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### Network # complex number

#### bhuvanesh

Aug 29, 2013
201
i am so much confused with complex exponential.so i need to break that down into basic and rethink about it

1) why do we need complex number why not we stay with real number alone and what are the impossible become after possible after discovery of complex number.Thank you in advance

#### Arouse1973

Dec 18, 2013
5,177
Think about the impedance of a circuit like an RC network for instance. How could we work out the complex impedance without it. You have your real part which is your resistor and the imaginary part which is your reactance. We can then work out the impedance which is complex.

#### hevans1944

##### Hop - AC8NS
Jun 21, 2012
4,739
Complex numbers are a mathematical construct, used to represent two orthogonal sets of numbers named "real" and "imaginary". There are many orthogonal sets possible. For example, the sine and cosine functions are orthogonal to each other. This is useful because there are many situations in the real world that can only be adequately described by orthogonal function sets. Complex impedance is one of them, consisting of a "real" resistance and an "imaginary" reactance. In Cartesian coordinates, resistance is described by numbers on the X axis while reactance is described by numbers on the Y axis. The two numbers (x,y) are on the complex plane containing the X-axis and the Y-axis. The distance from the origin (0,0 to x,y) is defined to be impedance. Everything else is you study about complex numbers is just different ways of looking at the same thing mathematically.

Note there is nothing "imaginary" about reactance. A capacitor, or an inductor, does affect the current through itself when a voltage is applied. Or, conversely, it affects the voltage across itself when a current is applied. If these voltages and currents are oscillating sinusoidal waveforms of constant frequency, eventually a steady state is reached where the current waveform is shifted exactly ninety degrees in phase with respect to the voltage waveform. This always occurs, no matter what value of capacitance or inductance is involved. It is an intrinsic property of the electromagnetic storage of energy in capacitors and inductors. One way to describe this phenomenon is complex arithmetic.

Why do we "need" complex numbers? Well, we don't. I was calculating impedance while still attending grade school, using rote formulas, without much understanding of why the formulas worked. It was only much later, when things became more complicated, that the introduction of complex numbers actually simplified calculations. The more mathematics you learn, the more you learn what you don't know. There is no end to it, but the beauty of mathematics (a purely human construct of the mind!) is its application to describing the world around us. Some things cannot be adequately described. How do you describe colors to a person born blind? Until you can see the colors, no abstract explanation will suffice to explain what colors are. Same with complex numbers. Until you can see the mathematical beauty of orthogonal relationships, and how these relate to the world around you, complex numbers might as well all be imaginary.

BTW, orthogonal sets are not limited to just pairs. The Universe we move around in can be described by a three-dimensional orthogonal set, unless you move around in it really fast. Then things get really complex.

• Arouse1973

#### bhuvanesh

Aug 29, 2013
201
i am sailing in boat moving 3 units in north moving 4 units in east having 3+4j as vector now i want to rotate it to 90 degree counter clockwise so i just multiply by i.Then i get the answer ,but i want to know it can be done in harder way,i mean how to do same rotation with using complex plane .THank you in advance

#### bhuvanesh

Aug 29, 2013
201
could you steer me to realize the beauty of orthogonal relationship ,please

#### Laplace

Apr 4, 2010
1,252
... steer me to realize the beauty ... Here is a plot of your sailboat vectors in various coordinate systems. The REAL plane (in blue) has been superimposed on the COMPLEX plane (in red) and the vectors are given in Cartesian coordinates, in polar form, as complex numbers, and in complex exponential form. You should apply trigonometry to verify the Cartesian-polar transformation. Then apply Euler's Formula to verify the complex number to complex exponential transformation.

Some formulas in complex variable analysis are less cluttered when expressed in complex exponential form. This may be a form of beauty to some. But beauty is in the eye of the beholder.

• Arouse1973

#### Merlin3189

Aug 4, 2011
250
i am sailing in boat moving 3 units in north moving 4 units in east having 3+4j as vector now i want to rotate it to 90 degree counter clockwise so i just multiply by i.Then i get the answer ,but i want to know it can be done in harder way,i mean how to do same rotation with using complex plane .THank you in advance
1- You are using the complex plain, with 1j being the unit vector East and 1 being the (orthogonal) unit vector North. (This is a reflection & rotation of the conventional axes, but you can use that.) You were going 3+4j, you rotate 90deg counterclockwise by multiplying by -j (because you chose unconventional base vectors: usually 1 is East, 1j is North) and are now going +4 -3j = 4 N and 3W
2- You could work out that it is SQRT(3^2 + 4^2)=5 in direction invtan(4/3) = bearing 53.1 deg
Then rotate 90 deg acw to bearing -36.9 deg or bearing 323.1 deg. You can resolve N & W components if you want.
3 - Matrix multiplication (4,3) x (0 1)(-1 0) = (-3, 4)
(Sorry this site doesn't handle maths like matrices. So by matrix (a b)(c d) I mean first row = a b and second row = c d )

Since you are interested in complex exponentials, you can also say your original course is 5 x e ^ (i x53.1 deg)
Then using your axes to rotate 90 deg acw is -90deg, so the new vector is 5 x e ^(i x -36.9deg)
If you want to resolve that, A e ^(i x w) = A cos(w) + i A sin(w)
So 5 x e ^(i x -36.9deg) = 5 cos(-36.9) +i 5 sin(-36.9) = 4 -3i (or 4 -3j )

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