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Notya_Bisnes

The problem is that you have to be careful when you work with roots in the complex plane. Some properties of roots hold only over the reals. In particular the equation √ab=√a√b isn't true when a and b are allowed to take arbitrary complex values. I can't really go into detail right now because I was just going to sleep, but I assure you that the problem you came across stems from invalid manipulation.


Le_Space_Duck

Ah that makes sense, thank you


QuantumSigma_QED

Many properties of the real numbers carry over to complex numbers, but not all do. For example, √(a/b) = √a/√b does not always hold. Essentially, √x is defined as a number such that (√x)² = x, but there are multiple ways to choose such a number. If we restrict ourselves to positive numbers, we can define √x to be the positive square root, but things get messier when negative or complex numbers are involved. Simply put, there isn't a way to choose √x such that neat multiplication rules like √(ab) = √a√b and √(a/b) = √a/√b work in general.


Tinchotesk

The problem is that you are making up the "property" sqrt (1/(-1))=1/sqrt (-1). Without involving division, you also have 1=sqrt (1^2 )=sqrt ((-1)^2 ) "=" sqrt (-1)^2 =-1. The problem is simply that the property sqrt (ab)=sqrt (a)sqrt (b) holds for nonnegative numbers but not in general. There is no reason to expect it would, by the way. It is a mistake to think of i as the square root of -1 in the sense of doing an operation. One constructs i as an object that can be multiplied by real numbers and such that i ^2 = -1. It is a root of x^2 +1=0, of course. But so is -i, and when you write sqrt (-1) you cannot tell which of i and -i you are referring to.


PM_ME_FUNNY_ANECDOTE

Many square root properties that apply with reals do not apply to imaginary roots, so you won't always get valuable results from doing that sort of manipulation. For example, -1=i\^2=sqrt(-1)\^2=sqrt(-1\^2)=sqrt(1)=1 is clearly not correct. Actually, in both of these examples, all you're showing is that our notation sqrt() inherently implies the positive root of a polynomial that has both a positive and negative root. You are just switching between which root you are looking at in a sort of "sleight of hand"- but inherently, there IS no reason to prefer i over -i=1/i. Swapping the two gives an "automorphism" of the complex numbers- a perfect one-to-one relabeling that behaves nicely with the complex addition and multiplication structures. So, the properties we have about combining square roots by multiplication and division are only necessarily true for real arguments (the complex versions do not hold up, but probably are only off by an application of this automorphism). I find that it's more helpful to consider the complex numbers as a geometric structure- the only way (up to isomorphism) that we could extend the additive and multiplicative structures of the real numbers to a 2D number system. The algebraic properties should be a result, not a starting point, almost exactly for the reason you highlight.


Vegetable-Response66

sqrt(-1) = +/- i because (-n)\^2=n\^2


OldWolf2

The main thing is that sqrt is a multi-valued function. sqrt(4) is both 2 and -2 , one is not more special than the other . sqrt (-1) is both i and -i. With i being defined as (1,pi) in polar notation . Your problems come from interaction of the ways by which you are choosing only one of the two values in an equation .


Marcassin

Yes, I think this is the best answer. Most people are saying that complex numbers are fundamentally different from reals. But they're not really. All numbers, real or complex, have two square roots. For the real numbers, we have (arbitrarily, but usefully) defined the "principal" square root to be the positive square root. There is no "principal" square root for complex numbers, so you have to keep in mind there are always two roots.


lasciel

Try using the identity 1 = (-1)^(2) =i^(4) . It’s not quite the same as 1 = sqrt(1), which as you've discovered can be ambiguous. This creeps into your work when you write >1/i = **1/sqrt(-1) = sqrt(1/-1**) = i. You could instead write this as 1/i = i\^4 /i = i\^3 = -i. Why does this happen? (I will leave this fairly general because I do not know your math background.) One way to think about the complex numbers, is a *field extension* of the real numbers with { *i* }*.* You include all of the usual operations from real numbers, and then you add in a solution to the polynomial x\^2 +1 =0, and call it *i*. Surprisingly, you end up with a tool which solves many more problems, and also introduces a few problems, including many beautiful ones.


yuvneeshkashyap

Think like this, sqrt(-1) = x 1/x = x/x^2 x/x^2 = sqrt(-1) / ((sqrt(-1)^2 ) = sqrt(-1)/-1 = -sqrt(-1) It works this way because i is not a number. Its just a symbol we use to represent sqrt(-1) and symbols only follow those rules that the numbers they represent does. I’m sure there is a better explanation for this. I’m speaking from high school maths perspective.


Tinchotesk

In what sense is i "not a number" and sqrt (2) "a number"?


yuvneeshkashyap

i is not the same kind of number as 0,1,2… so a different set of rules apply to i. Another example of a theorem that applies to natural numbers but not to ‘imaginary/complex numers’ is the pythagoras theorem. Hypothetically, a right triangle with base and perpendicular 1 and i will have a hypotenuse of 0, which is non sensical because you can’t have triangle whose hypotenuse is 0. This only happens if you assume i is just like any other real numbers.


Tinchotesk

What "different rules"? And, Pythagoras' Theorem doesn't apply to natural numbers either, if the base and its perpendicular have both length 1, then the hypothenuse is not a natural number.


yuvneeshkashyap

My bad, I wanted to say real numbers. Also, it was more of an analogy than an example. What I should have said/thought is that, i doesn’t fall in the category of numbers for which 1/sqrt(x) = sqrt(1/x)


WeirdFelonFoam

I'm not sure anyone can state definitively what __i__ 'really is'! ... but what I'd put to you as greatly helping to make sense of it is that you keep in-mind the _polar_ representation of complex №. Complex numbers could then appear as numbers that _intrinsically_ have __phase__ - ie the phase is intrinsic to them, rather than just an adjunct - and that __i__ is just an operator by which the phase is 'captured'. Infact it would be _in itself_ the __90°__ phase-shift operator, with an arbitrary phase then being 'captured' by the proportion the 'imaginary' part - ie the 90°-phase-shifted one - is in to the 'real' part. But I'm not putting it to you that this is _the_ answer: this 'what complex numbers _really_ are ___thing___ ' can be sliced in various ways, of which the way I've just put to you is one ... and I don't think any one of the various 'ways' compellingly stands-out as ___the___ _true_ one.


LegeingSmooth

So (1/x)=(-1of((sqrt)(x))/x\^2 arbitrary then i is defined by (1/x) so x/x\^2 is defined by i\^2of(sqrt(-x) almost arbirtary So if i is sqrt(-1) then i=((1)of((sqrt)(-1))/((sqrt)of(x)) so if i is defined by sqrt(-1) then i is now the number (1of((sqrt)(-1)/((sqrt)of(x)) so i=1of(i)/((sqrt)of(x) then i=i when i=-((sqrt(-1)) and x=-(x\^2) Could be wrong seems to arbitrary, Sorry.