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I have seen many explanations of $i$, this include comparing the understanding of the imaginary unit with negative numbers, fractions, etc. to solve polynomials and using the geometric interpretation of $i$ in the complex plane.

I´m starting to study complex analysis (basic differentiation at the moment) and noticed that everything makes sense if I just accept that $i^2=-1$, but that is my only problem, none of the explanation above totally clarify to me why $i$ works.

Real life examples such as $(s)(t)=d$ where $s$ stands for "speed", $t$ stands for "time" and $d$ stands for "distance", provide a good understanding on why $(-1)^2=(+1)$; (negative time would be seen as the time needed to reach distance $0$ at velocity $s$ (starting at a negative distance), or, if we are creative enough, we could imagine it as going at velocity $s$ just that time would be going backwards for $t$ seconds, or whatever time measure we want to use). Other example would be multiplying "deposit made" with "times made" to get "money in bank account".

And if we dare to say that $(-1)^2=(-1)$, not only it would be illogical, but the math would just break, taking a derivative or using the binomial theorem would give us crazy results, plus we could prove that any number is equal to any other number.

Parallel to the first example, we could imagine a ball rolling in a hill. Representing the ball accelerating towards the right direction as $+$, and representing a slope such that it gets higher as we go to left as $-$, we can say that.

$(-s)(+t)=(+a)$

Where $s$ stands for slope, $t$ stands for time and $a$ stands for acceleration. Notice that in this "system" $(-1)(+1)=(+1)$, $(+1)(+1)=(-1)$ and $(-1)(-1)=(-1)$. And like this any number of "systems" with the same properties can be created.

My idea is that, since we can create a "system" in which some positive unit squared, lets call it $i$, gives us a different negative unit, let´s call it $u$ ($i^2=-u$), and if $u$ satisfy the property of $(+1u)^2=(+1u)$, this unit $u$ would work just as $+1$ so we could just omit it and get imaginary and complex numbers.

The problem with this explanation is that it isn´t satisfying enough, I don´t even know if it´s right or if it just depends on how I see it, that´s why I´m asking for a better explanation.

Thank you for reading this unending text, I really would appreciate any other explanation of complex numbers.

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    $\begingroup$ What isn't satisfying about interpreting complex numbers as points on the plane and the corresponding operations as transformations? This tells you exactly why $i$ works! $\endgroup$ – Bernard W Aug 18 '16 at 2:22
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    $\begingroup$ You may be surprised to know that math does not break if we say that $(-1)^2=-1$, it just implies that the characteristic of our ring is $2$. $\endgroup$ – Matt Samuel Aug 18 '16 at 2:23
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    $\begingroup$ Reading the rest of your post, you have discovered for yourself the concept in abstract algebra of defining an algebra by generators and relations. It's one way to define the complex numbers. There are indeed more satisfying ways in this case, but not everything is a number. Look at matrices. We can have $AB=0$ where neither $A$ nor $B$ is zero. You could say, "But numbers in the real world don't behave that way!" I would reply that matrices are not real numbers. Some complex numbers are real numbers, but most are not. $\endgroup$ – Matt Samuel Aug 18 '16 at 2:38
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    $\begingroup$ I started having an easier time with the complex plane when I stopped thinking about $i$ as $\sqrt{-1}$ and started thinking about $z = x+i y$ as an ordered pair with a special sort multiplication rule. Once I was used to the algebra, it was straight froward to address $(\cos \theta + i \sin \theta)(\cos\phi + i \sin \phi) = \cos (\theta + \phi)+i \sin(\theta + \phi)$ and from there to $zw = (x+i y)(a+i b) = (|z|e^{i\theta})(|w|e^{i\phi}) = |z||w|e^{i(\theta+phi)}$ $\endgroup$ – Doug M Aug 18 '16 at 2:56
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    $\begingroup$ The real enlightenment comes when you disassociate mathematics from the universe that it approximately describes. You can do anything as long as there's no contradiction. Yes, math has some great practical applications. But there's more to it than that! $\endgroup$ – Matt Samuel Aug 18 '16 at 3:03
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A significant change in perspective seen in abstract algebra is that we can create any kind of mathematical structure we want, including algebraic structures with addition and multiplication, with the only caveat that once you set down some features, other features follow from them. For instance, once we write down the distributive property, it follows logically that $(-1)^2=1$ (way before derivatives or the binomial theorem). Sometimes imposing our own set of rules makes the whole thing trivial (you could end up making every element equal to every other element for instance, so there'd only be one "thing"). Other times it makes something interesting.

