key term - Re^(iθ)

Definition

The expression $re^{i\theta}$ is a fundamental representation of complex numbers in polar form. It combines the modulus (r) and argument (θ) of a complex number into a single, compact expression, allowing for easy manipulation and visualization of complex number operations.

5 Must Know Facts For Your Next Test

1. The expression $re^{i\theta}$ provides a compact way to represent a complex number in polar form, where $r$ is the modulus (or magnitude) and $\theta$ is the argument (or angle).
2. The polar form of a complex number is particularly useful for performing operations such as multiplication, division, and exponentiation, as these operations become simpler in the polar form.
3. The real and imaginary parts of a complex number $z = a + bi$ can be expressed in terms of $r$ and $\theta$ as $a = r\cos\theta$ and $b = r\sin\theta$.
4. The conversion between the rectangular form $(a + bi)$ and the polar form $(re^{i\theta})$ is given by the formulas: $r = \sqrt{a^2 + b^2}$ and $\theta = \tan^{-1}\left(\frac{b}{a}\right)$.
5. The polar form $re^{i\theta}$ is particularly useful in applications involving periodic functions, such as in the study of electrical circuits, signal processing, and quantum mechanics.

Review Questions

• Explain how the expression $re^{i\theta}$ represents a complex number in polar form.
  • The expression $re^{i\theta}$ represents a complex number in polar form, where $r$ is the modulus (or magnitude) of the complex number, and $\theta$ is the argument (or angle) of the complex number in the complex plane. This polar representation is particularly useful for performing operations on complex numbers, as it allows for simpler calculations compared to the rectangular form $(a + bi)$. The real and imaginary parts of the complex number can be easily extracted from the polar form using the formulas $a = r\cos\theta$ and $b = r\sin\theta$.
• Describe the advantages of using the polar form $re^{i\theta}$ over the rectangular form $(a + bi)$ for complex number operations.
  • The polar form $re^{i\theta}$ offers several advantages over the rectangular form $(a + bi)$ when performing operations on complex numbers. First, multiplication and division of complex numbers become simpler in the polar form, as these operations can be performed by multiplying or dividing the moduli and adding or subtracting the arguments. Second, the polar form is particularly useful for representing and working with periodic functions, as the argument $\theta$ can be easily interpreted as an angle or phase. Finally, the polar form provides a more intuitive visualization of complex numbers in the complex plane, as the modulus $r$ represents the distance from the origin, and the argument $\theta$ represents the angle with respect to the positive real axis.
• Analyze the relationship between the rectangular form $(a + bi)$ and the polar form $re^{i\theta}$ of a complex number, and explain how to convert between the two representations.
  • The rectangular form $(a + bi)$ and the polar form $re^{i\theta}$ of a complex number are related through the following formulas: $r = \sqrt{a^2 + b^2}$ and $\theta = \tan^{-1}\left(\frac{b}{a}\right)$. To convert from the rectangular form to the polar form, one can calculate the modulus $r$ using the Pythagorean theorem, and the argument $\theta$ using the inverse tangent function. Conversely, to convert from the polar form to the rectangular form, one can use the trigonometric identities $a = r\cos\theta$ and $b = r\sin\theta$. This ability to move between the two representations is crucial, as the polar form is often more convenient for performing operations on complex numbers, while the rectangular form is sometimes more intuitive for visualizing and working with complex numbers in the complex plane.

Related terms

Complex Number: A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, satisfying $i^2 = -1$.

Modulus:

The modulus of a complex number $z = a + bi$ is the distance of the point $(a, b)$ from the origin in the complex plane, and is denoted as $|z|$ or $r$.

Argument:

The argument of a complex number $z = a + bi$ is the angle between the positive real axis and the line segment connecting the origin to the point $(a, b)$ in the complex plane, and is denoted as $\theta$.