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the truth is i^2=-1 and that is correct
of course i<>(-1)^1/2 ,and if u have some problems whit this ask me or good book
:) :) :) :) :) :o :o :o :o :o
OK, Back to this again.Quote:
Originally posted by mmiill
the truth is i^2=-1 and that is correct
of course i<>(-1)^1/2 ,and if u have some problems whit this ask me or good book
:) :) :) :) :) :o :o :o :o :o
Yes, we all agree i^2=-1 . this is what I would call a property of i, and not a definition.
But the discussion {IMHO} is directed at why i<>(-1)^1/2. Obviously,
(-1)^1/2 Squared = 1, which satisfies the stated property of i,
so why can't i = radical(-1)?
:)
-Lou
Not Really, since i is a unit vector, then -i is -1*i, which does not rotate the mapping of the complex plane. It is just a seperate vector, opposite that of 1*i.Quote:
Originally posted by bugzpodder
the way complex numbers is defined, if you choose -i to be your i, you would still have the same number system, with a rotation of numbers in the complex plane. hope this explains better of my position. :D
i, symbolically, is assumed to be postive, and by the standard definition of i that I have in mind, the sqrt(-1) is i, and not -i, just like the sqrt(16) is some postive n.
urrgh i hated it when vbforums says i am not logged in and get rid of my msg all together! now i have to retype it again:
anywayz, yes it is not a rotation, it turns the number upside down on the complex plane.
sqrt(x) is not defined for x<0 because for example sqrt(-1) can be i or -i. if you really want to impose your own definition on sqrt(-1) to have it to return i only, then i have to say that i=sqrt(-1). however i believe i^2=-1 is a better definition than i=sqrt(-1) because there is a hole in the ladder (you must somehow prevent it in the definition):
i^2=sqrt(-1)*sqrt(-1)=sqrt(-1*-1)=sqrt(1)=1
but i^2=-1
you can argue that if you do it another way, you'll get -1 (such as the two sqrt() cancel out), but the above method is technically legal in the real number system, thus by my meaning as a hole.
by the way, UoT webpage here explains the matter a little bit better than I do: http://www.math.toronto.edu/mathnet/...s/guess12.html
Hmmm. Interesting.
The page you directed me to does actually say
So, by the etc, radical(-1) = i.Quote:
So, for example, one could say that what one means by {the copy on this page dropped this out==>} radical(-4) is 2i,
what one means by {ditto} radical(-9) is 3i, etc
Now, not knowing what page leads to this "This step is not the source of the fallacy." page, could mean much, but the page does clearly indicate that radical(-1) couldd = i.
So, By your own website, it is not MY definition of i, but actually a well recognized def of i that they adrressed, and recognize as "the square root symbol when applied to a negative number is one of its two complex square roots".
Which means, with -1 as the negative number, the square root of said number is one of its two complex square roots, meaning i is the square root os -1.
Otherwise, prove to me, NOT that we can't calculate sqrt(-1), but that sqrt(-1) does not exist.
Your own website claims sqrt(-1) exists on another page.
:)
-Lou
If we're talking about complex numbers, then it doesn't make any sense to say the domain of sqrt is +ve x. This only applies when you are confined to the Real numbers.
The equation i^2 = -1 doesn't define i, it merely states a property, as NotLKH said. The definition i = sqrt(-1) does make sense, as we already have definitions of sqrt(x) for x >= 0. With this one extra definition we can extend the domain of sqrt to all Real numbers, by virtue of sqrt(-x) = sqrt(-1.x) = sqrt(-1).sqrt(x) = i.sqrt(x).
To say it is using a complex number to define a complex number (can't remember who said that, sorry) is absurd. It defines a Complex number in terms of a function on a Real number. I'd like to see a definition of i that doesn't use a complex number (namely i) on the left hand side.
And to all the people who are whinging about multiple roots, i is defined as the positive root of -1. Therefore -i is the negative root.
That's how I see it and how I've been taught it throughout my life.
