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3.2: Geometry of Complex Numbers

  • Page ID
    9948
  • The most fundamental new idea in the study of complex numbers is the “imaginary number” \(j\). This imaginary number is defined to be the square root of −1:

    \[j = \sqrt{-1}\]

    \[j^2 = -1\]

    The imaginary number \(j\) is used to build complex numbers \(x\) and \(y\) in the following way:

    \[z = x + jy\]

    We say that the complex number \(z\) has “real part” \(x\) and “imaginary part” \(y\):

    \[z = \mathrm{Re}[z] + j \mathrm{Im}[z]\]

    \[\mathrm{Re}[z] = x; \mathrm{Im}[z] = y\]

    In MATLAB, the variable \(x\) is denoted by real(z), and the variable \(y\) is denoted by imag(z). In communication theory, \(x\) is called the “in-phase” component of \(z\), and \(y\) is called the “quadrature” component. We call \(z=x+jy\) the Cartesian representation of \(z\), with real component \(x\) and imaginary component \(y\). We say that the Cartesian pair \((x,y)\)codes the complex number \(z\).

    We may plot the complex number \(z\) on the plane as in Figure. We call the horizontal axis the “real axis” and the vertical axis the “imaginary axis.” The plane is called the “complex plane.” The radius and angle of the line to the point z=x+jy are

    \[r = \sqrt{x^2 + y^2}\]

    \[θ = \tan^{-1}{\left(\frac {y}{x}\right)}\]

    In MATLAB, \(\)r is denoted by abs(z), and \(θ\) is denoted by angle(z)

    complexPlane.PNGCartesian and Polar Representations of the Complex Number \(z\)

    The original Cartesian representation is obtained from the radius \(r\) and angle \(θ\) as follows:

    \[x = r\cos{θ}\]

    \[y = r\sin{θ}\]

    The complex number \(z\) may therefore be written as

    \[z = x + jy\]

    \[z = r\cos{θ} + jr\sin{θ}\]

    \[z = r(\cos{θ} + j\sin{θ})\]

    The complex number \(\cosθ+j\sinθ\) is, itself, a number that may be represented on the complex plane and coded with the Cartesian pair \((\cosθ,\sinθ)\). This is illustrated in Figure. The radius and angle to the point \(z=\cosθ+j\sinθ\) are 1 and \(θ\). Can you see why?

    complexUnitCircle.PNG
    The Complex Number \(\cosθ+j\sinθ\)

    The complex number \(\cosθ+j\sinθ\) is of such fundamental importance to our study of complex numbers that we give it the special symbol \(e^{jθ}\)

    \[e^{jθ} = \cosθ+j\sinθ\]

    As illustrated in the above Figure, the complex number \(e^{jθ}\) has radius 1 and angle \(θ\). With the symbol \(e^{jθ}\), we may write the complex number \(z\) as

    \[z = re^{jθ}\]

    We call \(z = re^{jθ}\) a polar representation for the complex number \(z\). We say that the polar pair \(r∠θ\) codes the complex number \(z\). In this polar representation, we define \(|z|=r\) to be the magnitude of \(z\) and \(\mathrm{arg}(z)=θ\) to be the angle, or phase, of \(z\)

    \[|z| = r\]

    \[/mathrm{arg}(z) = θ\]

    With these definitions of magnitude and phase, we can write the complex number \(z\) as

    \[z = |z|e^{j\mathrm{arg}(z)}\]

    Let's summarize our ways of writing the complex number z and record the corresponding geometric codes

    \[z = x + jy = re^{jθ} = |z|e^{j\mathrm{arg}(z)}\]

    In "Roots of Quadratic Equations" we show that the definition \(e^{jθ}=\cosθ+j\sinθ\) is more than symbolic. We show, in fact, that \(e^{jθ}\) is just the familiar function \(e^x\) evaluated at the imaginary argument \(x=jθ\). We call \(e^{jθ}\) a “complex exponential,” meaning that it is an exponential with an imaginary argument.

     

    Exercise \(\PageIndex{1}\)

    Prove \((j)^{2n}=(−1)^{n}\) and \((j)^{2n+1}=(−1)^{nj}\). Evaluate \(j^3,j^4,j^5\)

    Exercise \(\PageIndex{2}\)

    Prove \(e^{j[(π/2)+m2π]}=j,\; e^{j[(3π/2)+m2π]}=−j,\; e^{j(0+m2π)}=1\), and \(e^{j(π+m2π)}=−1\). Plot these identities on the complex plane. (Assume m is an integer.)

    Exercise \(\PageIndex{3}\)

    Find the polar representation \(z=re^{jθ}\) for each of the following complex numbers:

    1. \(z=1+j0\)
    2. \(z=0+j1\)
    3. \(z=1+j1\)
    4. \(z=−1−j1\)

    Plot the points on the complex plane.

    Exercise \(\PageIndex{4}\)

    Find the Cartesian representation \(z=x+jy\) for each of the following complex numbers:

    1. \(z=\sqrt2 e^{jπ/2}\)
    2. \(z=\sqrt2 e^{jπ/4}\)
    3. \(z=e^{j3π/4}\)
    4. \(z=\sqrt2 e^{j3π/2}\)

    Plot the points on the complex plane.

    Exercise \(\PageIndex{5}\)

    The following geometric codes represent complex numbers. Decode each by writing down the corresponding complex number \(z\):

    1. \((0.7,−0.1)\)
    2. \((−1.0,0.5)\)
    3. \(1.6∠π/8\)
    4. \(0.4∠7π/8\)

    Exercise \(\PageIndex{6}\)

    Show that \(\mathrm{Im}[jz]=\mathrm{Re}[z]\) and \(\mathrm{Re}[−jz]=\mathrm{Im}[z]\). Demo 1.1 (MATLAB). Run the following MATLAB program in order to compute and plot the complex number \(e^{jθ}\) for \(θ=i2π/360,i=1,2,...,360\):

    j=sqrt(-1) n=360 for i=1:n,circle(i)=exp(j*2*pi*i/n);end; axis('square') plot(circle)

    Replace the explicit for loop of line 3 by the implicit loop

    circle=exp(j*2*pi*[1:n]/n);

    to speed up the calculation. You can see from the Figure that the complex number \(e^{jθ}\), evaluated at angles \(θ=2π/360,2(2π/360),...,\) turns out complex numbers that lie at angle \(θ\) and radius 1. We say that \(e^{jθ}\) is a complex number that lies on the unit circle. We will have much more to say about the unit circle in Chapter 2.

    complexPlaneExample.PNG
    The Complex Numbers \(e^{jθ}\) for \(0≤θ≤2π\) (Demo 1.1)