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For example the roots of $z^3-1$ are one real and two complex. But I can only see the real root. So the complex roots are there to be able to say "the polynom has other solutions".

GniruT
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When we tried to solve equations such as $$x-2=0$$ We said, $2$ is it's solution.

Then there came equations such as $$x+2=0$$ then, we invented, "negative" integers, thus expanding our system from $\Bbb N$ to $\Bbb Z$.

Then there came equations such as $$2x-1=0$$ then, we invented fraction, and thus rational numbers, thus expanding our system from $\Bbb Z$ to $\Bbb Q$.

Then there came equations such as $$x^2-2=0$$ then, we invented, irrational numbers, thus expanding our system from $\Bbb Q$ to $\Bbb R$, a union of both of these, id est rational and irrational.

Then there came equations such as $$x^2+2=0$$ then, we invented, complex numbers, thus expanding our system from $\Bbb R$ to $\Bbb C$.

So the point is, it is just an expansion of our number system, which our mind isn't able to adapt, since we do not study about complex numbers from first grade. We get introduced to it in high school, and till then we had assumed that there can exist no number whose square can be negative.

You see my point now? You can plot complex numbers on complex plane, and easily visualize. Try to accept them, though it isn't easy to do so.

C'mon! Let's not be racist xD

Jaideep Khare
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  • So to see the solutions I have to plot the function on the complex plane? – GniruT Nov 29 '17 at 13:01
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    @GniruT Exactly. Do not see it as a weird thing. Just try to be good with them. Lets not be racist to $i$ :P – Jaideep Khare Nov 29 '17 at 13:02
  • I won't be :) But I don't know how to draw $z^3-1$ on the complex plane. Could you help me? I'm new to complex numbers :( – GniruT Nov 29 '17 at 13:06
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    Just to clarify a common misconception, real numbers were not invented to solve quadratic equations. They were invented to fulfill a rather non-algebraic need which is technically called completeness. But generally while introducing irrational numbers the usual textbooks mention only the algebraic need for solving equations like $x^2-2=0$ and never focus on the more important aspect of completeness. – Paramanand Singh Nov 29 '17 at 13:42
  • @ParamanandSingh Yes, you are right. I added that to explain this idea to OP, and therefore "feel" complex numbers. Because, if real numbers would have been originated from polynomial equations, we would be having no place for transcedental numbers such as $\pi$, cause they aren't algebraic. – Jaideep Khare Nov 30 '17 at 02:50
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Rafael Bombelli (c.1526-1572) observed that although the cubic equation $x^3 = 15x + 4$ has three perfectly respectable solutions, namely $4$ and $-2\pm\sqrt{3}$, when you try to solve it using Tartaglia's formula, you get this apparently nonsensical "solution": $$ x = \sqrt[3]{2 + \sqrt{-121}} + \sqrt[3]{2 - \sqrt{-121}}. $$ His inspired idea was that this was really the root $x = 4$ in disguise, because if you took a leap of faith, and treated expressions involving square roots of negative numbers as if they were actually numbers, you could interpret the above solution as having this "real" meaning (so to speak): $$ x = (2 + \sqrt{-1}) + (2 - \sqrt{-1}) = 4. $$ I believe this event began to convince many people that "imaginary" numbers had a "real" meaning.

Calum Gilhooley
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But you can see them.

The point $(x,y)$ in the Euclidean plane can be interpreted as the geometric representation of $x+iy$, and when this is done the Euclidean plane is called the complex plane.

In the complex plane, the cube roots of unity are the vertices of an equilateral triangle inscribed in the unit circle, one vertex of which is at $1=1+0i$.

Although I have not been able to find a good enough picture of the cube roots of unity, here is a picture of the fifth roots of unity, which are the vertices of an equilateral pentagon inscribed in the unit circle, one vertex of which is at $1=1+0i$.

Lee Mosher
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This feels like a "why do we even have to learn about this" sort of question. Often, answers to these can be of two varieties. One, is for the teacher to say "because I said so". The other, is to try to give some general idea of where these ideas are used (even if the student can't understand said application). I'll try for the latter here.

Complex analysis (the general topic that finding roots of this sort of polynomial), is a beautiful subject with a vast number of applications within and outside the field of mathematics. Within mathematics, the complex numbers are algebraically closed, which is useful. They are also used in number theory and topology a lot.

Useful applications that come up outside of mathematics include computing residues to compute very difficult real integrals, or using Euler's formula and everything that comes from that in Fourier analysis.

So, even if you can't "see the answer", don't write it off as not important. Keep studying and enjoy the hunt to find out where these things show up again!

dt688
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