OOPS, I missed that.Quote:
Originally posted by bugzpodder
i^2=sqrt(-1)*sqrt(-1)=sqrt(-1*-1)=sqrt(1)=1
[/url]
That is really interesting. I never really noticed it before,
However, since I've always used i=sqrt(-1) as the def of i, with
i^2 = -1 as its property, I've never had to worry about it either.
as i said before you can take two sides:
claim sqrt(-1) exists, but in the same time define i as the answer to that question (not -i). just becareful of that false example I showed. u might be caught in it.
or like me, just ditch sqrt(-1) at all.
just look at this page, see if you can find the error:
http://www.math.toronto.edu/mathnet/...econd1eq2.html
the error, according to the website, is a hidden way of the false example I've provided.
Caught it in my first attempt.
:)
-Lou
Why can't i exist?
If i is used in mathematics, and math exists, then i must exist.
However math is just abstract ideas used to represent concrete situations, so i suppose i doesn't really exist???
So how are we so sure that there can't be a sqrt of -1, even if it contradicts itself when we write i^2=-1.
Really, since it's a debatable issue, either definition is acceptable at this point in mathematical development, but I've been corrupted to believe that i^2=-1, from a source that no one contradicts. lol
lol i tried to contradict your "source" and when he replied my mind drew a blank. I don't know why it happened though. lol at least i had the courage to do that (maybe he won't like me anymore :()
depend on your definition of exist. there is no such thing as i apples but there is things such as roots of x^2=-1Quote:
Why can't i exist?
I exist, and I tell you i doesn't exist. does that make i exist or not? lol my point is this arguement is pretty poor.Quote:
If i is used in mathematics, and math exists, then i must exist.
there is actually two sqrt of -1Quote:
So how are we so sure that there can't be a sqrt of -1, even if it contradicts itself when we write i^2=-1.
but sqrt(x) when x<0 or x is an element of a complex number, sqrt(x) is not properly defined. (unlike when x>=0, sqrt(x) is defined to be the principal square root of x) you see, in complex numbers, there are no positives and negatives.
i=sqrt(-1) is an acceptable definition provided that you don't make the mistake as follows:
i*i=sqrt(-1)*sqrt(-1)=sqrt(-1*-1)=sqrt(1)=1
now the problems lies at sqrt(1), here it no longer should be the principal square root, but is actually the negative square root (since you've let the domain of x to be <0) thus thats what I mean by not properly defined (you wouldn't know which root to use).
Ha ha... if you have an array of 5i apples by 5i apples, you'd have -25 apples all up (but only if you multiplied). If you added them up, you'd apparently have 25i apples.Quote:
Originally posted by bugzpodder
depend on your definition of exist. there is no such thing as i apples but there is things such as roots of x^2=-1
Right! So, if you wish to denote i = (-1)(1/2)Quote:
Originally posted by bugzpodder
i=sqrt(-1) is an acceptable definition provided that you don't make the mistake as follows:
i*i=sqrt(-1)*sqrt(-1)=sqrt(-1*-1)=sqrt(1)=1
....
Then when you perform any calculations, do so where i2a = -1a
;)
OK.Quote:
Originally posted by Dreamlax
Ha ha... if you have an array of 5i apples by 5i apples, you'd have -25 apples all up (but only if you multiplied). If you added them up, you'd apparently have 25i apples.
Let me get this straight.
if you had an array, where each side was 5i APPLES??? in length,
then its area is -25.
Now, if each cell in that array contained the numbers 1 thru 25, then it would add to , hmm, (1+25)+(1+25)+(1+25)+(1+25)+(1+25)+(1+25)+(1+25)+(1+25)+(1+25)+(1+25)+(1+25)+(1+25)+13 = 12*26+13 = 52+260 + 13 = 312+13 = 323.
So, 323 = -25, by your logic.
Good, then I guess that 2=3 thread is resolved!
:D
You are assigning the size of the array as some number times a physical thing.
