Explanation of this complex number multiplication interpretation.

[njc_]

I have so many questions about this passage...(attached img)

”Multiplication of any complex number z by the complex number e^iθ can be interpreted geometrically as a counter-clockwise rotation through θ about the origin.” What does it mean to rotate “through” theta. Why can it be interpreted this way?

The second diagram has me confused also; the way it is written makes it look, at least to me, that the positional vector z is multiplied with the other complex number (4+2i) to give z. But isn’t z equal to (-1/2) + ((3^(1^2))/2)?

Please, if someone can explain this I would be tremendously greatful.

 

[Dr.Peterson]

I have so many questions about this passage...(attached img)

”Multiplication of any complex number z by the complex number e^iθ can be interpreted geometrically as a counter-clockwise rotation through θ about the origin.” What does it mean to rotate “through” theta. Why can it be interpreted this way?

The second diagram has me confused also; the way it is written makes it look, at least to me, that the positional vector z is multiplied with the other complex number (4+2i) to give z. But isn’t z equal to (-1/2) + ((3^(1^2))/2)?

Please, if someone can explain this I would be tremendously greatful.

Rotating something through a given angle means rotating it that far -- rotating through a 180 degree angle means rotating by 180 degrees.

In the picture, they are multiplying 4 + 2i by -1/2 + sqrt(2)/2 i to get an unstated product z, which turns out to be -2 + 2 sqrt(2) i - i + sqrt(2) i^2 = -2 + 2 sqrt(2) i - i - sqrt(2) = (-2 - sqrt(2)) + (2 sqrt(2) - 1)i. This is a rotation by 120 degrees. The multiplier is not shown on the picture.

 

[tkhunny]

I have so many questions about this passage...(attached img)

”Multiplication of any complex number z by the complex number e^iθ can be interpreted geometrically as a counter-clockwise rotation through θ about the origin.” What does it mean to rotate “through” theta. Why can it be interpreted this way?

The second diagram has me confused also; the way it is written makes it look, at least to me, that the positional vector z is multiplied with the other complex number (4+2i) to give z. But isn’t z equal to (-1/2) + ((3^(1^2))/2)?

Please, if someone can explain this I would be tremendously greatful.

You may also wish to keep in mind that the Complex Number are not "Well-Ordered". What you think of as multiplication in the Real Numbers may not make any sense at all if applied to the Complex Numbers. It's a substantially different animal. The magnitude of the multiplication may make sense, but the rest is part of a new language.