numbers set as well. Hoho.... Anyway I doubt it has confused anyone as these are just obscure definition problem regarding obscure property only (mathematical) scholar could have any interest about.
At least you realized your error. :)
Super Lloyd wrote:
Anyway I doubt it has confused anyone as these are just obscure definition problem regarding obscure property only (mathematical) scholar could have any interest about.
Well, I'm an applied math major. So I guess that means me. ;)
They dress you up in white satin, And give you your very own pair of wings In August and Everything After
I'm after everything
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Super Lloyd wrote:
it's a pure definition problem then?
NO, dude, it's not. Can you please tell me how you got the idea that PI is not a real number? Like I said, if a number is not real, it's square is negative. Is PI's square negative? Like I said, if a number is real, it's square is not negative. Is PI's square negative or not?
Super Lloyd wrote:
Some people would say tomatoes is a fruit, some people would say it is not.And then they each refer to their own definition.
Your analogy has nothing to do with the discussion.
Super Lloyd wrote:
Anyway I will stick to my definition
Please do. But since it's blatantly false, don't preach it to others, esp those who don't know and are trying to learn. Cheers, Vikram.
I don't know and you don't either. Militant Agnostic
-- modified at 1:18 Friday 17th March, 2006
Same as to David stone. Although it's funny your:
Vikram A Punathambekar wrote:
Like I said, if a number is not real, it's square is negative.
what about 1+i ? it square is 2i which is not negative, hence it is a real?
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David Stone wrote:
Observe: A Simple Proof that 22/7 exceeds Pi[^]. 22/7 is merely a convenient Diophantine approximation that people are taught in basic math so that they can have some frame of reference for Pi.
Good heavens! I am not even gonna click that link! You math types are weirdos! Regards, Nish
Nish’s thoughts on MFC, C++/CLI and .NET (my blog)
The Ultimate Grid - The #1 MFC grid out there!Nishant Sivakumar wrote:
You math types are weirdos!
Umm, in what way? ;P Cheers, Vikram.
I don't know and you don't either. Militant Agnostic
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Observe: A Simple Proof that 22/7 exceeds Pi[^]. 22/7 is merely a convenient Diophantine approximation that people are taught in basic math so that they can have some frame of reference for Pi.
They dress you up in white satin, And give you your very own pair of wings In August and Everything After
I'm after everything
Gee thanks!! I AM an idiot!! :doh: [edit] why's my post here? I replied to david's eyeopener for me!![/edit] If you need a hammer get C and shut up. If you need a nail gun get C++ and shut up. If you don't need *those* things (and good design should tell you) then by all means get a factory, factory, factory. --code-frog@codeproject -- modified at 0:11 Friday 17th March, 2006
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David Stone wrote:
Observe: A Simple Proof that 22/7 exceeds Pi[^]. 22/7 is merely a convenient Diophantine approximation that people are taught in basic math so that they can have some frame of reference for Pi.
Good heavens! I am not even gonna click that link! You math types are weirdos! Regards, Nish
Nish’s thoughts on MFC, C++/CLI and .NET (my blog)
The Ultimate Grid - The #1 MFC grid out there!:laugh: If you need a hammer get C and shut up. If you need a nail gun get C++ and shut up. If you don't need *those* things (and good design should tell you) then by all means get a factory, factory, factory. --code-frog@codeproject
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As Chris Munder said, it's a Transcendental number[^], much more uncommon than mere real number.
Somebody has probably said this already, and from the perspective of almost all numbers people think about in real life, it's absolutly true, but really the vast majority of real numbers are transcendental. So there are algebraic numbers, which are solutions to polynomial equations with rational coefficients. So 2 is algebraic(solution to x-2=0), Sqrt(2), which solves x^2-2=0, is, and so on. Then there are transcendental numbers, which are numbers that aren't solutions to ANY polynomial. Pi and e are two examples; both are pretty deep proofs, and pretty much have to be taken at face value unless you're a pro and have a spotter. Since there are as many polynomials as there are finite length sequences of rational numbers, there are as many algebraic equations as there are fractions and, in turn, as many algebraic equations as natural numbers. (by "as many as", I mean that we can pair each algebraic number with a natural number-1,2,3,4,...- so that each one from each group has exactly one partner from the other group.) Since the set of algebraic equations has the same size as the natural numbers, we say that it's a countable set. So for both algebraic equations and fractions, there is a way to count them up, 1,2,3, so that you hit every one of them. (I was blown away the first time I heard there was a way to count the fractions.) And since each algebraic eqation has a finite number of solutions, the set of all algebraic numbers is countable. You can also prove that the set of all real numbers is not countable. You can prove that any way you pair up natural numbers and real numbers, there will always be real numbers left over. A casual way to think about these sizes is that the size of the real numbers is as much bigger as infinity is from a some finite number. So in exactly the same way that (infinity - 10)=infinity, the uncountable real numbers-the countable algebraic numbers leaves an uncountable amount left over. So therefore "almost all" real numbers are not solutions to algebraic equations. And the "almost all" here means exactly the same thing as if I said that I'm thinking of a number between 1 and infinity, and you were to guess, that you have "almost no" chance of guessing correctly. So, "almost all" real numbers are transcendental. Was that kid sister safe?
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Marc Clifton wrote:
You can see how the series is infinite, but smaller and smaller.
I got all of that stuff. But it doesn't explain why the series exist in the first place. Fortunately, I think I got enough info from the fine folks at CP to help me set out a course to improve my understanding on it. I'm just not there yet. :) Jeremy Falcon
I think most people above didn't approach their explanation as follows: Pi is an "infinitely long" number due to its transcendental nature. I will explain why I put that in quotes. That is, there is no closed algebraic function whose solution yields exactly pi nor are there two whole numbers whose ratio expresses pi exactly. Consequently, we must use an approximation to pi. The precision of the approximation is limited to the computing power and time you want to spend on calculating pi. That is why the approximation to pi is infinitely long.
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Vikram A Punathambekar wrote:
By definition, i * i = -1
Thankyou! Someone who gets the definition correct! :-D I see most people say that i = sqrt(-1), which is NOT correct - it implies that i2 = 1
Ryan
"Punctuality is only a virtue for those who aren't smart enough to think of good excuses for being late" John Nichol "Point Of Impact"
Hang on. If the definition if i is i^2 = -1 then i = +/- sqrt(-1) Nothing wrong with i := sqrt(-1) cheers, Chris Maunder
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