What is the best way to introduce for students the new mathematical definition : ‎$i^2=-1 ‎$?

Question

I'm really interesting to know the best or good way for introducing to my student the best way to know the mathematical definition ‎$i^2=-1 ‎$, even now the the easiest way to let my student check the titled definition is to correct the mistake that is :$\displaystyle\sqrt{a}\sqrt{b}=\displaystyle\sqrt{a\cdot b}$ is true only when $a$, and $b$ are positive real numbers, then are there others?

My question here is:

What is the best way to introduce for students the new mathematical definition : ‎$i^2=-1 ‎$?

Answer 1

I like to use "paradoxes" to illustrate my point and catch the attention of my students.

My standard technique for this case is:

$$1 = \sqrt{1} = \sqrt{(-1)(-1)} = \sqrt{-1}\sqrt{-1} = (-1)^{\frac12 2} = -1.$$

Then we discuss what went wrong.

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On the other hand, I introduce $i^2 =-1$ using geometry. I start with the number line and notice that multiplying by $-1$ rotates the unit length by 180 degrees. And then if multiplying by a variable $i$ represents rotation by 90 degrees, we obtain $i^2 = -1$ by composing two rotations. Then, we compute $\sqrt{-i}$ which leads to a discussion on the fundamental theorem of algebra.

Answer 2

Tell them that $i=\sqrt{-1}$, and ask them to find $i,i^2,i^3,$ and $i^4$

Answer 3

a) I introduced (direct) similarities as a composite of rotation and scaling

b) I made the (dancing) demonstration of $-1$ as half turn

c) I asked them if they could find a square root for it (quarter turn)

The byproduct of this method is that it is very expressive intuitively that there are two square roots.

Answer 4

You could define a complex number $a+ib$ by the pair $(a, b)$ and define addition in the usual way and multiplication by $(a,b)\cdot(c,d)=(ac-bd,ad+bc)$.