Problem 41 - Algebra and Trigonometry

Question

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ -20 $$

Step-by-Step Solution

Answer

The polar form of \(-20\) is \(20\text{cis} \pi\).

Step 1: Identify the Complex Number Components

The given complex number is \(-20\). This number can be thought of in the complex plane as \(-20 + 0i\) where the real part is \(-20\) and the imaginary part is 0.

Step 2: Calculate the Magnitude

The magnitude (or modulus) \(r\) of a complex number \(a + bi\) is given by \(r = \sqrt{a^2 + b^2}\). Here, \(a = -20\) and \(b = 0\), so the calculation is:\[r = \sqrt{(-20)^2 + 0^2} = \sqrt{400} = 20\]

Step 3: Determine the Argument

The argument \(\theta\) of a complex number is given by \(\tan^{-1}(\frac{b}{a})\). For \(-20 + 0i\), this does not directly apply because the imaginary part is 0. Instead, since the number lies on the negative real axis, the argument is:\[\theta = \pi\]This is because the angle corresponding to the negative real axis is \(\pi\) radians.

Step 4: Write the Polar Form

The polar form of a complex number is \(r(\cos\theta + i\sin\theta)\). Substitute the values of \(r = 20\) and \(\theta = \pi\):\[20 (\cos \pi + i \sin \pi)\]This simplifies to:\[20 (-1 + 0i) = 20 (-1)\]Thus, the polar form of the complex number is 20\(\text{cis} \pi\).

Key Concepts

Polar Form

The polar form of a complex number represents it in a different perspective which is particularly useful in trigonometry and complex analysis. Unlike the standard rectangular form, which expresses a complex number as \(a + bi\), polar form uses the magnitude \(r\) and the angle \(\theta\) to define the same number.

• Magnitude (or modulus) \(r\) describes how far the number is from the origin of the complex plane.
• Angle \(\theta\) indicates the direction of the number from the positive real axis.

In polar form, a complex number \(z = a + bi\) is represented as:\[ r(\cos \theta + i \sin \theta) \]Where \(r = \sqrt{a^2 + b^2}\) and \(\theta\) depends on the quadrant in which the complex number lies. For easy notation, it’s often written as \(r \text{cis} \theta\), where "cis" stands for "cosine plus i sine."
This form is especially valuable when multiplying or dividing complex numbers, as it simplifies the process by allowing operations on magnitudes and angles separately.

Argument of a Complex Number

The argument of a complex number is the angle \(\theta\) formed by the real part of the number and the positive real axis on the complex plane. It's expressed in radians and ranges typically from 0 to \(2\pi\).Depending on the location of the complex number in the complex plane, \(\theta\) can be calculated using various strategies:

• If both parts are positive, the angle can be directly calculated using \( \tan^{-1}(\frac{b}{a}) \).
• When the real part is negative, remember to adjust \(\theta\) by adding \(\pi\) radians, as the point lies on the opposite side of the plane.
• If the imaginary part \(b = 0\), like in our example with \(-20\), and the number lies on the negative real axis, the argument is exactly \(\pi\) radians.

Understanding how the argument changes with the position of the complex number helps grasp how angles control the orientation of numbers in the complex plane.

Complex Plane

The complex plane is a two-dimensional plane used to represent complex numbers visually. It consists of two perpendicular axes:

• The horizontal axis represents the real part of complex numbers.
• The vertical axis represents the imaginary part.

In this plane, each complex number \(z = a + bi\) corresponds to a unique point \((a, b)\). Plotting these points provides a visual representation useful for addition, subtraction, multiplication, and understanding the geometric interpretation of complex numbers.

For instance, a complex number like \(-20 + 0i\) is positioned directly on the real axis. Its position corresponds only to a real magnitude as the imaginary component tips the scale, quite literally, upward only when \(beq 0\).

With the complex plane, the entire realm of complex number operations is vividly illustrated, showing how shifts along any axis result from real or imaginary changes. This makes it a necessary tool for both visualizing concepts like rational roots and simplifying deeper theoretical ideas in mathematics.