Formally, algebraic structures are defined as sets with various operations on them. (Special constants may be considered "nullary" operations, something like negation or taking inveres is a kind of "unary" operation, addition and multiplication are "binary" operations, etc.)

Let's try an example. Suppose we want to create an algebraic structure that has addition and multiplication, in which both operations are associative, every element has an additive inverse, and multiplication distributes over addition. Such an algebraic structure is called a ring. Let's say our elements look like $2\times 2$ arrays of real numbers, addition is defined "entrywise," and then define a multiplication operation by

$$ \begin{bmatrix} a & b \\ c & d\end{bmatrix}\begin{bmatrix} e & f \\ g & h\end{bmatrix}=\begin{bmatrix} ae+bg & af+bh \\ ce+dg & cf+dh\end{bmatrix}.$$

We can check that this actually distributes over addition. These are of course matrices, and we know about their having addition and multiplication.

Another example. Let's take the integers, and keep all of the true equations but don't know any inequalities. In particular, we don't know that integers written by different symbols are actually different. Let's say we impose the relation $(-1)^2=-1$. It follows from the distributive property that $(-1)^2=1$, so by the transitive property of equality we get $1=-1$, and adding $1$ to both sides yields $0=2$.

This isn't actually a logical contradiction! I leave it as an exercise to deduce from this that every even number is equal and every odd number is equal, so there are really only two numbers in this new system, namely $0$ and $1$. Zero is still a multiplicative absorption element, and an additive identity, and one is still the multiplicative identity. These are precisely the integers mod $2$, which since every nonzero element (namely $1$) has a multiplicative inverse (namely $1$ itself) is a field and hence is called $\mathbb{F}_2$ since it has $2$ elements.

Let's try another example of inventing our own algebraic structure. First, we start with the underlying set $\{\circ,\square,\triangle,\heartsuit\}$. Let's only have an addition operation (which is commutative), defined by the following addition table:

$$\begin{array}{l|llll} + & \circ & \square & \triangle & \heartsuit \\ \hline \circ & \circ & \square & \triangle & \heartsuit \\ \square & \square & \circ & \heartsuit & \triangle \\ \triangle & \triangle & \heartsuit & \circ & \square \\ \heartsuit & \heartsuit & \triangle & \square & \circ \end{array} $$

In other words, $\circ$ is an additive identity ($\circ+x=x=x+\circ$ for any $x\in\{\circ,\square,\triangle,\heartsuit\}$) and adding any two elements from $\{\square,\triangle,\heartsuit\}$ yields the third one. This is an algebraic structure of our own creation! It turns out, viewing $\mathbb{F}_2$ as a set of scalars, this is a two-dimensional vector space over $\mathbb{F}_2$, where $0x=\circ$ and $1x=x$ for all $x\in\{\circ,\square,\triangle,\heartsuit\}$, and any two elements from $\{\square,\triangle,\heartsuit\}$ form a basis.

What's the point of all this? To impress upon you the fact that we can make our own algebraic structures. We are not tied down by the condition that it has to mean anything. In fact, insofar as an algebraic structure is just a set with some operations defined on it, most operations (in a set-theoretic sense) will not satisfy nice conditions like associativity, commutativity, distributivity (if there's more than one), existence of identity or inverses, etc., and most algebraic structures will never be touched on and have no meaningful interpretation we are aware of.

But we do tend to work with algebraic structures that are meaningful or useful for some purpose, at least potentially, and we shouldn't be narrowminded when it comes to determining how to interpret them. So let's talk about complex numbers now.

Historically, mathematicians knew that polynomial equations like $x^2+1=0$ did not have solutions. But some noticed that, while in the process of solving cubic equations, if we pretended there were such solutions (namely, the one $i$ with $i^2=-1$), then we could perform algebraic manipulations in order to obtain solutions to cubic equations, even if the solutions were real numbers! (This is consistent with the famous quote: "the shortest path between two truths in the real domain passes through the complex domain.")

