Problem 19 - College Trigonometry

Question

Find a polar representation for the complex number \(z\) and then identify \(\operatorname{Re}(z)\), \(\operatorname{Im}(z),|z|, \arg (z)\) and \(\operatorname{Arg}(z)\). $$ z=4-2 i $$

Step-by-Step Solution

Answer

Real part: 4, Imaginary part: -2, Magnitude: \(2\sqrt{5}\), Argument: \(-0.4636\) radians, Polar form: \(2\sqrt{5}(\cos(-0.4636) + i\sin(-0.4636))\).

Step 1: Identify the Real and Imaginary Parts

For the complex number \(z = 4 - 2i\), the real part \(\operatorname{Re}(z)\) is 4 and the imaginary part \(\operatorname{Im}(z)\) is -2.

Step 2: Calculate the Magnitude

The magnitude (or modulus) of \(z\), denoted \(|z|\), is found using the formula \(|z| = \sqrt{x^2 + y^2}\), where \(x\) is the real part and \(y\) is the imaginary part of \(z\). Thus, \(|z| = \sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}\).

Step 3: Determine the Argument

The argument \(\arg(z)\) of the complex number is the angle formed with the positive x-axis. It can be calculated using \(\tan^{-1}(\frac{y}{x})\), where \(y = -2\) and \(x = 4\). So, \(\arg(z) = \tan^{-1}\left(-\frac{2}{4}\right) = \tan^{-1}\left(-\frac{1}{2}\right)\). This gives approximately \(-0.4636\) radians.

Step 4: Determine the Principal Argument

The principal argument \(\operatorname{Arg}(z)\) is the value of the argument that falls within the interval \((-, ]\). Since \(-0.4636\) is already within this range, \(\operatorname{Arg}(z) = -0.4636\).

Step 5: Express the Polar Form

The polar form of \(z\) is expressed as \(|z|(\cos \theta + i\sin \theta)\), where \(\theta\) is the argument. Therefore, \(z = 2\sqrt{5}(\cos(-0.4636) + i\sin(-0.4636))\).

Key Concepts

Polar Representation of Complex Numbers

The polar representation of complex numbers is a powerful way to express a complex number using magnitude and angle instead of real and imaginary parts. For the complex number given, \(z = 4 - 2i\), we can convert it to polar form. This requires calculating:

• The magnitude \(|z|\)
• The argument \(\arg(z)\)

Once these values are determined, the polar form can be expressed as \( |z| (\cos \theta + i\sin \theta) \), where \( \theta \) is the argument of \(z\). In our case, with a magnitude of \(2\sqrt{5}\) and an argument of approximately \(-0.4636\) radians, the polar form is \( 2\sqrt{5}(\cos(-0.4636) + i\sin(-0.4636)) \). This representation links complex numbers to polar coordinates, proving useful in multiplication or division, as well as in transformations.

Real and Imaginary Parts

Complex numbers are made of two parts: a real part and an imaginary part. It can be written in the form \(a + bi\), where \(a\) represents the real part and \(b\) the imaginary part. For the complex number \(z = 4 - 2i\),

• The real part \(\operatorname{Re}(z)\) is 4
• The imaginary part \(\operatorname{Im}(z)\) is -2

These components are essential, as they allow us to compute other properties like the magnitude and argument. Recognizing and separating these parts of a complex number is a fundamental skill in complex number arithmetic.

Magnitude of a Complex Number

The magnitude of a complex number, also known as its modulus, tells us about its "size" or "length" in the complex plane. The formula for the magnitude is \(|z| = \sqrt{x^2 + y^2}\), where \(x\) is the real part and \(y\) is the imaginary part. This formula is reminiscent of the Pythagorean theorem, which calculates the hypotenuse of a right triangle.
For \(z = 4 - 2i\), the magnitude calculates as follows:

• \(|z| = \sqrt{4^2 + (-2)^2} = \sqrt{16 + 4} = \sqrt{20} = 2\sqrt{5}\)

Understanding magnitude is crucial as it represents the distance from the origin to the point \((4, -2)\) in the complex plane.

Argument of a Complex Number

The argument of a complex number is the angle it forms with the positive real axis. This angle is crucial for the angular perspective in polar representation.
To find the argument, we use the formula \(\tan^{-1}(\frac{y}{x})\), with \(y\) being the imaginary and \(x\) the real part. Using \(z = 4 - 2i\), we compute:

• \(\arg(z) = \tan^{-1}\left(-\frac{2}{4}\right) = \tan^{-1}\left(-\frac{1}{2}\right)\)
• The result is approximately \(-0.4636\) radians

It is noteworthy that the angle is measured in radians and reflects a rotation from the positive x-axis, going counter-clockwise for positive values. Understanding argument is vital when considering rotations and orientations of complex numbers in the complex plane.