As someone once said, the best maths is the maths you can use without reverting to deriving everything from first principles. But you need to understand the first principles so you don't violate them.
Example -1=1, without misuse of zeros, and in an example which can foul up frequency domain analysis of electronic circuits:
1: sqrt(-1) = sqrt(-1)
2: therefore sqrt(-1/1) = sqrt(1/-1)
3: therefore sqrt(-1)/sqrt(1) = sqrt(1)/sqrt(-1)
4: therefore sqrt(-1)*sqrt(-1) = sqrt(1)*sqrt(1)
5: therefore -1 = 1
QED
Almost everybody can correctly determine that the error is going from 2 to 3, but they can't say what the error is. I've only come across one person who could give the reason.
Back before Christsmas I had PMs requesting that I show the error. Given the difficulty of formatting maths on this forum, it will be difficult, but I'll do my best.
Firstly, as any fule knos, "i" represents current, so I'll use the traditional "j" to be sqrt(-1), and I'll use "w" instead of omega=2*pi*f. Secondly, this kind of manipulation can arise when doing frequency domain analysis of circuits, where term such as
1/
(R+jwL) are common. Thirdly, I'll note that MrFlibble's comment about using the Euler form is equivalent to using the complex conjugate described below.
The key is recognising that a full representation of a complex number is a+jb, and that sqrt(-1) is a special case when a=0 and b=1.
If we want to "get rid of the complex denominator" / "move the j to the top line" in
1/
(a+jb) we have to multiply top and bottom by the complex conjugate (a-jb).
(
1/
(a+jb))*(
(a-jb)/
(a-jb))
Multiplying out the numerator and denominator gives
(a-jb)/
(a2-jab+jab+b2)or
(a-jb)/
(a2+b2)Now for the special case where a=0 and b=1, we can see that
1/
j =
-j/
1 , and that's why
sqrt(1/-1) != sqrt(1)/sqrt(-1)
And that's why despite "the best maths being the maths you can use without reverting to deriving everything from first principles",
you need to understand the first principles so you don't violate them.
I hope that's all legible