2.9: Complex Numbers
- Page ID
- 84384
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)By Carey A. Smith
The imaginary unit number is i.
i = sqrt(-1) or i^2 = -1
In engineering, j is often used instead of i for the imaginary unit number.
In MATLAB, you can use either i or j.
Therefore, i and j should not be used as variable names. Instead, use ii or jj or another letter.
In MATLAB, 2*i can be written as 2i, 3*i as 3i, etc. Try this.
A complex number has both a real part and an imaginary part.
a = 2 + i
b = 1 - 4i
c = -2 + 3j
Solution
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% (1 pt) Write a MATLAB m-file script that does the following:
% (1 pt) Define this complex number:
z1 = (2 + 3i)
% (2 pts) Separate the real and imaginary parts like this:
z1_real = real(z1)
z1_imag = imag(z1)
% (1 pt) Compute the conjugate of z1 with this line of code:
z2 = conj(z1) % This changes the sign on the imaginary part
% (1 pt) Compute:
z1*conj(z1) % Note that this = square of the real part + square of the imaginary part
% (1 pt) Compute the length of this vector:
z1_length = sqrt(z1*conj(z1))
- Answer
-
z2 = conj(z1) = 2 - 3i
z1*conj(z1) = 13
z1_length = 3.6056
% (3 pts) A complex number can be plotted by letting zx = real(z1) and zy = imag(z1)
% Plot this complex number with these lines of code:
figure;
plot(z1_real,z1_imag,'*')
grid on % Draw x & y grid lines
xlim([-1,4]) % Set the minimum and maximum x values
ylim([-1,4]) % Set the minimum and maximum y values
- Answer
-
Add texts here. Do not delete this text first.
Draw a line from (0,0) to the point with the following code:
hold on % Keep whats been plotted and add to it
plot([0,z1_real],[0,z1_imag])
% [0,z1_real] = x coordinates of start and and end of the line
% [0,z1_imag] = y coordinates of start and and end of the line
Solution
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