Recall that in addition to rectangular form, \(a+bi\), we can also represent complex numbers in polar form:
\(z=r(cos\theta+isin\theta)\),
where \(r\) is called the modulus and \(\theta\) is called the argument of the complex number.
Sometimes, you might see \(z=r(cos\theta+isin\theta)\) abbreviated as \(z=rcis\theta\).
To add/subtract complex numbers in polar form, follow these steps:
1. Convert all of the complex numbers from polar form to rectangular form (see the Rectangular/Polar Form Conversion page).
2. Perform addition/subtraction on the complex numbers in rectangular form (see the Operations in Rectangular Form page).
3. After evaluating the sum/difference, convert it back into polar form for your answer (see the Rectangular/Polar Form Conversion page).
The following example will demonstrate how to apply these steps to find the sum/difference of complex numbers.
See this video for the solution:
To multiply/divide complex numbers in polar form, multiply/divide the two moduli and add/subtract the arguments. More specifically, for any two complex numbers, \(z_1=r_1(cos(\theta_1)+isin(\theta_1))\) and \(z_2=r_2(cos(\theta_2)+isin(\theta_2))\), we have:
\(z_1 z_2=r_1r_2[cos(\theta_1+\theta_2)+isin(\theta_1+\theta_2)]\)
\(\frac{z_1}{z_2}=\frac{r_1}{r_2}[cos(\theta_1-\theta_2)+isin(\theta_1-\theta_2)], z_2 \ne 0\)
Solution:
1. Identify the moduli and arguments. In this example, we have:
\(r_1=5,r_2=7,\theta_1=\frac{5\pi}{2},\theta_2=\frac{3\pi}{4}\)
2. Substitute these values into the above produce and quotient formulas:
\(z_1 z_2=(5)(7)[cos(\frac{5\pi}{2}+\frac{3\pi}{4})+isin(\frac{5\pi}{2}+\frac{3\pi}{4})]\)
\(\frac{z_1}{z_2}=\frac{5}{7}[cos(\frac{5\pi}{2}-\frac{3\pi}{4})+isin(\frac{5\pi}{2}-\frac{3\pi}{4})]\)
3. Evaluate the product and quotient by simplifying both expressions:
\(z_1 z_2=35[cos(\frac{13\pi}{4})+isin(\frac{13\pi}{4})]\)
\(\frac{z_1}{z_2}=\frac{5}{7}[cos(\frac{7\pi}{4})+isin(\frac{7\pi}{4})]\)