Problem 32 - Precalculus Mathematics for Calculus

Question

Write the complex number in polar form with argument \(\theta\) between 0 and \(2 \pi\). $$1-i$$

Step-by-Step Solution

Answer

The polar form of the complex number is \( \sqrt{2}(\cos(\frac{7\pi}{4}) + i\sin(\frac{7\pi}{4})) \).

Step 1: Identify Real and Imaginary Components

The complex number can be written in standard form as \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. For the complex number \( 1-i \), \( a = 1 \) and \( b = -1 \).

Step 2: Calculate the Magnitude

To find the magnitude \( r \) of the complex number, use the formula: \[ r = \sqrt{a^2 + b^2} \]Substituting \( a = 1 \) and \( b = -1 \), we get:\[ r = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

Step 3: Calculate the Argument

The argument \( \theta \) of a complex number \( a+bi \) can be found using:\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \]Substitute \( b = -1 \) and \( a = 1 \):\[ \theta = \tan^{-1}\left(\frac{-1}{1}\right) = \tan^{-1}(-1) \]The tangent of \(-1\) corresponds to an angle of \(-\frac{\pi}{4}\), but since we're in the fourth quadrant, we must adjust \( \theta \) to be between 0 and \( 2\pi \):\[ \theta = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4} \]

Step 4: Write the Polar Form

Now that we have both the magnitude \( r = \sqrt{2} \) and the argument \( \theta = \frac{7\pi}{4} \), we can write the polar form of the complex number:\[ z = r(\cos(\theta) + i\sin(\theta)) \]Thus, the polar form of \( 1-i \) is:\[ \sqrt{2}(\cos(\frac{7\pi}{4}) + i\sin(\frac{7\pi}{4})) \]

Key Concepts

Polar Form

Polar form is an alternative way of representing complex numbers, apart from the standard rectangular form, which is typically written as \(a + bi\), where \(a\) and \(b\) are real numbers. In polar form, a complex number is expressed in terms of its magnitude and angle (often referred to as the argument). This form is particularly useful in multiplying and dividing complex numbers as it simplifies these operations.

In polar form, a complex number \(z\) is represented as \(z = r(\cos(\theta) + i\sin(\theta))\), where \(r\) is the magnitude of the complex number and \(\theta\) is the angle it makes with the positive real axis—known as the argument. This form is also widely represented as \(z = r e^{i\theta}\), using Euler's formula, which relates complex exponentials to trigonometric functions.

When a complex number like \(1-i\) is converted to polar form, it becomes \(\sqrt{2}(\cos(\frac{7\pi}{4}) + i\sin(\frac{7\pi}{4}))\). The magnitude \(r\) is \(\sqrt{2}\), and the angle or argument \(\theta\) is \(\frac{7\pi}{4}\), lying within the required interval of \(0\) to \(2\pi\).

• This representation is especially handy when dealing with powers and roots of complex numbers.
• The polar form directly correlates to vectors in the Argand plane, making it a fascinating area of study in both mathematics and physics.
• Transforming into polar coordinates emphasizes the geometric interpretation of complex numbers.

Magnitude of Complex Number

The magnitude of a complex number is a measure of its distance from the origin in the complex plane. Imagine plotting the complex number like \(1-i\) on a graph, where the x-axis represents the real part and the y-axis represents the imaginary part. The magnitude is simply the length of the line segment connecting the origin to this point.

To find the magnitude \(r\) of a complex number \(a + bi\), use the formula:
\[ r = \sqrt{a^2 + b^2} \]
For the complex number \(1-i\), the values \(a = 1\) and \(b = -1\) yield:
\[ r = \sqrt{1^2 + (-1)^2} = \sqrt{2} \]. This simple, yet powerful formula effectively gives you the hypotenuse of a right triangle with sides \(a\) and \(b\).

• The magnitude \(\sqrt{2}\) represents the diagonal from the origin to the point (1,-1) in the Argand plane.
• Knowing the magnitude is crucial for converting complex numbers into their polar form.
• This distance measure is a scalar, indicating size but not direction.

Argument of Complex Number

The argument of a complex number is one of its most interesting aspects, fundamentally representing the angle made with the positive direction of the real axis in the complex plane. This angle, denoted as \(\theta\), helps give direction to the complex number when expressed in polar coordinates.

For a complex number \(a + bi\), the argument is typically calculated using the arctangent of the ratio between the imaginary and real parts. The formula is:
\[ \theta = \tan^{-1} \left( \frac{b}{a} \right) \]
However, care must be taken to adjust \(\theta\) according to the quadrant in which the complex number lies, as the arctangent function alone does not distinguish angles beyond \([-\frac{\pi}{2}, \frac{\pi}{2}]\).

In our example \(1-i\) with \(b = -1\) and \(a = 1\), the angle initially calculates as \(-\frac{\pi}{4}\). Given that this number is in the fourth quadrant, where angles range from \(\frac{3\pi}{2}\) to \(2\pi\), the correct argument is adjusted to
\[ \theta = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4} \].

• The argument provides the necessary directional component to fully describe a complex number’s orientation and is essential in polar form representation.
• Understanding the nature of \(\theta\) helps navigate the geometric visualization of complex arithmetic.
• In practical applications, the argument also plays a role in determining rotational transformations and phase shifts.