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Question
Dear Prof Scott

http://en.wikipedia.org/wiki/Complex_number

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, where i2 = −1. In this expression, a is the real part and b is the imaginary part of the complex number. Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point (a, b) in the complex plane.


1. Can this be also considered as a variant of Complex Number where the vertical axis Y is used for the real part and the horizontal axis for the imaginary part ?.
i.e. The complex number ai + b can be identified with the point (a, b) in the complex plane. The complex plane will be the x -axis
instead of the conventional y-axis. a and b are real numbers, only the real and imaginary coefficients are interchanged with the axis.

Examples : 2i+3 , 4i-5, -3i+6 etc

The complex variant conjugate for example 4i + 9 will be -4i+9

The plotting of the complex numbers variant will also vary since the complex plane is taken as the x-axis.

2. In this case for the complex number variant, the complex number arithmetic viz multiplication, subtraction, division, addition , exponentiation etc will also be impacted ?.

4. Do you feel this complex number variant can be also considered in math applications viz control theory, geometry, electricl engineering etc ?.


Awaiting your reply,

Thanks & Regards,
Prashant S Akerkar

Answer
1. Yes, it can be done that way, but nobody does it much.
It would be about as common as in algebra when the y axis was called the x-axis and
the x-axis was called the y-axis.  That would be like, as an example, x = y - 3y + 2.
It just isn't done.

When a number is squared, in polar format, the magnitude is squared and the angle is doubled.
When they are expressed backwards, a little thought must be taken since the angle would have to be measured from the y-axis rather than the x-axis.  It seems like it would still be measured in a counter clockwise direction, but I've never thought about that.

I don't believe addition / subtraction would be affected, since both put the two real parts together and the two imaginary parts together.

It could still be used, but the basic concept of what it looks like would be changed.

That is like asking if math can be done in France.  If the person were a Frenchman, it would be easy.  To the typical american, it would be difficult.

All it comes down to is reversing the graph.  As I think about it, I believe that the angle would still be measure from the x-axis (now the vertical axis), but in a clockwise direction).

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Scott A Wilson

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I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?

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My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

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Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

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I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

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