Is there a way to prove that i²=-1?

Question

I have 4 questions regarding the imaginary and complex numbers. (And some ideas)

My questions are about the way that I’m trying to come up with a proof to the equation i²=-1 (and from there maybe also the equation sqrt(-1)=i or -i) without of course first accepting them to be true axiomatically.

I would like to hear your ideas! The following is not any kind of proof itself, but just some ideas that I am having and I would like to know if they are correct or not and what are your ideas about them.

---

We start by having 2 axes, one for the real numbers and one (at right angles to it) for the imaginary ones, forming the complex plain. But because we have not defined any complex numbers yet we should better avoid naming it "complex plain" just yet.

xx' axes are the axes of the real numbers

yy' axes are the axes of the imaginary numbers

*(note 1: we should not call "i" or any other imaginary number, "imaginary" yet. At least not before proving them to be so)

*(note2: angles are measured counterclockwise from the x’x axes)

I will first take i and give this new definition where: i=1<90° (or i=(1,90°))

1) i=1<90°

That notation means that i is a number on top of a vector with a length of 1 unit and 90° degrees angle counterclockwise from the x’x axes.

I will also take -1 and make it a vector on my complex plain. -1 of course is a number on a vector with 180° of angle from the x’x axes and a length of 1 unit. Therefore -1 can be written as a vector of (1<180°). This can also be written as -1=(1,180°) but I'll just stick with the complex polar form for the rest of this approach.

*(note3: the symbol < means "angle" by the way)

2) -1=1<180°

Then I will proceed on squaring this new i number.

Squaring i will also square the vector (1<90°) and we all know that squaring this vector in polar form equals to ( 1*1 < 90°+90° ) which equals to (1<180°)

i²=(1<90°)² =>

i²=(1*1 <90°+90°) =>

i²=(1<180°)

Therefore I ended up with this equation i²=(1<180°)

3) i²=(1<180°)

i² therefore is a new number upon our new vector (1<180°) that has a length of 1 and an angle of 180° counterclockwise from the xx' axes

Using (2) We know that -1=1<180°

So squaring the i, will lead us upon the real number -1

So to recap the whole thing we did:

i²=(1<90°)² => Step 1

i²=(1*1 < 90°+90°) => Step 2

i²=(1<180°) => using (2) Step 3

i²=-1 Step 4

sqr(-1)=±i ??? Step 5

And here are my 4 questions:

1) Is it correct to go from step 1 to step 2 ?

2) Can we therefore prove that i²=-1 from the 3rd step i²=(1<180°) ?

3) Can we then say that from i²=-1 we can get to sqr(-1)=±i step 5 ?

4) And even further, can we then define the imaginary and complex numbers and base their new definitions (and maybe the whole ℂ) in this new equation i=1<90° instead of this i²=-1 or ±i=sqr(-1) equation?

---

Main goal for me is to get a better understanding on the whole concept of imaginary numbers, and I have to be honest with you, that the way I was introduced in school to the imaginary numbers and being told that people just accepted this idea just because it worked is not the best way, the already open mind of a student, to be introduced in complex analysis and beyond.

Are imaginary numbers created or invented? A 'meaning' can mean a lot of things, but most importantly a 'meaning' in mathematics is a way to logically connect 2 and more things together and see how they cooperate. Algebra and geometry for example.

You name it!

Thanks for your time.

Answer 1

I think the difficulty of your formulation is that:

• You don't really understand what you're doing when you square a polar point.
• You don't really understand why $-1=(1, 180^\circ)$ as a polar point.
• You leave us without a definition of addition of complex numbers because it's not necessary for the proof.

First, you're multiplying two polar points, which doesn't really make sense. Therefore, we need a different formulation of polar points.

Don't think of numbers as points. Think of them as transformations. With numbers, you can dilate and you can rotate. A number that dilates by a factor of $a$ and rotates $b^\circ$ is called $(a, b^\circ)$ as a polar point. Now, we want $i$ to be the rotation of $90^\circ$, so we get: $$i=(1, 90^\circ)$$ Now, what does it mean to multiply transformations? It means we do the first transformation, then the second. If we have $(a, b^\circ)$ then $(c, d^\circ)$, we dilate by $a$, rotate $b^\circ$, dilate by $c$, and rotate by $d^\circ$. Now, when we dilate by two different factors, we multiply the factors together, so the dilation factor is now $ac$. Also, when we rotate by $b^\circ$ and then $d^\circ$, we can really just add the angles together and say we're rotating by $(b+d)^\circ$. Thus, we get: $$(a, b^\circ)\cdot (c, d^\circ)=(ac, (b+d)^\circ)$$ Now, since $i=(1, 90^\circ)$, if we want to find $i\cdot i$, we get: $$(1, 90^\circ)\cdot (1, 90^\circ)=(1, 180^\circ)$$

