Question

Find and plot the complex conjugate of each number. $$ 5(\cos 0+i \sin 0) $$

Short answer

The complex conjugate is 5. Both the number and its conjugate are located at (5, 0) on the complex plane.

Step-by-step solution

- Identify the components

The given complex number is in polar form: \( 5(\cos 0 + i\sin 0) \). We identify that \( r = 5 \), \( \theta = 0 \).

- Convert to rectangular form

Use the formulas \( x = r \cos \theta \) and \( y = r \sin \theta \): \( x = 5 \cos 0 = 5 \), \( y = 5 \sin 0 = 0 \). So, the complex number is \( 5 + 0i \).

- Find the complex conjugate

The complex conjugate of \( a + bi \) is \( a - bi \). Therefore, the complex conjugate of \( 5 + 0i \) is \( 5 - 0i \), which simplifies to \( 5 \).

- Plot the number and its conjugate

Plot the complex number and its conjugate on the complex plane: Both the number and its conjugate correspond to the point (5, 0).

Key concepts

complex conjugate

A complex conjugate is a concept where we invert the sign of the imaginary part of a complex number. Suppose we have a complex number in the form of \( a + bi \). The complex conjugate would be \( a - bi \). This is useful because certain calculations, especially in equations and integrals, become simpler using the conjugate. In our exercise, we had the complex number 5 in polar form: \( 5(\text{cos 0} + i\text{sin 0}) \). After converting it to rectangular form (5 + 0i), the conjugate becomes 5, as the imaginary part is zero.
Understanding complex conjugates helps in visualizing reflections of points over the real axis in the complex plane.

polar form

The polar form of a complex number expresses the number as a magnitude and an angle. It is given by: \( r (\text{cos}(\theta) + i\text{sin}(\theta)) \). Here, \( r \) is the magnitude (or modulus) and \( \theta \) is the angle (or argument). In our exercise, we started with the polar form \( 5(\text{cos} 0 + i \text{sin} 0) \), where \( r = 5 \) and \( \theta = 0 \).
This form is especially useful in multiplying and dividing complex numbers, as well as converting them back to rectangular form.

rectangular form

The rectangular form of a complex number represents it as \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. This form is derived using trigonometric identities from polar coordinates. For the given number \( 5(\text{cos} 0 + i \text{sin} 0) \), converting to rectangular form involves:
\

• \( x = r \text{cos}(\theta) \)
• \( y = r \text{sin}(\theta) \)
\ From our exercise:
\( x = 5 \text{cos}(0) = 5 \)
\( y = 5 \text{sin}(0) = 0 \)
Thus, the number is written as \( 5 + 0i \). This form simplifies arithmetic operations such as addition and subtraction between complex numbers.

complex plane

The complex plane is a visual representation of complex numbers with two perpendicular axes. The horizontal axis represents the real part, while the vertical axis represents the imaginary part.
In our exercise, the complex number \( 5 + 0i \) and its conjugate plot as (5, 0). Each complex number corresponds to a unique point in this plane. By plotting, we can easily understand operations like finding conjugates, where the conjugate reflects over the real axis.
Using the complex plane helps to grasp the geometric implications of algebraic transformations on complex numbers. This is not only crucial for problem-solving but also for deeper insights into their properties and behaviors.