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It is a common known fact that i is defined as a number that satisfies the quality of $i^2 = -1$ or in other words, $i = \sqrt-1$. The intuition of this is often said to be that multiplying by $i$ represents a $90^{\circ}$ rotation on the complex plane. This would make since because multiplying by the same square root once takes away the square root and a second time gives back a square root. For example, $ \sqrt50\times\sqrt50 = 50 $ and $ 50 \times \sqrt50 = 50\sqrt50$ and in the case of imaginary numbers since there is a $1$ the coefficient will not be written because the $1*1=1$ or a $-$ will be written because $-1*1=-1$. Also multiplying by a negative rotates by $180^{\circ}$. So putting this together mutliplying by the $\sqrt-1$ will oscillate between taking the square root, multiplying a negative, and placing a square root (there will not be a number in front of the square root besides $1$ or $0$).

However, to me this sounds like a hack by manipulating the properties of negatives and square roots. This would be fine for practical purposes but in pure mathematics it seems to be not defined well. Shouldn't there be a deeper explaination; an explanation that defines the number by its properties but not assuming the properties of the square root extends to negatives. If not is the reason we define the imaginary unit this way is we only have one operator that is defined with these properties (couldn't there another type of operator satisfying the same properties)? I understand that it useful in finding the roots of polynomials and it seems the definition works but that still doesn't make the definition general enough.

Thanks,

Jackson

Also before anyone downvotes this because it is another question about the intuition of imaginary numbers I would like to point out no one has pointed out that the imaginary number's definition is just manipulating properties that are assumed to be true even though it could possibly be defined with another operator satisfying these properties.

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closed as primarily opinion-based by quid, Peter, Matt Samuel, user223391, Christopher May 31 '15 at 23:29

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ you could look at the group presentation of the complex numbers check this out euclideanspace.com/maths/algebra/realNormedAlgebra/complex/… $\endgroup$ – alkabary May 31 '15 at 17:41
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    $\begingroup$ Another interesting thing to note is that under the definition $i^2=-1$ we don't have $i$ uniquely satisfying it, we could equally be using $-i$ as our imaginary unit and call it $j$. $\endgroup$ – nathan.j.mcdougall May 31 '15 at 17:43
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    $\begingroup$ From a pedagogical perspective, the name imaginary is very misleading. Why not introduce them as a system for multiplying pairs of numbers with curious properties? $\endgroup$ – copper.hat May 31 '15 at 17:48
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    $\begingroup$ Have you ever seen the complex plane defined as $\mathbb R[x]/(x^2+1)$, which essentially means "the complex plane is the same as the real numbers, plus an extra element, $x$, which satisfies $x^2+1=0$ and otherwise acts like we'd expect under addition and multiplication". Of course, $x$ is essentially $i$ (or $-i$), but this emphasizes that $i$ is indeed defined purely by its algebraic properties - and is made by taking $\mathbb R$ and saying, "Hey, $x^2+1$ is sad because it doesn't have a root. We should give it one!" $\endgroup$ – Milo Brandt May 31 '15 at 17:52
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    $\begingroup$ "What if we do this" is an absolutely great definition, it drives so much modern math. We would never be able to do new things if we never asked that. $\endgroup$ – anon May 31 '15 at 18:10
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The geometry of multiplying complex numbers is usually a depiction, not a definition. I don't understand why you claim it's "not defined well," though. It's perfectly defined. If you need any clarification on how it works, you can always ask a question. Also, please do be aware that the geometry of multiplying complex numbers is much richer than just multiplying by $i$. Read and learn some more about them and you will find how the geometry works with polar form. Every complex number can be written as $z=re^{i\theta}$, where geometrically $r$ is the absolute value and the phase $\theta$ describes the angle between the positive real axis and the number (as a vector). Then multiplying two numbers multiplies their absolute values and adds their phases.

There is a paradigm shift that mathematics has made historically, and that you might still need to make on your journey: math isn't just discovery, it's also part invention. With science and technology, whenever our world wants for a device or machine that does a certain job, we make one to do it. Math can be the same way! If you want a number system with certain properties, you construct it. We wanted a number system containing $\Bbb R$ in which we can solve all polynomial equations, and so we simply made one. The fact that everything works out perfectly is proof enough that we are talking about a real thing. Whether we want to still call this real thing a number system is an important and "morally correct" choice we've made based off of all of the facts.

One can always factor polynomials down to quadratic and linear factors over $\Bbb R$, the one issue with going further and factoring all the way occurs whenever we try to factor by completing the square: negative numbers are not squares in $\Bbb R$. So $x^2=-1$ is the prototypical equation to introduce a new solution to, call it $i$. As it happens, once you introduce $i$, every number in the new number system looks like $a+bi$, since higher powers of $i$ can simplified. And yes this definition is general enough - as a result of this definition, we can add/subtract/multiply/divide complex numbers and we can solve any polynomial equation, just as we wanted.

