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While teaching complex numbers for those who are encounter for the first time (usually 10th grader and 11th grader), I get the question like

"Can even squared number give negative results? How come there be square root of negative numbers? How much length does $i \; \mathrm{cm}$ represent in meter scale?"

in every year and in every section. I have been answering

"Well, this is just a construction such that it gives solution to equation $x^2 + 1 = 0$ which must have two solution by use of Fundamental theorem of algebra but since we do not have that in Real numbers, we are forced to construct $i$ that way." Then I go to history accepting the difficulty of some numbers like $\sqrt 2 $ by ancient Greek mathematicians (just a red herring). Then I continue "Natural number is used to represent Money while Real number is used to measure length, complex number, while not representing length or weight, it my represent some other things than money or length that has applications in physics, math and engineering and so it is valid to take construct a symbol $i^2 = -1$."

I don't know if I am speaking too much out of my limited knowledge. What is the best way this knucklehead to explain those knucklehead who have been brainwashed in secondary and lower-secondary school that you cannot take square root of negative numbers, while still being correct and make them accept it easily?

EDIT: I have last question. Are some problems that is impossible to solve without use of Complex Numbers or all problems involving complex numbers have other workarounds?

EDIT-ADDED:: What would mathematics be like without the use of complex numbers? Perhaps, if I mention this clearly, maybe it would be helpful for students. One example is given by @Semiclassical on the comments in the first answer. What other things cannot be absolutely solved without the use of complex numbers?

Mula Ko Saag
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    There is a Mathematics Educators StackExchange that this question would likely be appropriate for, if it's not a duplicate there. –  Nov 02 '14 at 12:22
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    I like your "this is just a construction such that". Think about negative numbers : how could you explain a very young person that $4$ apples minus $5$ apples give a result of $-1$ apple ? I face this problem and I use to say simple things as yours. I thank you for the post. Cheers :-) – Claude Leibovici Nov 02 '14 at 12:25
  • If you wanna take this path, perhaps you might want to insist further by saying that they should abandon the meaning they attribute $-1$ and think of it merely as a symbol, then $i^2=-1$ might not be so difficult to accept. This might be catalyst for further questions, but that's how the cookie crumbles if people start digging towards foundations. Edit: Perhaps you can get some ideas from this question. I particularly like Peter Smith's answer in conjunction with DeVito's comment. – Git Gud Nov 02 '14 at 12:38
  • I would say just tell them that a complex number can't be measured in anyway unless it's imaginary part is 0.Complex numbers are useful in many theoretical aspects but have no practical use like any other set – kingW3 Nov 02 '14 at 13:05

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I think the conceptual difficulty for students is not brainwashing about square roots of negative numbers, but the fact that what a number is, from a naive point of view, is the result of either (a) counting or (b) measuring, and these "imaginary numbers" are not either one. So we're asking the students to accept a serious departure from their fundamental concept of what numbers are.

If you wanted to address this aspect directly, one approach would be to recall how operations with real numbers can also be used for describing and reasoning about certain kinds of movement of points on a line. Addition represents shifting right or left; multiplication by positive numbers represents scaling; multiplication by negative numbers represents scaling and flipping. On the number line, the point we call $r$ is the point that $0$ moves to if you do the operation "adding $r$"; it's also the point that $1$ moves to if you do the operation "multiplying by $r$". Algebraic rules express facts about these operations; e.g., the commutativity of addition represents the fact that you can do a sequence of left/right shifts in any order and get the same result.

Once all that has been brought to mind, then you can introduce movements of points on a plane: shifting right/left/up/down, scaling, and rotating. Complex "numbers" describe these operations in the same way that real numbers represent motions of points on a line. The fact that $i^2=-1$ is just the fact that rotating by $90^\circ$ twice is rotating by $180^\circ$, which flips the real line.

I think this approach builds a fairly comfortable bridge from the familiar world of real numbers to the world of complex numbers, without the abstract and difficult-to-motivate step of saying, "What would happen if we just assumed $-1$ could have a square root?"

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    Oh, and I wanted to add: it's very much not a knucklehead question. It took hundreds of years for mathematicians to accept these "numbers"! Students may well not be able to reproduce that acceptance in an hour's lecture. –  Nov 02 '14 at 13:53
  • I have a last question. Are there some questions which cannot be solved without using complex numbers? – Mula Ko Saag Nov 02 '14 at 15:45
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    @MulaKoSaag: A historical example is Cardano's formula for the roots of a cubic polynomial. This includes cases where, in order to use the formula and find real roots, one would need to compute complex roots in the process. This should be contrasted with the case of the quadratic formula, where one complex square roots only show up when there are complex roots. So going from quadratic to cubic forced people to take the complex numbers seriously for computations. – Semiclassical Nov 02 '14 at 16:02
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The way I first learned about $i$ was that it was called $j$ and multiplication by $j$ meant rotating the plane $90^\circ$ counterclockwise. Every complex number $a+bj$, where $a$ and $b$ are real, has an absolute value $|a+bj|=\sqrt{a^2+b^2}$ and an angle of rotation $\theta$ whose cosine is $a/|a+bj|$. To multiply by $a+bj$ is to dilate by $|a+bj|$ and rotate by $\theta$. After I learned this as rotating and dilating, several years passed before I found out that it is possible to introduce multiplication by complex numbers without talking about rotating and dilating. And there are disadvantages to learning it without learning the geometry.

Michael Hardy
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In a geometric setting, if $1$ is $1$cm long, then $i$ is $1$ cm long. The real and imaginary parts in a complex number have the same dimension, and expressions that involve them are homogenous, like $\|z\|=\sqrt{x^2+y^2}$.

The geometric representation of complex numbers as 2D points can be helpful and rescue the centimeters.

Once you have defined what addition and multiplication in the complex plane are, it becomes acceptable that $i^2=-1$.

What they need to accept is not the existence of imaginary quantities (do real numbers "exist" ?), but the fact that we define an algebra on a superset of the real numbers.