Even without i, that is meaningless.
:(
Huh? 5*5 = 25, 5i*5i = -25....
Each apple would be i (so that the total for that row would be 5i). Since there would be 25 items in total ( from |25i| ) then adding them up would be like i+i+i+i+i.... or 25*i.
I just got up to check my e-mail (waiting for an important one), I might go back to bed if I'm disproved again.
Again, you are confusing dimensions with that which is contained in an array.Quote:
Originally posted by Dreamlax
Huh? 5*5 = 25, 5i*5i = -25....
Each apple would be i (so that the total for that row would be 5i). Since there would be 25 items in total ( from |25i| ) then adding them up would be like i+i+i+i+i.... or 25*i.
I just got up to check my e-mail (waiting for an important one), I might go back to bed if I'm disproved again.
Now, talk about an impossibility!
How do you Dim an array, where its X Ubound is 5i Apples, and its Y Ubound is 5i Apples?
Or even, where Arr(X,Y) has an Ubound of its X = 5I, and the same for Y?
How,How,How,How,How,How,How,?????
:(
Right NotLKH, thats why i^2=-1 is a better difinition!
;P
heh!Quote:
Originally posted by bugzpodder
Right NotLKH, thats why i^2=-1 is a better difinition!
;P
lol!
BUT,... Thats a property, and not a definition!
:D
i only has one property that is relative to the real numbers, that is i^2=-1
btw new people may get tricked by my example so i^2=-1 is definately better than i=sqrt(-1)
Well, I can't disagree with that!Quote:
Originally posted by bugzpodder
i only has one property that is relative to the real numbers, that is i^2=-1
btw new people may get tricked by my example so i^2=-1 is definately better than i=sqrt(-1)
:D
Except,
"new people"?
How New?
:p
anyone who doesn't know the example lol.
Who said we were programming? I can have an array of saucers on a table if I wanted to!Quote:
Originally posted by NotLKH
Again, you are confusing dimensions with that which is contained in an array.
Now, talk about an impossibility!
How do you Dim an array, where its X Ubound is 5i Apples, and its Y Ubound is 5i Apples?
Or even, where Arr(X,Y) has an Ubound of its X = 5I, and the same for Y?
How,How,How,How,How,How,How,?????
:(
Uhm...
(-1)^(1/2)=i
because (-1)^(1/2) is the sqrt(-1) which is i.
and...
i^2 = -1
and i^3 = -i
and i^4 = 1
and... whatever....
all i really is, is just a nice way to write sqrt(-1)
because replace sqrt(-1) with an i and you get the same result
so i = sqrt(-1)
and since it's always best to show equality with the varible alone, i = sqrt(-1) would be technically the best
sqrt(-1)*sqrt(-1) = -1
just like
sqrt(2)*sqrt(2)=2
and .999.... <> 1
duh.
1)Why??Quote:
and .999....<>1
duh
2) For everyone that doesn't know, a very useful HTML tag is the superscript- good for powers.
e.g:
i [ s u p ] 2 [ / s u p ] (without the spaces)
= i2
Also subscript - good for Chemistry.
H [ s u b ] 2 [ / s u b ] (without the spaces)
= H2
also, sqrt(2)*sqrt(2)=sqrt(2*2)=sqrt(4)=2Quote:
Originally posted by DiGiTaIErRoR
all i really is, is just a nice way to write sqrt(-1)
because replace sqrt(-1) with an i and you get the same result
so i = sqrt(-1)
and since it's always best to show equality with the varible alone, i = sqrt(-1) would be technically the best
sqrt(-1)*sqrt(-1) = -1
just like
sqrt(2)*sqrt(2)=2
but, sqrt(-1)*sqrt(-1)=sqrt(1)=1
i=sqrt(-1) isn't any good PRECISELY BECAUSE OF THIS!