A person could ask, "but what if some polynomials simply don't have roots"? One could analogously ask, "but what if there really is no game with the rules of chess? What if we're incorrect in believing that knights must move two spaces and then one orthogonal?" But there is such a game with such-and-such rules: we made it. There can be potential issues in math of constructing a new structure, and in the process in order to keep everything consistent you end up collapsing the structure into something completely trivial and degenerate, essentially because the thing you wanted was simply not meant to be. Or, you may want to construct a new structure with a set of properties, but there it isn't possible to get all of the properties satisfied simultaneously. Gladly, the case of complex numbers is not one of those situations.

Note "assuming properties of square roots extend to negatives" is a subpar way of thinking about it; instead we should be thinking about it as "we wanted a number system in which we could still follow the usual rules of arithmetic but successfully solve more polynomial equations, and we were able to construct one." Sometimes we introduce facts like $0!=1$ or $x^0=1$ as "assuming the properties of factorials and powers extend to $0$," but a better way of thinking about it is that we want to define it in such a way that the properties do extend, because wouldn't that be nice.

After we construct $\Bbb C$ we are able to determine, as a matter of investigation, what properties still carry over. The rules of arithmetic for addition, subtraction, multiplication and division still work, and the rules for integer exponents still work, for instance. The rules for radicals and rational exponents we know for fact do not actually still work without caveats. For instance the familiar rule $\sqrt{ab}=\sqrt{a}\sqrt{b}$ is not true for all real numbers $a$ and $b$, so not only are we not assuming that properties of square roots carry over willy-nilly, namby-pamby, we know that they don't.

There are other ways of defining and constructing $\Bbb C$ of course.

  1. Given any quadratic polynomial $ax^2+bx+c$ that cannot be factored with real numbers, we can define $\Bbb C$ as the quotient ring $\Bbb R[x]/(ax^2+bx+c)$. If you study some abstract algebra, you will find polynomial rings and quotient rings are very useful in constructing rings (generalized number systems) with desired properties. In particular, what this construction does is "adjoin" a root of the polynomial $ax^2+bx+c$. One major drawback here is that there is no geometry, no notion of the sizes of or distances between complex numbers.

  2. One can also define $\Bbb C$ as a subset of the algebra $M_2(\Bbb R)$ of $2\times 2$ real matrices comprised of elements of the form $(\begin{smallmatrix}\phantom{-}a & b \\ -b & a\end{smallmatrix})$ . Every such matrix can be written as $a(\begin{smallmatrix}1 & 0 \\ 0 & 1\end{smallmatrix})+b (\begin{smallmatrix}\phantom{-}0 & 1 \\ -1 & 0 \end{smallmatrix})$, which is no coincidence: $(\begin{smallmatrix}1 & 0 \\ 0 & 1\end{smallmatrix})$ is the identity and $(\begin{smallmatrix}\phantom{-}0 & 1 \\ -1 & 0 \end{smallmatrix})$ is a $90^\circ$ rotation. So this is very close to the usual, familiar understanding of complex numbers and has geometry in it.

  3. Finally, if you want an elegant definition through properties instead of an at-first-artificial-seeming construction, consider the fact that $\Bbb C$ is the smallest uncountable algebraically closed field containing the integers, or equivalently the smallest algebraically closed field containing the reals. This is a useful and correct way of thinking about what the $\Bbb C$ "is," but it's even more pedagogically and practically flawed than the first definition: not only do we not have any geometry or distances, we don't even know what the elements of $\Bbb C$ look like or how they interact in this definition!

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  • $\begingroup$ This might be a question for another post but is there anyway I can learn this without running into a paradox? The paradox is that to learn abstract algebra, linear algebra is usually considered a prerequisites but to learn linear algebra the understanding of complex numbers is necessary but complex numbers are defined by abstract algebra. $\endgroup$ – Jackson H May 31 '15 at 18:21
  • $\begingroup$ @JacksonH You don't need to know linear algebra or complex numbers to start learning abstract algebra. $\endgroup$ – anon May 31 '15 at 18:26
  • $\begingroup$ Again this might be another question for another post but what book would you personally recommend for my situation? $\endgroup$ – Jackson H May 31 '15 at 18:27
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    $\begingroup$ @JacksonH I hardly remember when I first learned abstract algebra, but the courses I took on it were split into two parts - in the first part we studied groups, and in the second part we studied rings, fields and Galois theory. You can jump right into the rings and fields without really learning about groups. The book we used was Gallian, although any sufficiently-introductory book on abstract algebra should be just as good. One can also get the jist of the arithmetic and geometry of vectors and matrices in linear algebra without assuming any knowledge of complex numbers. $\endgroup$ – anon May 31 '15 at 18:47
  • $\begingroup$ (cont'd) Indeed I think one should start with real numbers to begin with. The exploration of more advanced topics in linear algebra might be served by abstract algebra, and conversely some fundamental things in abstract algebra can be served by methods and results linear algebra, but you do not need to worry about these where you are. (For instance, the structure theory / classification of finitely-generated abelian groups and the generalized eigenspace decomposition + Jordan normal forms of linear maps can be wedded together into one overarching fact about modules over PIDs.) $\endgroup$ – anon May 31 '15 at 18:47

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