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Complex numbers

$i$

Define $i$ as $\sqrt{- 1}$ , so $i^{2} = - 1$ .

Properties:

$\sqrt{- r} = i \sqrt{r}$
For example, $i \sqrt{3} = \sqrt{- 3}$
- Why?
  
  $\begin{array}{lll} {(i \sqrt{3})}^{2} & = & i^{2} {(\sqrt{3})}^{2} \\ = & i^{2} 3 \\ = & - 1 \cdot 3 \\ = & - 3 \\ = & {(\sqrt{- 3})}^{2} \end{array}$

Example: solve quadratic equation

Solve $3 x^{2} + 10 = - 26$ .

\begin{array}{rcl} 3 x^{2} + 10 & = & - 26 \\ 3 x^{2} & = & - 36 \\ x^{2} & = & - 12 \\ x & = & \pm \sqrt{- 12} \\ x & = & \pm i \sqrt{12} \\ x & = & \pm i \sqrt{4} \sqrt{3} \\ x & = & \pm 2 i \sqrt{3} \end{array}

Standard form

a + b i

Where $a$ , $b$ are real numbers. $a$ is the real part and $b i$ is the imaginary part.

Plotting complex numbers

Arithmetic

Adding/subtracting: Just think of $i$ as a variable, like $x$ :

\begin{array}{rcl} (4 + 5 i) + (3 + 2 i) & = & (4 + 3 + 5 i + 2 i) \\ = & (7 + 7 i) \end{array}

Multiplying: Think of $i$ as a variable, except that it can do tricks.

\begin{array}{rcl} (4 + 5 i) (3 + 2 i) & = & 4 \cdot 3 + 4 \cdot 2 i + 3 \cdot 5 i + 5 \cdot 2 i^{2} \\ = & 12 + 8 i + 15 i + 10 i^{2} \\ = & 12 + 23 i + 10 i^{2} \\ = & 12 + 23 i + (- 10) \\ = & 2 + 23 i \end{array}

Here is the same multiplication using the distributive law rectangle:

$\times$	$4$	$5 i$
$3$	12	$15 i$
$2 i$	$8 i$	$10 i^{2} = - 10$

Complex conjugate of $a + b i$ is $a - b i .^{}$

Multiplying by the conjugate is guaranteed to produce a real number:

\begin{array}{rcl} (a + b i) (a - b i) & = & a^{2} - a b i + a b i - b^{2} i^{2} \\ = & a^{2} - b^{2} (- 1) \\ = & a^{2} + b^{2} \end{array}

Example: dividing complex numbers.

Write $\frac{5 + 3 i}{1 - 2 i}$ in standard form. Solution:

\begin{array}{rcl} \frac{5 + 3 i}{1 - 2 i} & = & \frac{5 + 3 i}{1 - 2 i} \cdot \frac{1 + 2 i}{1 + 2 i} \\ = & \frac{5 + 13 i - 6}{1 + 4} \\ = & \frac{13 i - 1}{5} \\ = & - \frac{1}{5} + \frac{13 i}{5} \end{array}

Absolute value:

| a + b i | = \sqrt{a^{2} + b^{2}}

Example:

\begin{array}{rcl} | 3 + 4 i | & = & \sqrt{3^{2} + 4^{2}} \\ = & \sqrt{9 + 16} \\ = & \sqrt{25} \\ = & 5 \end{array}

Note that the absolute value can also be written in terms of the complex conjugate:

| a + b i | = \sqrt{(a + b i) (a - b i)}

Here's the same example, using the conjugate form:

\begin{array}{rcl} | 3 + 4 i | & = & \sqrt{(3 + 4 i) (3 - 4 i)} \\ = & \sqrt{9 + 16} \\ = & \sqrt{25} \\ = & 5 \end{array}

Geometrically, the absolute value of a real number is the distance of that real number from zero on the number line. Similarly, $| a + b i |$ is the distance of $a + b i$ from 0 in the complex plane, using the Pythagorean theorem to find the hypotenuse of a right triangle with legs of length $a$ and $b$ .