I just noticed, the error comes about from useing the sqrt(-1) asQuote:
Originally posted by bugzpodder
also, sqrt(2)*sqrt(2)=sqrt(2*2)=sqrt(4)=2
but, sqrt(-1)*sqrt(-1)=sqrt(1)=1
i=sqrt(-1) isn't any good PRECISELY BECAUSE OF THIS!
a proper function, but since sqrt(-1) cannot be evaluated thru the
normal properties of sqrt, then the associated properties of the sqrt function cannot be relied upon. So, trying to do a proof useing sqrt(-1) will produce imprecisions similar to that when we've seen division by zero in proofs.
So, My Suggestion, lets define the rule of priority, no Matter What,
Once you see sqr(-n), you MUST substitute i*Sqr(n), before you
do anything else.
;)
Hmm, I would have to say that i is implicitly defined as i2 = -1, because of what Bugz said. I don't think i can be explicitly defined accurately...
i agree with you guys. new people who does not know this may follow exactly what I did if they were given i=sqrt(-1)... (not saying that they will every time but there is a chance that they might)
Edits (hereinafter) are in BoldQuote:
Originally posted by NotLKH
So, My Suggestion, lets define the rule of priority, no Matter What,
Once you see sqr(-n), you MUST substitute i*Sqr(n), before you
do anything else.
I was taught to use i more like a variable, and that Sqrt(-n) [where n is positive] should be rewritten with Sqrt(n×i2) which allows you to square rooting it easier, since the power of half will cancel the power of two.
Are there any limitations or problems doing it the way I do? A lot of the stuff I post on here which I learnt from school seems to have a talent of being wrong.
if you are taught in school its most likely right than wrong. but there are a few things your teacher may not know (that goes with experience rather than teaching skills) that happens we are all human but i just want to stress again that its most likely right than wrong.
Well, it's kinda like in Chemistry where they deliberately teach you wrong as to not confuse you, like electron configurations.
First of all they say the e.c. of Na is (2, 8, 1) then later they tell you that it's wrong and to write it as (1s2 2s2 1p6 3s1) or something... I can't exactly remember.
Actually:Quote:
sqrt(2)*sqrt(2)=sqrt(2*2)=sqrt(4)=2
sqrt(2)*sqrt(2)=sqrt(2*2)=sqrt(4)=+/-2
So, you could sort of say, using this, that sqrt(2) -real number- squared can give a negative number (-2). Which is wrong
=> This logic is not really valid. (At least, after reading it for 10 min. it is not. Maybe after longer i might have changed my mind. Just a thought)
yea thats why sqrt(4) is defined to be the principal square root which is 2. however it isn't a reason that you do not acknoledge the other square root with is -2, since (-2)*(-2)=4
plus the domain of sqrt(x) defines x to be a real >=0 so technically you can't even put sqrt(-1) because of the definition
I was intrigued by Notlkh's signature and hence..........
Now, my exposure is to Math is limited to High School and what little I ahve picked up along the way later, and no way do I claim myself an authority:o
That said, my two paise...........
Assumptions:
1) A Definition can be only one of two: It has to be constructed arbitarily from one's whim and fancy OR inducted/deducted from an already established definition or observation.
2) Mathematical proof is solely dependent on consistency within the given framework.
Axioms:Code:Definitions:
1: {Arbitary} i is point on the real plane such
that it is a distance y, 90 degrees from the point x.
Further, x is identified with the point (0,0) and y with point (0,1)
2: {Derived from an existing entity viz. POWER} The nth root of A
is that which when raised to power n results in A.
Using Definition One:Code:1) In a plane; (x,y) * (u,v) = (xu - yv, xv + yu)
i = (0,1)
Using Axiom One:
i * i = (0,1) * (0,1) = (-1, 0)
Ergo: i 2 = -1
Using Definition Two on the above result
n = 2 ; A = -1, ergo: Square Root of -1 = i
QED
Neither of the definitions I have proposed are wrong. Therefore, the above proof(s) are admissable, given my assumptions. As to their being right, I'll look forward to your bouquets and brickbats.:)