At first this seemed strange. But with the advent of modern algebra it's not so mysterious. We can invent a new number system by adjoining a symbol $i$ to the real numbers, assuming most of the usual properties of addition and multiplication hold, and then imposing the rule $i^2=-1$. From this one relation we can deduce any power of $i$, including negative powers. If addition is to work, we need to be able to multiply $i$ by real numbers, and then add those to real numbers too, so our new system will contain "complex" numbers of he form $a+bi$. From there we may deduce the complex numbers of the form $a+bi$ are closed under addition (just combine like terms), multiplication (use the distributive property and $i^2=-1$), and even division by nonzero numbers (rationalize denominators).

The mathematicians of old simply had a (by our standards) narrow view of what a "number" was. To them, numbers had interpretations in the form of counting things, arranging things in order, and measuring size, and of course there is no such thing as multiplying $i$ by itself $i$ times (but neither is there any such thing as multiplying $\sqrt{2}$ by itself $\sqrt{2}$ times). From natural numbers (including or not including $0$), to integers, to rational numbers, to real numbers, mathematicians have historically expanded what they consider a number, and interpretations of earlier numbers don't necessarily apply to later invented ones.

Today, there are many things called numbers. There are $p$-adic numbers, surreal numbers, infinitessimals and infinities in nonstandard analysis, infinite cardinals and ordinals in set theory, and so on. Loosely speaking, elements of a algebraic system are called numbers if they sufficiently resemble the other algebraic systems we already recognize as numbers, in which case we may take the liberty of calling the algebraic system a number system.

Now, if $i$ doesn't measure the length or volume or speed etc. of anything then what kind of physical interpretation might it have? IIRC Gauss (and possibly others) originally came up with the idea to depict complex numbers $a+bi$ as points $(a,b)$ in the Cartesian plane. Some sources may formally define complex numbers as ordered pairs, and then write down the formula for adding or multiplying any two ordered pairs in a way that matches what we've talked about so far, so e.g. $(0,1)(0,1)=(-1,0)$.

We can then speak of the norm of a complex number as if it were a 2D vector, $|a+bi|=\sqrt{a^2+b^2}$. We may check that this is multiplicative, i.e. $|zw|=|z||w|$ identically. Once we expand the domain of the exponential function using its Taylor series (or even the compound interest limit definition), we may derive de Moivre's formula $e^{i\theta}=\cos\theta+i\sin\theta$. By normalizing, we find that every complex number $a+bi$ may be written in the polar form $re^{i\theta}$, where $r$ is its length and $\theta$ is the angle it makes with the positive real axis (interpreted as a 2D vector again).

In this way, multiplication by $re^{i\theta}$ as a function on the complex plane is a combination of radially scaling everything by a factor of $r$ and rotating everything $\theta$ counterclockwise. This geometric understanding is the closest to a direct physical interpretation I think you can hope for. Some things in physics that have to do with, say, rotations (like oscillations, hence signals in Fourier analysis or electromagnetism) will therefore have to do with complex numbers.

Even without a physical interpretation we can recognize they are useful. Again, to repeat the quote form before: "the shortest path between two truths in the real domain passes through the complex domain." The prime number theorem for instance was first proved and is still considered most elegantly proved using methods of complex analysis. If you want to do linear algebra, sometimes you end up solving polynomial equations, and if we expand our number system from reals to complex numbers we can solve all of them which is immensely useful (e.g. eigenstuff).

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$i$ is just a gadget for keeping track of quarter turns in the plane.

Say you're facing east. Instead of east, we'll call it $1$. Doesn't mean anything, it's just a name.

You make a counterclockwise quarter turn. Now you're facing north. We call it $i$. Just a name.

After another quarter turn we're facing west, which we call $-1$. Notice that you are now facing in the opposite direction from where you started; and you got there with two quarter turns. If we define "multiplication" as simply composing rotations, then the notation $i^2 = -1$ makes perfect sense. It just says that if you make two quarter turns in a row, you end up facing in the exact opposite direction from where you started.

Now another quarter turn brings us to a point we call $-i$; and another turn brings us back to $1$. So $i^4 = 1$. From that we can easily calculate any power of $i$. For example if someone asks us what is $i^{17}$ we know that $i^{16}$ must be $1$ so $i^{17} = i$.

