2.1: Algebra with Complex Numbers
- Page ID
- 106806
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The imaginary unit \(i\) is defined as the square root of -1: \(i= \sqrt{-1}\). If \(a\) and \(b\) are real numbers, then the number \(c= a+ib\) is said to be complex. The real number \(a\) is the real part of the complex number \(c\), and the real number \(b\) is its imaginary part. If \(a=0\), then the number is pure imaginary. All the rules of ordinary arithmetic apply with complex numbers, you just need to remember that \(i^2=-1\) For example, if \(z_1= 2+3i\) and \(z_2=1-4i\):
- \(z_1+ z_2=3-i\)
- \(z_1- z_2=1+7i\)
- \(\dfrac{1}{2} z_1+z_2=2-\dfrac{5}{2} i\)
- \(z_1z_2=(2+3i)(1-4i)=2-12 i^2-5i=14-5i\) (remember that \(i^2 = -1\)!)
- \(z_1^2=(2+3i)(2+3i)=4+9 i^2+12i=-5+12i\)
In order to divide complex numbers we will introduce the concept of complex conjugate.
The complex conjugate of a complex number is defined as the number that has the same real part and an imaginary part which is the negative of the original number. It is usually denoted with a star: If \(z = x + iy\), then \(z^∗ = x − iy\)
For example, the complex conjugate of \(2-3i\) is \(2+3i\). Notice that the product \(zz^*\) is always real:
\[\label{complexeq:eq1}(x+iy)(x-iy)=x^2-ixy+ixy+y^2=x^2+y^2. \]
We’ll use this result in a minute. For now, let’s see how the complex conjugate allows us to divide complex numbers with an example:
Given \(z_1= 2+3i\) and \(z_2=1-4i\) obtain \(z_1/z_2\)
Solution
\[\dfrac{z_1}{z_2}=\dfrac{2+3i}{1-4i} \nonumber \]
Multiply the numerator and denominator by the complex conjugate of the denominator:
\[\dfrac{z_1}{z_2}=\dfrac{2+3i}{1-4i}\dfrac{1+4i}{1+4i} \nonumber \]
This “trick” ensures that the denominator is a real number, since \(zz^*\) is always real. In this case,
\[\begin{align*} (1-4i)(1+4i) &=1-4i+4i-16i^2\\[4pt] &=17. \end{align*} \]
The numerator is
\[\begin{align*} (2+3i)(1+4i) &=2+3i+8i+12i^2 \\[4pt] &=-10+11i \end{align*} \]
Therefore,
\[\begin{align*}\dfrac{z_1}{z_2} &=\dfrac{2+3i}{1-4i} \\[4pt] &=\displaystyle{\color{Maroon}-\dfrac{10}{17}+\dfrac{11}{17}i} \end{align*} \]
Calculate \((2-i)^3\) and express your result in cartesian form (\(a + bi\))
Solution
\[ \begin{align*} (2-i)^3 &= (2-i)(2-i)(2-i) \\[4pt] (2-i)(2-i) &=4-4i+i^2 \\[4pt] &=4-4i-1 \\[4pt] &=3-4i \\[4pt] (2-i)(2-i)(2-i) &=(3-4i)(2-i) \\[4pt] &=6-3i-8i+4i^2 \\[4pt] &=6-11i+4(-1) \\[4pt] &=\displaystyle{\color{Maroon}2-11i} \end{align*} \]
The concept of complex conjugate is also useful to calculate the real and imaginary part of a complex number. Given \(z = x+iy\) and \(z^*=x-iy\), it is easy to see that \(z+z^*=2x\) and \(z-z^*=2iy\). Therefore:
\[\label{eq2} Re(z)=\dfrac{z+z^*}{2} \]
and
\[Im(z)=\dfrac{z-z^*}{2i} \nonumber \]
You may wonder what is so hard about finding the real and imaginary parts of a complex number by visual inspection. It is certainly not a problem if the number is expressed as \(a+bi\), but it can be more difficult when dealing with more complicated expressions.
- Dividing complex numbers: http://tinyurl.com/lkhztm5
External Links:
- Multiplying and dividing complex numbers: http://www.youtube.com/watch?v= KPSj4-76eEc
- Dividing complex numbers: http://www.youtube.com/watch?v=9I4QsSV1XDg