Complex numbers#

What you need to know

  • Complex numbers generalize oridnary 1D numbers to 2D To descirbe complex numbers we need two compoenents called real and imaginary parts. Complex numbers appear naturally in quantum mechanics and are essential part of physics.

  • The imaginary unit: \(i = \sqrt{-1}\) This symbol, \(i\), represents the imaginary component of a complex number, where \(i^2 = -1\).

  • Complex numbers as roots of polynomial equations. For example, quadratic equations can have two complex roots, showcasing their relevance in solving equations.

  • Cartesian and polar representations Complex numbers can be expressed in Cartesian form \((x + iy)\) or polar form \((re^{i\phi})\), where \(r\) is the magnitude, and \(\phi\) is the phase (angle).

  • Euler’s formula: A beautiful and fundamental equation \(re^{i\phi} = r(\cos \phi + i\sin \phi)\), elegantly connects complex numbers to trigonometric functions. uler’s equation makes it much easier to manipulate complex numbers, especially in their polar form, simplifying multiplication and division.

  • Rotation in the complex plane Multiplying by a complex number in polar form corresponds to a rotation. For instance, \(re^{i\phi}\) rotates a vector of length \(r\) counterclockwise by an angle \(\phi\), while \(re^{-i\phi}\) rotates it clockwise by \(\phi\).

  • Complex conjugate: Flipping the imaginary part The conjugate of a complex number \(z = x + iy\) is given by \(z^* = x - iy\).

  • Multiplying by the conjugate When you multiply a complex number by its conjugate, \(z \cdot z^*\), the result is a real number equal to the square of its distance from the origin in the complex plane: \(|z|^2 = x^2 + y^2\).

Complex number living in 2D#

  • A complex number \(z\) is a kind of 2D number that lives in 2D space and requires two components for its full specification. Watch the video to get a visual feel for why we need complex numbers.

Introducing imaginary number \( i \)#

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Fig. 98 Visualizing complex numbers on cartesian plane#

  • Fundamental theorem of algebra. says that polynomial equations like quadratic, cubic, quartic etc must have numbers of roots equal to the highest power. E.g quadratic must have two roots. Complex numbers ensure the existence of roots for polynomials.

  • This equation must have two solutions/roots. How do we visualize them?

\[x^2+1=0\]
\[x_{1,2}= \pm\sqrt{-1} = \pm i\]
  • The answer: we will need two dimensions! quantifying how much real and how much imaginary component the number has.

  • What is the definition of \( i \)? The imaginary number \( i \) is defined solely by the property that its square is \(−1\), that is: \(i\cdot i=-1\).

  • How does \(i\) change what I know about math of real variables? Imaginary numbers extend the real number system \(\mathbb{R}\) to the complex number system \(\mathbb{C}\).

applied photoelectric

Fig. 99 Complex numbers live in 2D: You must specify real and imaginary components to fully define a complex number#

Cartesian vrepresentation#

complex nums on plane

Fig. 100 Visualizing complaex numbers in cartesian coordinates.#

Cartesian Representation

\[z = x+iy\]
  • \(x=Re(z)\) real component

  • \(y=Im(z)\) imaginary component

Example: Identify real and imaginary parts

Find real and imaginary components of the following complex numbers \(z_1 = 3 +2\), \(z_2 = -2i\), \(z_3=1.1\)

Solution

  • \(Re(z_1) = 3\), \(Im(z_1)=2\)

  • \(Re(z_2) = 0\), \(Im(z_2)=-2\)

  • \(Re(z_3) = 1.1\), \(Im(z_1)=0\)

Polar representation#

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Fig. 101 Visualizing complaex numbers in polar coordinates.#

  • Polar representation expresses complex numbers in terms of radius from origin \(r\) and angle of counterclockwise rotation \(\phi\).

  • Using trigonometry we have: \(x = r\cos\phi\) and \(y = r\sin\phi\) which can be plugged into cartesian representation:

Polar Representation via sin and cos

\[z= x+iy = rcos\phi + ri\sin\phi =r(cos\phi+isin\phi)\]
  • \(r=\sqrt{x^2+y^2}\) distance from origin.

  • \(\phi\) rotation angle in complex plane.

  • What’s the big deal with this polar representation? We will find that life is much much easier in polar represnation when deriving new expressions or manipulating complex numbers.

