Complex Numbers

  • Complex Numbers

Complex Numbers


The Imaginary Number

An imaginary number is any number with a factor of $\sqrt{-1}$ or $i$. Note that $i^n$ has four possible values:

  • $i^1 = i$
  • $i^2 = (\sqrt{-1})^2 = -1$
  • $i^3 = (\sqrt{-1})^3 = -i$
  • $i^4 = (\sqrt{-1})^4 = 1$

Example:

  1. $\sqrt{-81} = \sqrt{81(-1)} = 9\sqrt{-1} = 9i$
  2. $i^{74}$ $$\begin{align} i^{74} & = (i^4)^{18} \cdot i^2 & \text{(use Law I and Law II of Exponents)} \\ & = (1)^{18} \cdot (-1) \\ & = -1 \\ \end{align} $$
  3. $i^{131}+i^{13}+i^{25}$
    4. $$\begin{align} &i^{131}+i^{13}+i^{25}\\ & = (i^4)^{32} \cdot i^3 + (i^4)^3 \cdot i + (i^4)^6 \cdot i & \text{(use Law I and Law II of Exponents)} \\ & = (1)^{32} \cdot (-i) + 1^3 \cdot i + 1^6 \cdot i \\ & = -i + i + i\\ & = i \\ \end{align} $$


Complex Number

Complex number is a binomial composed of real and imaginary numbers in the form $a+bi$, where $a$ and $b$ are real numbers. $a$ is called the Real part and $bi$ is the Imaginary part.

$$a + bi $$

Addition and Subtraction of Complex Numbers

To add or subtract two or more complex numbers, get the sum or differences of the real parts and the sum or differences of the imaginary parts.

$$(a + bi) \pm (c + di) = (a \pm c) + (b \pm d) i $$

Example:

  1. $ (10 +2i) + (12 +3i) $ $$\begin{align} & (10 +2i) + (12 +3i)\\ & = (10 + 12) + (2 + 3)i \\ & = 22 + 5i \\ \end{align} $$
  2. $ (3 - 2i) - (-4 + 7i) $ $$\begin{align} & (3 - 2i) - (-4 + 7i) \\ & = ( 3 - (-4) ) + ( -2 - 7 ) i \\ & = 7 - 9i\\ \end{align} $$

Multiplication of Complex Numbers

When multiplying two complex numbers, use FOIL method (First, Outer, Inner, Last).

$$ \begin{align} (a+bi) \cdot (c+di) & = ac + ad i + bc i + bd i^2 \\ & = ac + (ad+bc) i + bd (-1) \\ & = (ac - bd) + (ad+bc)i \\ \end{align} $$

Example:

  1. $ (10-3i)(-2+5i) $ $$\begin{align} & (10-3i)(-2+5i) \\ & = [(10 \cdot -2) - (-3 \cdot 5)] + [(10 \cdot 5) + (-3 \cdot -2)] i\\ & = [-20 - (-15)] + [50 + 6] i\\ & = -5 +56i \\ \end{align} $$
  2. $ (4+3i)(-5i) $ $$\begin{align} & (4+3i)(-5i) \\ & = [4 \cdot -5i] + [3i \cdot (-5i)] \\ & = -20i + (-15) i^2 \\ & = 15 - 20i \\ \end{align} $$

Conjugate of Complex Numbers

A complex number $c-di$ is said to be a conjugate of $a+bi$ if $a=c$ and $b=d$ but the signs differ in the imaginary part. The product of two conjugates is a Real number.

$$ (a-bi) \text{ is a conjugate of } (a+bi)$$ $$ \begin{align} (a+bi)(a-bi) & = a^2 - b^2 i^2 \\ & = a^2 + b^2 \\ \end{align} $$

Example:

  1. $ (3-3i)(3+3i) $ $$\begin{align} & (3-3i)(3+3i) \\ & = 3^2 + 3^2\\ & = 18\\ \end{align} $$
  2. $ (y + \sqrt{-5})(y - \sqrt{-5}) $ $$\begin{align} & (y + \sqrt{-5})(y - \sqrt{-5}) \\ & = (y + i\sqrt{5})(y - i\sqrt{5})\\ & = y^2 + (\sqrt{5})^2 \\ & = y^2+5 \\ \end{align} $$

Division of Complex Numbers

To get the quotient of a complex number, rationalize the equation by multiplying the conjugate of the denominator to both the numerator and the denominator.

$$ \frac{a+bi}{c+di} = \frac{a+bi}{c+di} \cdot \frac{c-di}{c-di} = \frac{(a+bi)(c-di)}{c^2 + d^2}$$

Example:

  1. $ \frac{3}{ \sqrt{5} - \sqrt{-5} } $ $$\begin{align} & \frac{3}{ \sqrt{5} - \sqrt{-5} } \\ & = \frac{3}{ \sqrt{5} - i\sqrt{5} } \cdot \frac{ \sqrt{5} + i\sqrt{5} }{ \sqrt{5} + i\sqrt{5} } \\ & = \frac{ 3 \left( \sqrt{5} + i\sqrt{5} \right) }{ (\sqrt{5})^2 + (\sqrt{5})^2 } \\ & = \frac{ 3\sqrt{5} }{10} (1+i)\\ \end{align} $$
  2. $ \frac{16x^2 +9}{ 4x+3i } $ $$\begin{align} & \frac{16x^2 +9}{ 4x+3i } \\ & = \frac{16x^2 +9}{ 4x+3i } \cdot \frac{4x-3i}{4x-3i} \\ & = \frac{(16x^2 +9)(4x-3i)}{ (4x)^2+(3)^2 } \\ & = \frac{(16x^2 +9)(4x-3i)}{ 16x^2 +9 } \\ & = 4x-3i \\ \end{align} $$