In this post we will try to find some light in those strange complex numbers. There is a nice way to present them as rotations and stretchings of vectors, where complex numbers concepts and properties are all very natural, look for that later in this post.
Let's begin by telling my first experiences with complex numbers.
I think my first impressions were based on those mysterious names, after all we would learn about "complex numbers", about the number " i " that was the "square root of -1", that the teacher said before it didn't existed, and also about the "imaginary" numbers in opposition to the more "real" numbers learned before.
The names "complex numbers" and "imaginary numbers" were not a good choice of words from the pedagogical point of view. Well, "i" is the number that squared is -1, what is the logic in that? Lets make some experiments.
For example: 32 = 9 and 52 = 25 so positive numbers squared are positive. (-3)2 = (-3)x(-3) = 9, because negative times negative is positive, so negative numbers squared are also positive!
Concluding that all numbers squared are positive! So no number squared is negative! And in particular no number squared is -1! at worst it is 0.
So they actually invented a number " i " not a "normal number" that squared would be -1, that is what my teacher said to me.
After some algebra facts they let you know that you can plot a complex number like " 3+ 4i " in an x-y axis were the point (3,4) is, unfortunately that was not so helpful. Later on that class my teacher showed me lots of even more bizarre properties of complex numbers.
All that together was a dazzling sequence of "out of the blue" concepts and relations. Actually, complex numbers have a pretty decent history, real flesh and bone people invented it you know, and that history is not "out of the blue" I may tell something of that story latter.
Now I will present complex numbers in a different and I hope better way, first lets explore vectors.
A vector can represent something as velocity or force on a body and lots of other concepts. Lets think of it as a force on an object (which I am not drawing) to fix ideas.
When we apply two forces on an object what happens?
We have a resulting force! That depends on the strength of the original ones and on their direction. There is a way to add them, say u = AB v = AC move the vector u (maintaining direction) to the endpoint of the other C making a vector CD the sum u + v is AD that is the parallelogram law.
If we add v + v we have a vector twice the size in the same direction or v + v = 2v, also, v + v + v = 3v is the vector in the same direction three times longer. 1.5v is the vector with 1.5 the size, -v = -1v is the vector with opposite direction and same size (v + -v = 0), and -5v has opposite direction and is 5 times the size of v.
So when we multiply by a positive number we "stretch" the vector, or maybe shrink it if we multiply by a number smaller than 1, and when we multiply by a negative number the vector inverts its direction and is stretched appropriately.
If we multiply by 3 and after that multiply by 2 we get a vector six times larger, or just 2(3v) = 6v.
Imagine what we get with -3(4v), 5(-1v) and -3(-2v)? Think geometrically!!
-12v, -5v and 6v respectively
Obs. A negative number times a negative number should be positive since we invert the vector two times! - a good justification for that rule/theorem in my opinion.
If we put another vector w somewhere else multiplying by real numbers would also work exactly in the same way.
We can also think that multiplying by -1 rotates vectors by 180o degrees, in that spirit we could rotate vectors by other angles like 90o or 120o . Let's say the number "j" rotates vectors by 90o degrees counterclockwise.
If we multiply our vector by j two times we rotate it by:
90o + 90o = 180o that is -v, or
j*jv = -v = -1v
So j*j = j2 should be -1, and we have a number that squared is -1! j is really our imaginary number i, time for more experiments.
2i rotates 90o and multiplies or stretches by 2
2i*2i rotates by 90o + 90 o =180 o and multiplies by 2x2=4 or 2ix2i = -4.
-i rotates by 90o and inverts the direction so it rotates by 270o .
i3 = i*i*i rotates by 90o + 90o + 90o = 270o also, so i3 = -i.
i4 = i*i*i*i rotates 4*90o = 360o so i4 = 1 = i2 * i2 .
You can guess i5 , i6 , and i37 , note that the algebra coincides with our geometrical ideas.
All properties of complex numbers can be understood in this geometric "rotation-stretching" way, more on that on next post.
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