Now, we want to find $(1, 180^\circ)$ as a separate number. For this, we're going to have to define what it means when you multiply something by a real number. There's a pretty straightforward answer to this: You just multiply the dilation factor by that number. Thus, you get: $$r\cdot (a, b^\circ)=(ra, b^\circ)$$

For positive numbers, all we're doing is multiplying the dilation factor by $r$. We're not rotating, which is the same thing as rotating by $0^\circ$. Thus, we get: $$r=(r, 0^\circ) \text{ if } r \geq 0$$

However, what does this mean for negative numbers? What is $(-1, 0^\circ)$? How do we dilate something by a factor of $-1$? One answer is that this means to go in the opposite direction. The beginning of this page has a pretty good explanation.

Now, notice what we've done here. By multiplying by $-1$, we've basically just rotated the point $180^\circ$. Therefore, we get: $$-1=(1, 180^\circ)$$ Thus, from above: $$i^2=i\cdot i=(1, 180^\circ)=-1$$

Now, we still need to define what square root really means to find what $i$ is. Therefore, I'll define square root as: $$x=\sqrt y \implies x^2=y$$ Now, as you pointed out, the problem is that there are two solutions here: $i$ and $-i$. We need to figure out what the "principal" square root is in order to decide between these two. Now, just because it would be simpler, we'd like $i$ to be the principal square root. $i$ is the polar point on top, so we could define square root that way: $$x=\sqrt y \implies x^2=y \text{ and x is on top}$$ However, what does it mean to be "on the top"? It means that we're in Quadrants I and II, which means we have angles between $0^\circ$ and $180^\circ$, so we get: $$x=\sqrt y \implies x^2=y \text{ and } x=(a, b^\circ) \text{ where } a \geq 0 \text{ and } 0^\circ \leq b < 180^\circ$$ Thus, since $i$ is the solution to $x^2=-1$ that is on the top, so we get: $$i=\sqrt{-1}$$

Now, we have a definition of imaginary numbers where $i=\sqrt{-1}$ with a theory of multiplication! Hooray! However, how do we get from a point like $(1, 30^\circ)$ to a complex number? Also, how do we go from $x+yi$ to a polar point?

$$x+yi=(x, 0^\circ)+y(1, 90^\circ)=(x, 0^\circ)+(y, 90^\circ)$$

To do this, we need to add two polar points. Now, since we're thinking in polar points, this is going to be pretty hard, but one way we can do this is by converting to a coordinate system that lends itself more to addition: Cartesian coordinates. As we learn in pre-calc, we convert polar to Cartesian as follows: $$(a, b^\circ)=(a\cos b^\circ, a\sin b^\circ)$$ Once we have this, it is pretty easy to define addition: $$(x_1, y_1)+(x_2, y_2)=(x_1+x_2, y_1+y_2)$$ Now, we can find $x+yi$ as a Cartesian coordinate: $$x+yi=(x, 0^\circ)+(y, 90^\circ)=(x, 0)+(0, y)=(x, y)$$ Now that we can convert between complex and Cartesian, using conversions between Cartesian and polar, we can convert between complex and polar. Here's a few examples: $$2+2i=(2, 2)=(\sqrt 8, 45^\circ)$$ $$(2, 30^\circ)=(2\cos 30^\circ, 2\sin 30^\circ)=(\sqrt 3, 1)=\sqrt 3+i$$

Notice that if we have just any polar point: $$(a, b^\circ)=(a\cos b^\circ, a\sin b^\circ)=a\cos b^\circ+a\sin b^\circ i=a \ \text{cis} \ b^\circ$$ Therefore, we don't really need to use polar points anymore since we can just use $a \ \text{cis} \ b^\circ$. Thus, just to review, our geometric interpretation of polar coordinates boils down to the following two definitions: $$(x_1+y_1i)+(x_2+y_2i)=(x_1+y_1)+(x_2+y_2)i$$ $$a \ \text{cis} \ b^\circ\cdot c \ \text{cis} \ d^\circ=ac \ \text{cis} \ (b+d)^\circ$$

Now, finally, we have a theory of adding and multiplying complex numbers based off of geometry. However, there's still one last thing to show: Using these geometric definitions of complex numbers, do all of the field axioms still hold?

The field axioms are all of the arithmetic rules you learned in elementary school, such as adding by $0$, multiplying by $1$, negative numbers, reciprocal numbers (fractions), commutative properties, and associative properties. In order to have a working system of complex numbers here, we need to show that all of these properties are true based off our definitions.