# Complex Number Basic Operations

🡑 Complex Numbers

### Cartesian Form

Complex addition and subtraction is easily done in Cartesian form. Simply treat the imaginary number i as a variable.

(a + jb) + (c + jd) = (a + c) + j(b + d) \\ (a + jb) – (c + jd) = (a – c) + j(b – d)

### Polar Form

When dealing with polar form, it’s easiest to first convert to Cartesian form and add or subtract from there.

## Multiplication and Division in Polar Form

### Cartesian Form

For multiplication, treat the imaginary number i as a variable.

(a + jb) \times (c + jd) = (ac – bd) + j (bc + ad) \\

For division, multiply both numerator and denominator by the conjugate of denominator. This leaves the denominator with a real number, simplifying the division process.

\begin{aligned} \frac{(a + jb)}{(c + jd)} &= \frac{(a + jb)}{(c + jd)} \times \frac{(c – jd)}{(c – jd)} \\ &= \frac{(ac + bd) + j (bc – ad)}{(c^2+d^2)} \\ &= \frac{ac + bd}{c^2+d^2} +j\frac{bc – ad}{c^2+d^2} \\ \end{aligned}

### Polar Form

Complex multiplication and division is easily done in polar form:

r_{1} \angle \theta_{1} \times r_{2} \angle \theta_{2} = r_{1}r_{2} \angle(\theta_{1} + \theta_{2}) \\ r_{1} \angle \theta_{1} \div r_{2} \angle \theta_{2} = \frac{r_{1}}{r_{2}} \angle(\theta_{1} – \theta_{2})

Therefore, when dealing with Cartesian form, it’s easiest to first convert to polar form and multiply or divide from there.

## Examples

Find (2-j8) + (-5+j4)

#### Solution

\begin{aligned} (2-j8)+(-5+j4) &= (2-5) + j(-8+4) \\ &= -3 -j4 \end{aligned}

### Complex Multiplication – Cartesian

Find (2-j8) \times (-5+j4)

#### Solution

\begin{aligned} (2-j8)\times(-5+j4) &= [(2)(-5) – (-8)(4)] + i [(-8)(-5) + (2)(4)]\\ &= 22+j48 \end{aligned}

### Complex Division – Cartesian

Find (10-j5) \times (-1+j2)

#### Solution

\begin{aligned} \frac{(10 – j5)}{(-1 + j2)} &= \frac{(10 – j5)}{(-1 + j2)} \times \frac{(-1 – j2)}{(-1 – j2)} \\ &= \frac{(-10 – 10) + j (-5 – 10)}{(1^2+2^2)} \\ &= \frac{-20}{5} +j\frac{-15}{5} \\ &= -4 -j3 \\ \end{aligned}

### Complex Multiplication – Polar

Find 2\angle{\pi} \times 3\angle{-\frac{\pi}{3}}

#### Solution

\begin{aligned} 2\angle{\pi} \times 3\angle{-\frac{\pi}{3}} &= (2\times3)\angle{(\pi+(-\frac{\pi}{3}))}\\ &= 6\angle{\frac{2\pi}{3}}\\ \end{aligned}

### Complex Division – Polar

Find 4\angle{\frac{\pi}{6}} \div 2\angle{-\frac{\pi}{3}}

#### Solution

\begin{aligned} 4\angle{\frac{\pi}{6}} \div 2\angle{-\frac{\pi}{3}} &=(4\div2)\angle{(\frac{\pi}{6}-(-\frac{\pi}{3}))}\\ &=2\angle{\frac{\pi}{2}}\\ \end{aligned}