๐ Complex Numbers in Polar Form: Product and Quotient Rules
Complex numbers, often represented as $a + bi$ where $a$ and $b$ are real numbers and $i$ is the imaginary unit ($\sqrt{-1}$), can also be expressed in polar form. This form offers a more intuitive way to understand their magnitude and direction in the complex plane. Understanding the product and quotient rules for complex numbers in polar form simplifies multiplication and division considerably.
๐ Historical Context
The development of complex numbers dates back to the 16th century, with mathematicians like Gerolamo Cardano grappling with solutions to cubic equations. However, it was mathematicians such as Abraham de Moivre and Carl Friedrich Gauss who formalized the concept and provided a geometric interpretation, leading to the use of polar representation.
๐ Key Principles: Polar Form Representation
A complex number $z = a + bi$ can be written in polar form as $z = r(\cos\theta + i\sin\theta)$, where:
- ๐ $r = |z| = \sqrt{a^2 + b^2}$ is the modulus (or magnitude) of $z$. It represents the distance from the origin to the point $(a, b)$ in the complex plane.
- ๐งญ $\theta = \arg(z)$ is the argument of $z$. It represents the angle between the positive real axis and the line connecting the origin to the point $(a, b)$ in the complex plane. \(\theta\) is typically measured in radians.
โ Product Rule
If $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$, then their product $z_1z_2$ is given by:
$z_1z_2 = r_1r_2[\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]$.
In simpler terms, to multiply two complex numbers in polar form, multiply their moduli and add their arguments.
- โ๏ธ Multiply the moduli: $r = r_1 * r_2$.
- โ Add the arguments: $\theta = \theta_1 + \theta_2$.
โ Quotient Rule
If $z_1 = r_1(\cos\theta_1 + i\sin\theta_1)$ and $z_2 = r_2(\cos\theta_2 + i\sin\theta_2)$, then their quotient $\frac{z_1}{z_2}$ (where $z_2 \neq 0$) is given by:
$\frac{z_1}{z_2} = \frac{r_1}{r_2}[\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)]$.
In simpler terms, to divide two complex numbers in polar form, divide their moduli and subtract their arguments.
- โ Divide the moduli: $r = \frac{r_1}{r_2}$.
- โ Subtract the arguments: $\theta = \theta_1 - \theta_2$.
๐ก Example 1: Product Rule
Let $z_1 = 2(\cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3}))$ and $z_2 = 3(\cos(\frac{\pi}{6}) + i\sin(\frac{\pi}{6}))$.
Then, $z_1z_2 = (2)(3)[\cos(\frac{\pi}{3} + \frac{\pi}{6}) + i\sin(\frac{\pi}{3} + \frac{\pi}{6})] = 6(\cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2})) = 6(0 + i) = 6i$.
โ๏ธ Example 2: Quotient Rule
Let $z_1 = 8(\cos(\frac{5\pi}{6}) + i\sin(\frac{5\pi}{6}))$ and $z_2 = 2(\cos(\frac{\pi}{3}) + i\sin(\frac{\pi}{3}))$.
Then, $\frac{z_1}{z_2} = \frac{8}{2}[\cos(\frac{5\pi}{6} - \frac{\pi}{3}) + i\sin(\frac{5\pi}{6} - \frac{\pi}{3})] = 4(\cos(\frac{\pi}{2}) + i\sin(\frac{\pi}{2})) = 4(0 + i) = 4i$.
โ๏ธ Conclusion
Using polar form simplifies complex number multiplication and division. Remember to multiply (or divide) the moduli and add (or subtract) the arguments. This approach is particularly useful in fields such as electrical engineering, physics, and signal processing.