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 : ‎$i^2=-1 ‎$?

Actually $\sqrt a \sqrt b = \sqrt {ab}$ for lots of nonreal $a,b;$ for example, $a,b$ in the open right half plane. — zhw., Jan 28 '17 at 21:43
ok, I know that by the way i meant the mistick for using that in complex number "New ensemble for students" — zeraoulia rafik, Jan 28 '17 at 21:44
Thanks for that , but may there are other way than what posted as answer for the question you montioned — zeraoulia rafik, Jan 28 '17 at 21:53

score 4 · Accepted Answer · answered Jan 28 '17 at 22:25

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.

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.

score 2 · Answer 2 · answered Jan 28 '17 at 22:42

2

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

answered Jan 28 '17 at 22:42

pshmath0

10,565

score 0 · Answer 3 · answered Jan 28 '17 at 21:40

0

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

answered Jan 28 '17 at 21:40

suomynonA

6,895

score 0 · Answer 4 · answered Jan 28 '17 at 21:56

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.

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

4 Answers4