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Introduction to Matlab & Complex Numbers

Peter Norgaard

February 2nd, 2011

Like many programming languages, Matlab assigns values to variablesusing the standard evaluate right-hand side, then assign to left-hand side

format. For example, type the following, hitting return after each line,

x = 1 % x is assigned the value 1

y = 4*x + 1 % the RHS is evalued as 5, which is assigned to y

Each line is executed independently, and is independent of previous lines, soif we change x, the value of y remains unchanged.

x = 1 % y is still equal to 5

By default, Matlab sets i =1. However, this can be (accidentally)

overwritten by the user with a simple statement like

i = 2

In such cases, you can either assign it the imaginary number value again, orclear the variable assignment, in which case it reverts to its original default.

i = sqrt(-1) % the imaginary number is assigned to i

clear i % i goes back to its default value

You can write any complex number using the sum of the real an imaginaryparts,

z = 2 + 5i % z = 2 + 5*i also works

To separate out the real and imaginary parts, use the real() and imag()functions,

x = real(z) % returns 2

y = imag(z) % returns 5

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Matlab has an incredible number of available functions, including a large set

that operates on complex numbers. Weve already seen real() and imag().Here are some others that can be very useful:

abs(z) % gives the modulus of the complex number

angle(z) % gives the phase angle of z in polar coordinates,

Note that angle() returns the answer in the range (, ]. So we can obtainthe polar coordinates (r, ) of a complex number, and then use them toreconstruct z.

r = abs(z)

theta = angle(z) % in the range (-pi, pi]

z = r*exp(i*theta) % returns 2.0000 + 5.0000i

To get the complex conjugate of a complex number, use the function conj()

z 1 = 2 + 5 i

z2 = conj(z1) % returns 2.0000 - 5.0000i

Standard operations like addition (+), subtraction(-), multiplication(*) anddivision (/) can all be performed on complex numbers:

z1+z2 % returns 4

z1-z2 % returns 10i

z1*z2 % returns 29

z1/z2 % returns -0.7241 - 0.6897i

To find the nth root of a number, we do something special. The functionnthroot() is only intended for real roots. Instead, we cast the nth root asan algebraic equation. For example, to find the quartic roots of 1, take thefourth power of both sides ofz =

41 to get z4 = 1. Now use the symbolic

math toolbox function solve() to obtain the solution for z,

z_a = solve(z^4 = 1,z)

which returns the four solutions, {1, 1 i, i}. Note that these are not inorder of increasing phase angle. In some cases the symbolic solution will be

very long... use the function double() to recast it as a floating point number.

Although you cant just use Matlab to do your homework (showing yourwork is definitely required), you can use it to quickly check your results.Good luck!

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