One fine day, She-Hulk met She-Nerd and asked her what complex numbers were and why do we need to learn it? Do

Question

One fine day, She-Hulk met She-Nerd and asked her what complex numbers were and why do we need to learn it? Can you explain what complex numbers are in a summarised form and explain why do we need to learn it.
Examples would be good

Answer 1

Ok, She-hulk... with complex numbers we're no longer trapped in one dimension, or even two, or even three! They give us an easy way to figure out movement in different directions, even thru hyperspace.



if you have only natural numbers you can only jump from one house to the next, going in the front door. That is you can go to 1, 2, 3, 4, etc.

If you have rational numbers you can go halfway thru someone's yard, or in the back door, but you must stay on the same street.

If you have complex numbers, you can go anywhere in town! Say you want to go three blocks west and 2 blocks north That's 3 + 2i. Say you want to go diagonally across 2 vacant blocks then up into a treehouse. That's 2 + 2i + hj, where h is the height of the treehoues measured in the same units as you measure blocks. Say you then go into the tree house and find a portal into another universe. Easy, just add another component to your vector. Voila! Multidimensional space! You might even find a He-Hulk there who can help you add 1 + 1 and make 3! Or you might find your parallel universe counterpart Dr. Bertha Banner who found a cure for the Hulk condition, letting you keep your brains even while you morph into the brawny She-Hulk! Then you wouldn't have to wake up in the morning with your clothes all torn and wonder where you've been and if you've done anything horrible. You'd know that you've been out saving the world, and you'd remember it all!



Complex numbers give us an easy way to compute with vectors, because they behave just like polynomials. So that's why it would be good to learn complex numbers.



(PS yeah, I know there's a lot more to complex numbers than that, but I'm talking to She-Hulk here! I gotta keep it in terms she can understand in her Hulk condition, otherwise she might smash me to smithereens! When she turns back into Dr. Bertha Banner she can appreciate the other things people have said here.)



Edited later: Wow, I looked up She-Hulk and found that she is an actual comic book character, not just someone made up for the purpose of this question. Seems she was, like Hulk, not very intelligent at first, but later her mind grew into her role and she became a pilot and an attorney. (Hulk's mind grew as the character developed also; at first he could barely speak; later, he philosophises and prays.) So in her later form she would understand all of these explanations, and she'd especially appreciate the use of complex numbers to explain how she can cross the "fourth wall" out of the comic panel and into another. Cool!

Answer 2

From my comic book days, I seem to recall that She-Hulk retained the intellect of her alter-ego, who was an attorney. Since she'd been through the four years of college, and then more in law school, I imagine she would have taken the math course which would have explained these.







Anyways,

Complex numbers are numbers that have a Real component and an "Imaginary" component. It's a number plus something times the square root of negative one.



Complex numbers are used in calculations involving LC or RLC circuits. You'll get that in physics.

Answer 3

Complex numbers are numbers with the unit defined as the square root of-1. Although real numbers require a line to represent them, complex numbers require an entire plane.



Dunno why we need to learn them.

Answer 4

You need complex numbers so that all simple equations, polynomials, have solutions.



Ex:



X^2+1 = 0



Needs solutions. With complex numbers all such equations have solutions. That's called the Fundamental Theorem of Algebra.



For the complex solution to the 2 dimensional heat diffusion equation, the real part is isotherms, or constant temperature lines, the complex part is constant heat flow lines. They are everywhere perpendicular (orthogonal). Same with electostatics. The real and imaginary parts of the solution give you constant voltage surfaces and constant electric field lines.

Answer 5

"In mathematics, a complex number is a number of the form

a + bi

where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. The real number a is called the real part of the complex number, and the real number b is the imaginary part. Real numbers may be considered to be complex numbers with imaginary part set to zero, that is, the real number a is equivalent to the complex number a+0i.



For example, 3 + 2i is a complex number, with real part 3 and imaginary part 2. If z = a + bi, the real part (a) is denoted Re(z), and the imaginary part (b) is denoted Im(z).



Applications:

The words "real" and "imaginary" were meaningful when complex numbers were used mainly as an aid in manipulating "real" numbers, with only the "real" part directly describing the world. Later applications, and especially the discovery of quantum mechanics, showed that nature has no preference for "real" numbers and its most real descriptions often require complex numbers, the "imaginary" part being just as physical as the "real" part.





Control theory

In control theory, systems are often transformed from the time domain to the frequency domain using the Laplace transform. The system's poles and zeros are then analyzed in the complex plane. The root locus, Nyquist plot, and Nichols plot techniques all make use of the complex plane.



In the root locus method, it is especially important whether the poles and zeros are in the left or right half planes, i.e. have real part greater than or less than zero. If a system has poles that are in the right half plane, it will be unstable, all in the left half plane, it will be stable, on the imaginary axis, it will have marginal stability. If a system has zeros in the right half plane, it is a nonminimum phase system.



Signal analysis

Complex numbers are used in signal analysis and other fields as a convenient description for periodically varying signals. The absolute value |z| is interpreted as the amplitude and the argument arg(z) as the phase of a sine wave of given frequency.



If Fourier analysis is employed to write a given real-valued signal as a sum of periodic functions, these periodic functions are often written as complex valued functionswhere ω represents the angular frequency and the complex number z encodes the phase and amplitude as explained above.



In electrical engineering, the Fourier transform is used to analyze varying voltages and currents. The treatment of resistors, capacitors, and inductors can then be unified by introducing imaginary, frequency-dependent resistances for the latter two and combining all three in a single complex number called the impedance. (Electrical engineers and some physicists use the letter j for the imaginary unit since i is typically reserved for varying currents and may come into conflict with i.) This use is also extended into digital signal processing and digital image processing, which utilize digital versions of Fourier analysis (and Wavelet analysis) to transmit, compress, restore, and otherwise process digital audio signals, still images, and video signals.



Improper integrals

In applied fields, the use of complex analysis is often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this, see methods of contour integration.



Quantum mechanics

The complex number field is also of utmost importance in quantum mechanics since the underlying theory is built on (infinite dimensional) Hilbert spaces over C. The more limited original formulations of Schrödinger and Heisenberg are also in terms of complex numbers.



Relativity

In special and general relativity, some formulae for the metric on spacetime become simpler if one takes the time variable to be imaginary.



Applied mathematics

In differential equations, it is common to first find all complex roots r of the characteristic equation of a linear differential equation and then attempt to solve the system in terms of base functions of the form f(t) = ert.



Fluid dynamics

In fluid dynamics, complex functions are used to describe potential flow in 2d.



Fractals

Certain fractals are plotted in the complex plane e.g. Mandelbrot set and Julia set."



From Wikipedia, the free, online encyclopedia at:

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



Hope this helps!

Answer 6

The imaginary number 'i' = square root of -1.

You cannot take the square root of negative numbers, so we factor out the -1 nad call it 'i'. Which allows us to continue with calculations.



It's used extensively in electrical engineering.



Your best bet for now is to accept them and the day you need to use them in a practical examples,you will see that they are a god send.

Answer 7

Complex numbers are numbers with a real part and an imaginary part. They're used for doing things in mathematics, like finding roots of numbers. They're used in quantum mechanics, relativity, and fluid dynamics.

Answer 8

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