Complex Number Basic Operations

πŸ‘‘ Complex Numbers

Addition and Subtraction

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

Complex Addition Practice

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}