Of course all we've done is give new names to the compass directions and redefine multiplication as composition of quarter turns. We can formalize this idea mathematically.

The real line is a $1$-dimensional vector space with basis $\{1\}$. Any vector -- that is, any real number -- may be thought of as a "scalar" real number $\alpha$ times the basis vector $1$. For example $\pi$ in vector space notation is $\pi(1)$ where the notation is a scalar times the standard basis vector.

Geometrically, the scalar $\pi$ stretches vectors by a factor of $\pi$, and likewise for any scalar real $\alpha$. If $\alpha$ is negative it not only stretches (or shrinks) a $1$-vector, it reverses the vector's direction. Real numbers can therefore be viewed as geometric operations on lengths.

Viewed this way, multiplication by the number $-1$ is a geometric operation that reverses the direction of a vector. When you multiply the scalar $-1$ by the vector $5$ you get the vector $-5$ which has the same magnitude and the opposite direction.

Once you start thinking of multiplication as a geometric operation that stretches or shrinks or flips a vector on a line, it's natural to ask if there might be some geometric operation that if you do it twice in a row, you get the $-1$ flip operation. For a while you'd be stuck, because no real number will do this trick. Eventually you'd realize: Ah! What if we go into the plane and rotate a vector a quarter turn? The square root of the $-1$ operator is the quarter turn operator in the plane. If we call that rotation $i$, then $i^2 = -1$ and this is obvious.

[We generally privilege the counterclockwise rotation as $i$ and the clockwise one as $-i$ but this is completely arbitrary].

A final remark is that if you've taken linear algebra, then multiplication by the real number $\alpha$ is a linear transformation from the reals to the reals with matrix $(\alpha)$.

Then a counterclockwise quarter turn of the plane is a linear transformation $T : \mathbb R^2 \rightarrow \mathbb R^2$ that takes the standard basis vectors $(1,0)$ to $(0,1)$, and $(0,1)$ to $(-1, 0)$.

Its matrix is therefore $\begin{bmatrix} 0& -1\\ 1 & 0 \end{bmatrix}$. If you square that you get $\begin{bmatrix} -1& 0\\ 0 & -1 \end{bmatrix}$. If you call the first matrix $i$ and you note that the second one is the additive inverse of the identity matrix, then the notation $i^2 = -1$ is simply a true statement about matrices, with $1$ notating the $2 \times 2$ identity matrix.

All this means is that multiplication by $i$ is a linear transformation that rotates the plane a quarter turn. Everything else follows from that.

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Just think about complex multiplication as rotation and dilation. Multiplication by $i$ corresponds to counterclockwise rotation by $\frac{\pi}{2}$, or $90$ degrees. So $i^2$ corresponds to rotating the arrow starting at the origin and pointing to $1$ counterclockwise by $\pi$, or $180$ degrees. What does this arrow point to after this rotation?

This video should make it clear how to geometrically interpret complex numbers.

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Someone I know once joked about, partway through teaching a complex analysis course, apologizing that he made a grave mistake and that everyone should replace every "$i$" with "$-i$". This is because, as you say, $i$ is a thing that when you square it you get $1$. Whenever you have such an $i$, $-i$ also has this property! So, as far as the algebra of having a solution to $x^2=-1$ is concerned, they are indistinguishable.

There are a few concepts related to what you are talking about. One is the idea of "adjoining an element to a ring." The complex numbers are $\mathbb{R}[\sqrt{-1}]$, which is to say "the smallest ring containing both $\mathbb{R}$ and $\sqrt{-1}$." There are other interesting adjunctions, such as $\mathbb{Q}[\sqrt{2}]$. I've also seen things such as adjoining an element $\epsilon$ which is not a real number whose square is zero: this is known as non-standard analysis.

Another is that one point of $i$ is that it makes it so that every polynomial has a root. The way I think about it (as a student of algebraic topology) is that adding a square root of $-1$ gives a two-dimensional number system, and that gives circular paths. The fundamental theorem of algebra follows from the fact that polynomials are continuous functions, so you can see what they do to circular paths, and see how for large radii the path must wrap around the origin a number of times equal to the degree of the polynomial, and for small radii, zero times around the origin, so somewhere in between, there must be some path which passes through the origin (hence a root).

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