i and inconsistent functions and properties

Since early days, well maybe grade 9 or later, we all encounter the mystical number “i”, we get familiar with real numbers fairly easily, they are mostly intuitive (but there is a lot of unintuitive stuff) but i eludes a lot of people. What is it? How does it relate to real numbers? How do we compare them? Why is it functions get so much more complicated with i? etc.

 

I will come to describe how complex numbers are constructed at a later date, but for now we agree that complex numbers are in the form of a+bi with a and b being real numbers and leave out the strict formalities here.

 

A lot of cranks want to claim this is inconsistent because they claim that i=\sqrt{-1}=\sqrt{\frac{1}{-1}}=\frac{\sqrt{1}}{\sqrt{-1}}=\frac{1}{i}=-i, which is absurd as a non-zero number cannot equal to its negative self so its inconsistent and wrong!

 

Seems convincing, no? There are a whole lot of errors here that I will work through. First and foremost to get rid of the notion that i=\sqrt{-1} is the DEFINITION! Which it isn’t, we define i with this relation i^2=-1, this might seem nitpicky and it is but for very good reason that I will return to soon. The most glaring issue is the step \sqrt{\frac{1}{-1}}=\frac{\sqrt{1}}{\sqrt{-1}}. This step is where it all breaks down. They use \sqrt{x} in the same fashion as if it was real numbers. Let me explain more detailed.

When we write \sqrt{x}, or any equivalence, we mean a specific type of function with a specific domain, that is the set of all inputs that it is defined for. We often omit writing there cause context can salvage most of it and in most cases, it just doesn’t matter but here it does. Normally we have \sqrt{}:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}, that is we have squareroot defined as a function from the non-zero real numbers to the non-zero real numbers. Why is that so? Because in real numbers, we cannot make sense of squareroot for negative numbers, it is just non-sensical, this is what complex numbers are attempting to fix but that is not relevant right now. Now to distinguish them I will write {}^+\sqrt{} for \sqrt{}:\mathbb{R}_{\geq 0}\to\mathbb{R}_{\geq 0}. This function will have certain properties we can prove, like the familiar {}^+\sqrt{xy}={}^+\sqrt{x}{}^+\sqrt{y} which is what they use.

The astute reader will know we are still working on real numbers, what happens when we input a complex number into our {}^+\sqrt{}, NOTHING! We cannot do it! It has no meaning for complex numbers, what we need to do is EXPAND our function {}^+\sqrt{} such that it can take complex numbers. But notice when we expand it, it no longer is the same function! However, for evident reasons, while we can expand it in infinitely many ways, it makes intuitive sense and align with our desires that the expansion will be such that if we still put in just the real numbers, and ignore any complex number, it would coincide with our original {}^+\sqrt{}, in mathematics we would write something like \sqrt{}|_{\mathbb{R}_{\geq 0}}={}^+\sqrt{}, which means that when we restrain the domain to positive real numbers, we cannot tell the two functions apart cause they always yield the same thing.

 

There is a natural way to do this and we do define it as such, I won’t go into the details here, but we do get \sqrt{}:\mathbb{C}\to\mathbb{C} such that they coincide and we can get a meaningful way to get \sqrt{-1}=i. This is what I talked about before, we cannot define i in that manner because to put -1 into our squareroot, we must expand its domain and to do it, we need the complex numbers so it becomes circular if one tries to define it as such. If we define it as the square, the problem vanishes. Now, when we expand a function, a natural question arises, does the properties we had the luxury of before remain? Are they preserved? The general question is….no they are not. Sometimes it might remain and in certain expansions, but in general you cannot assume it. It is something you must PROVE that it is conserved, you cannot just ASSUME it. And that is the issue, they assume the distribution of squareroto remains and the fact of the matter is, what they prove is not what they want. Then want to go like this

  1. Assume complex numbers work
  2. Show that squarerooting gives a contradiction
  3. Therefore our assumption of complex numbers was false.

When in fact, what they end up proving is only

  1. Assume squareroot retain its former properties
  2. Show it leads to a contradiction
  3. THerefore the assumption of squareroot is false.

This is a keydifferens because the complex numbers as a field has only addition and multiplication and there, it is perfectly consistent. Squareroot is not a “native” function to complex numbers, it is one we add as an addition to perform computation and solve things but it is not innate. You cannot make algebraic structures with it being “native” beacuse most often, it requires a very restricted image or domain to be workable. To demonstrate that complex numbers as a field is inconsistent or wrong, you have to use addition and multiplication, nothing else. Any proof involving any additional functions will only invalidate that singular functions definition.

 

Always be careful with the domain of functions and how it changing changes the function, when they overlap and are expansions we mathematicians, out of sloth, omit it cause we assume it is salvagable by the knowledgable.

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