• no real number multiplied by itself
    • gives a negative result
  • no real square root of a negative number
    • \( (-2) \times (-2) = 4 \)
  • so define an entity designating \( \sqrt{-1} \)
    • \( i = \sqrt{-1} \)
    • \( i^2 = -1 \)
    • \( (-i)^2 = -1 \)
  • in engineering, \( j = \sqrt{-1} \)
    • to avoid confusing with current symbol \(i\)

imaginary and complex numbers

  • some imaginary numbers
    • \( 4i \)
    • \( 3.74i \)
    • \( i\pi \)
    • put \( i \) after numbers but before variables and constants
  • complex numbers are imaginary and real numbers together
    • general form: \( g = a + ib \)
    • real part: \( a = Re(g) \)
    • imaginary part: \( b = Im(g) \)
complex conjugate: a complex number’s imaginary part sign reversed
  • given complex number \( g = a + ib \)
    • complex conjugate is \( g^* = a + ib \)
    • every \(i\)’s sign must be reversed
  • multiplication of a complex number with its complex conjugate gives a positive number
    • the result is called the modulus of the complex number
    • \( \lvert g \rvert^2 = g^{*} g = g g^{*} \)
    • \( \lvert g \rvert = +\sqrt{\lvert g\rvert^2} \)
    • \( \lvert g \rvert = \sqrt{a^2 + b^2} \)

complex number identities

  • \( \lvert g \rvert^2 = g g^{*} = (a+ib)(a-ib) \)
    • \( = a^2 - iab + iba - i^2b^2 = a^2 + b^2 \)
    • \( \lvert g \rvert^2 = a^2 + b^2 \)
  • \( g = \frac{1}{c+id} = \frac{c}{c^2 + d^2} -i\frac{d}{c^2 + d^2} \)
    • reciprocal of a complex number can be re-arranged into a complex number

euler’s formula and the complex plane

euler’s formula

  • \( \exp{(i\theta)} = \cos \theta + i \sin \theta \)
  • this notation: major reason for use of complex numbers in engineering
    • exponentials replace sines and cosines
    • simple power additions replace cumbersome trigonometric algebra
  • note that: \( \exp{(-i\theta)} \equiv \exp{(i[-\theta])} \)
    • extending this to euler’s formula
      • \( \exp{(-i\theta)} = \cos{(-\theta)} + i \sin{(-\theta)} \)
  • complex conjugate:
    • \( [\exp{(i\theta)}]^* = \exp{(-i\theta)} \)
  • modulus:
    • \( \exp{i\theta}\exp{(-i\theta)} = \exp{(i\theta - i\theta)} = \exp{(0)} = 1\)

polar form of complex number

  • \( g = \lvert g \rvert \exp{(i\theta)} \)

complex plane - argand diagram

  • not a physical plane
  • a mathematical plane, similar to the \(x-y\) plane
    • real axis: horizontal line
    • imaginary axis: vertical line
  • any complex number
    • \( g = a + ib = \lvert g \rvert \exp{(i\theta)} \)
    • \( \theta = \arg{(g)}\)

complex-plane

multiplication in polar form

  • to multiply two complex numbers
    • multiply moduli
    • add angles
  • example:
    • \( g = \lvert g \rvert \exp{(i\theta)} \)
    • \( h = \lvert h \rvert \exp{(i\phi)} \)
    • \( g \times h = \lvert g \rvert \lvert h \rvert \exp{(i[\theta + \phi])} \)

nth roots of unity

  • the number \( \exp{\left(\frac{2\pi i}{n}\right)} \)
    • when raised to the nth power is 1 (unity)
    • \( \left[\exp{\left( \frac{2\pi i}{n} \right) } \right]^n = \exp{ \left( \frac{2\pi i}{n} \right) } = \exp{(2\pi i)} = 1\)
  • many different complex numbers give 1 when raised to nth power
    • this specific is the nth root: \( \sqrt[n]{1} = \exp{ \left( \frac{2\pi i}{n} \right)} \)

standard results for complex numbers

  • \( (gh)^{ * } = g^{ * }h^{ * } \)
  • \( \left( \frac{1}{gh} \right)^{*} = \frac{1}{ g^{*} h^{*} } \)
  • \( \left( \frac{g}{h} \right)^{*} = \frac{g^{*}}{h^{*}} \)