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Papabravo said:What is wrong is that in the second line you assumed that i was equal to 1/i and that is not true. The algebra of complex numbers is different than the algebra of real numbers. That is why we use a different symbol so that we won't be tempted to make inconsistent assertions like that one. The problem with your original assertion is that when combining real numbers and complex numbers the order of operations is important AND many of those operations are NOT commutative.
Papabravo said:In order to substantiate the claim that the two algebras are the same you need to start with elements of the set which are different. Now it is true that the algebra of real numbers is contained within the algebra of complex numbers. It is the set of all complex numbers with the inaginary part set equal to zero. There is a much closer relationsip between vector algebra and complex algerbra.
43617373 said:How is a number being obtained from a negative in the square root, or am I forgetting a large part of my algebra.
I invite the other members to reread your original post to see if you made precisely that claim. It sure sounds like you did to me, but what do I know, I'm just a graduate engineer.3iMaJ said:I made no such claim. I only said there certain rules that numbering systems follow. Yes the real numbers are a subset of the complex numbers, but that in no way implies that there are two different kinds of algebra?
And a side note, real numbers can be just as easily be manipulated using vector algebra as complex numbers, although they typically aren't because there is no reason to complicate it to such a degree.
In the end its all algebra, all following the same rules.
Papabravo said:I invite the other members to reread your original post to see if you made precisely that claim. It sure sounds like you did to me, but what do I know, I'm just a graduate engineer.
Nigel Goodwin said:My daughters doing A level Further Maths, and apparently you need to be 'weird' to be any good at it - she fits right in!
According to the teachers, A level Further Maths is more difficult than a normal maths degree, and where top universities require applicants to have grade A's in their subjects, even a grade E is acceptable if it's Further Maths.
Needless to say, I haven't got the faintest clue what she's doing with it! - means she can't ask me to help
I did offer to help if she took Physics, but she took Chemistry, Maths, Further Maths and Music Technology - so I'm safe!
quixotron said:Further maths??
A question often arises what are complex numbers useful for?
The "imaginary" number i is a consequence of how we define the operations + and * in our new field we're calling the complex numbers. A complex number is simply a number that has operations + and * that satisfy several properties that were listed earlier.
Its VERY important to note, and this was the point made in the previous post is that the sqrt(.) function is only defined for real x >= 0.
This is incorrect because the square root function IS NOT defined for x < 0
Crisis averted.
Ratchit said:3iMaJ,
I don't know why you are proving binary properties of associativity, communitivity, and distributivity on the unary operator i.
I don't believe so. Anyone who has worked in engineering knows how useful duplex numbers are.
Actually, there is no "imaginary" number. That is a misnomer. The number most folks call imaginary is actually a number that is orthogonal to the real number taken as the reference. For instance, in electrical engineering, the energy in AC circuits within the magnetic and electrostatic fields represented by "imaginary" numbers is every bit as real as the energy disipated by the resistive component.
Well, every negative number has two square roots. I already showed that -1 has a square root of plus or minus 1/_90.
Well, I say that that isn't so for the above reason.
Crisis that never was. Ratch
And I'd like to point out that i is not an operator. The operators of the field are + and *. i is simply (0,1) its a number within the complex set, not an operator.
You'll also note that I put "imaginary" in quotations just as I've done here. That is commonly what they're called, even though it is technically incorrect.
Reminding you (0,1)(0,1) = i^2 = -1.
HOWEVER sqrt(-16) w/o separating it into sqrt(-1)*sqrt(16) is not defined for the square root function.
The mathematical definition of the square root is that its the inverse function of f(x) = x^2 for all real x >= 0. In other words the square root function is a mapping of non-negative real numbers to R+ union 0. The key there is non-negative.
Quit with all the orthogonal nonsense, of course a purely real number is orthogonal to a "imaginary" by construction.
(a,b) is our ordered pair construction, so (a,0)*(0,b) = (0,0). Which is by definition orthogonal. This true because of how we constructed our field, and not for any other reason.
(0,1) is 1<90 because of the mapping of the complex numbers to the complex plane. Your orthogonality isn't a consequence of the mapping like you seem to claim, but rather a consequence of the construction of the field.