  • This dramatic simplification is thanks to “magical” Euler’s formula that turns trig functions into exponentss!

Euler’s formula

\[\cos{\phi} + i \sin{\phi} = e^{i\phi}\]

Polar Representation via complex exponential

\[z = re^{i\phi}\]
  • \(r=\sqrt{x^2+y^2}\) distance from origin.

  • \(\phi\) rotation angle in complex plane.

Converting from Carteisan to Polar#

  • Now having established Euler’s formula we need a recepie to go back and forth between cartesian and polar representations.

  • Extract r The value \(r\) is the Euclidean distance of vector \((x,y)\) from the origin.

\[r = \sqrt{x^2 + y^2}\]
  • Extract \(\phi\) The value \(\phi\) is the angle of with respect to the real axis. The tangent of \(\phi\) is \(\left(\frac{y}{x}\right)\). Therefore using simple trigonometry we can backcalculate angle and sin, cos tan functions from cartesia representatn

\[ \tan{\phi} = \frac{y}{x} \]
\[ \phi = \tan^{-1} \Big(\frac{y}{x} \Big) \]

Example: Convert to polar

Write complex number \(z = -1-2i\) in polar form

Hot Tip use atan2 or arctan2 in your calculator which correctly identifies quadrant for angle.

Solution

  • We need to extract \(r\) and \(\phi\) to write \(z= re^{i\phi}\)

\[r = \sqrt{(-1)^2+(-2)^2} = \sqrt{5}\]
\[\phi = arctan2(-2, -1) \approx -2.034\]
  • We can always add \(2\pi\) to angle we get from arctan2 to make angle positive. Reminder that \(\pm 2\pi \) leave trig functions unchanged.

\[\phi = -2.034\rightarrow -2.034+2\pi = 4.248\]
\[z = \sqrt{5}e^{4.248i}\]
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Fig. 102 Visualization of Euler’s formula \(e^{i\omega t}\) as a function of \(t\). The helix is formed by plotting points for various values of \(\omega\) and is determined by both the cos and sin components of the formula. One curve represents the real component \(cos\omega\) of the formula, while another curve, rotated 90 degrees around the \(t\) axis (due to multiplication by \(i\) represents the imaginary component \(sin\omega\)#

Complex Conjugate#

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Fig. 103 Visualizing conjugate of complex number as a mirror image or rotation backwards#

Complex Conjugate

\[\bar z = x-iy\]
\[\bar z = re^{-i\phi}\]

Extracting absolute value#

  • Multiplying complex number by its conjugate always results in positive real number!

  • Geometrically this means rotating back to real axis!

  • Product of complex number and its conjugate returns the distance from the origin in complex plane.

\[|z|^2 = \bar z \cdot z = (x-iy)\cdot (x+iy) = x^2+y^2\]
\[|z|^2 = \bar z \cdot z = re^{-i\phi}\cdot re^{i\phi}=r^2\]

Expressing sin and cos via complex exponentials#

  • Using Euler’s formula we can take the sum and difference of \(z\) and \(\bar z\) to express sin and cos in terms of compelx exponentials.

  • These representations of sin and cos are super powerful in simplifying various integrals and for deriving experssions.

\[ \cos{\phi} = \frac{e^{i\phi} + e^{-i\phi}}{2} \]
\[ \sin{\phi} = \frac{e^{i\phi} - e^{-i\phi}}{2i} \]

Applications of complex numbers#

Problems#

Problem 1: Multiplication#

Multiply the two complex numbers \(z_1 = 3 + 4i\) and \(z_2 = 1 - 2i\).

Problem 2: Find conjugate#

Find the complex conjugate of \(z = -3 + 5i\).

Problem 3: from polar to cartesian#

Convert the complex number \(z = 5e^{i\frac{\pi}{4}}\) to its Cartesian form.

Problem 4: Division#

Divide the complex numbers \(z_1 = 3 + 4i\) by \(z_2 = 1 - 2i\).

Problem 5: Find Modulus#

Find the modulus of the complex number \(z = 7 + 24i\).

Problem 6: Multiplication#

Multiply the complex numbers \(z_1 = 2e^{i\frac{\pi}{6}}\) and \(z_2 = 3e^{i\frac{\pi}{3}}\).

Problem 7: from cartesian to polar#

Express the complex number \(z = -4 + 4i\) in polar form.