Polar and Exponential Form §

Introduction §

Polar and exponential forms are crucial in understanding complex numbers. These forms provide alternative ways to represent complex numbers, often simplifying multiplication, division, and finding powers and roots.

Polar Form §

A complex number $z=a+bi$ can be represented in polar form as $z=r(\cos \theta +i\sin \theta )$, where $r$ is the magnitude (or modulus) of the complex number, and $\theta$ is the argument (or angle).

Calculating $r$ and $\theta$ §

• Magnitude ($r$): It is the distance from the origin to the point in the complex plane, calculated as $r=\sqrt{a^2+b^2}$.
• Argument ($\theta$): This is the angle formed with the positive x-axis, determined by $\theta =\arctan \left(\frac{b}{a}\right)$, considering the quadrant where the complex number lies.

Exponential Form §

The exponential form of a complex number uses Euler’s formula, $e^{i\theta }=\cos \theta +i\sin \theta$. Thus, a complex number $z=r(\cos \theta +i\sin \theta )$ can also be written as $z=r{e}^{i\theta }$.

Advantages §

• Simplifies Multiplication and Division: Multiplication and division become simpler as they turn into addition or subtraction of exponents.
• Powers and Roots: Calculating powers and roots of complex numbers becomes more straightforward using the exponential form.

Historical Context §

The concept of representing complex numbers in polar and exponential forms was significantly developed in the 18th century. Leonhard Euler, a Swiss mathematician, made substantial contributions to these concepts, particularly through Euler’s formula.

Examples §

1. Convert $1+i$ into polar and exponential forms.
2. Find the product of $2e^{i\pi /4}$ and $3e^{i\pi /3}$ using exponential form.

Conclusion §

Understanding polar and exponential forms of complex numbers is pivotal in advanced mathematics, particularly in fields such as signal processing and quantum mechanics.

Test Questions §

1. STARTI [Basic] Question: Convert the complex number $3+4i$ into polar form. Back: $5(\cos \arctan (\frac{4}{3})+i\sin \arctan (\frac{4}{3}))$. ENDI
2. STARTI [Basic] Question: Express $e^{i\pi /6}$ in standard complex form. Back: $\frac{\sqrt{3}}{2}+\frac{1}{2}i$. ENDI
3. STARTI [Basic] Question: Calculate the product of $2e^{i\pi /4}$ and $3e^{i\pi /3}$. Back: $6e^{i5\pi /12}$. ENDI

Mathematics | Complex numbers | Euler’s Formula