Problem 58 - Algebra and Trigonometry

Question

\(57-64=\) Write \(z_{1}\) and \(z_{2}\) in polar form, and then find the product \(z_{1} z_{2}\) and the quotients \(z_{1} / z_{2}\) and 1\(/ z_{1}\) . $$ z_{1}=\sqrt{2}-\sqrt{2} i, \quad z_{2}=1-i $$

Step-by-Step Solution

Answer

Product: \(2\sqrt{2} e^{-i\pi/2}\); Quotient: \(\sqrt{2}\); Reciprocal: \(\frac{1}{2} e^{i\pi/4}\).

Step 1: Convert Complex Numbers to Polar Form

To convert a complex number into polar form, we need to find its magnitude and argument. For \(z_1 = \sqrt{2} - \sqrt{2}i\):The magnitude \(r_1 = |z_1| = \sqrt{(\sqrt{2})^2 + (-\sqrt{2})^2} = \sqrt{2+2} = \sqrt{4} = 2\).The argument is \(\theta_1 = \tan^{-1}\left(\frac{-\sqrt{2}}{\sqrt{2}}\right) = \tan^{-1}(-1) = -\frac{\pi}{4}\).Thus, in polar form, \(z_1 = 2\left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right)\).For \(z_2 = 1 - i\):The magnitude \(r_2 = |z_2| = \sqrt{1^2 + (-1)^2} = \sqrt{2}\).The argument \(\theta_2 = \tan^{-1}\left(\frac{-1}{1}\right) = \tan^{-1}(-1) = -\frac{\pi}{4}\).Thus, in polar form, \(z_2 = \sqrt{2}\left(\cos\left(-\frac{\pi}{4}\right) + i\sin\left(-\frac{\pi}{4}\right)\right)\).

Step 2: Calculate the Product of z1 and z2

To find the product of two complex numbers in polar form, multiply their magnitudes and add their arguments.Magnitude of product: \(|z_1z_2| = 2 \times \sqrt{2} = 2\sqrt{2}\).Argument of product: \(\theta_1 + \theta_2 = -\frac{\pi}{4} + \left(-\frac{\pi}{4}\right) = -\frac{\pi}{2}\).Thus, the product \(z_1z_2 = 2\sqrt{2}\left(\cos\left(-\frac{\pi}{2}\right) + i\sin\left(-\frac{\pi}{2}\right)\right)\).

Step 3: Calculate the Quotient z1/z2

To find the quotient of two complex numbers in polar form, divide their magnitudes and subtract their arguments.Magnitude of quotient: \(|z_1 / z_2| = \frac{2}{\sqrt{2}} = \sqrt{2}\).Argument of quotient: \(\theta_1 - \theta_2 = -\frac{\pi}{4} - \left(-\frac{\pi}{4}\right) = 0\).Thus, the quotient \(z_1 / z_2 = \sqrt{2}\left(\cos(0) + i\sin(0)\right) = \sqrt{2}\).

Step 4: Calculate the Reciprocal of z1

To find the reciprocal of a complex number in polar form, take the reciprocal of its magnitude and negate its argument.Magnitude of reciprocal: \(|1/z_1| = \frac{1}{2}\).Argument of reciprocal: \(-\theta_1 = \frac{\pi}{4}\).Thus, the reciprocal \(1/z_1 = \frac{1}{2}\left(\cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right)\right)\).

Key Concepts

Complex Numbers

Complex numbers are a sophisticated yet fundamental concept in mathematics. A complex number is typically expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) represents the imaginary unit. The imaginary unit \(i\) is defined by the property \(i^2 = -1\). Complex numbers extend the idea of one-dimensional numbers (real numbers) to a two-dimensional plane using the complex plane.

In this plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis. This representation helps in visualizing complex numbers as points or vectors in two-dimensional space. For instance, the complex number \(z = x + yi\) can be seen as a vector from the origin to the point \((x, y)\) in the complex plane. This geometric interpretation is especially useful when dealing with complex multiplication and division.

Magnitude

The magnitude of a complex number is a measure of its length or size in the complex plane. It's somewhat analogous to the absolute value for real numbers but extended to two dimensions. The magnitude of a complex number \(z = a + bi\) is found using the formula: \(|z| = \sqrt{a^2 + b^2}\).

This formula stems from the Pythagorean theorem, as the magnitude is essentially the distance from the origin \((0, 0)\) to the point \((a, b)\). In the provided exercise, the magnitudes of the complex numbers \(z_1 = \sqrt{2} - \sqrt{2}i\) and \(z_2 = 1 - i\) are computed to be \(2\) and \(\sqrt{2}\) respectively, showing how these complex numbers extend from the origin in the complex plane.

Argument

The argument of a complex number is the angle formed between the positive real axis and the line segment connecting the origin to the point \((a, b)\) in the complex plane. It defines the direction of the complex number. The argument is usually measured in radians.

For a given complex number \(z = a + bi\), the argument \(\theta\) is calculated using: \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\). It’s important to note that the range of the inverse tangent function often needs adjustment depending on the quadrant where the complex number lies. In the solution provided, both complex numbers \(z_1\) and \(z_2\) have an argument of \(-\frac{\pi}{4}\), indicating their direction in the complex plane.

• This angle helps in converting complex numbers from rectangular form \((a + bi)\) to polar form \((r(\cos \theta + i\sin \theta))\).
• Polar form is beneficial for easily visualizing and performing operations such as multiplication and division.

Complex Multiplication

Complex multiplication can be understood more intuitively using the polar form of a complex number. When multiplying two complex numbers in polar form, their magnitudes are multiplied together, and their arguments are added.

Given two complex numbers \(z_1 = r_1(\cos \theta_1 + i\sin \theta_1)\) and \(z_2 = r_2(\cos \theta_2 + i\sin \theta_2)\), their product \(z_1z_2\) is given by: \(|z_1z_2| = r_1 \times r_2\) and the angle \(\theta_1 + \theta_2\). This simple addition of angles results from the properties of exponential growth represented in polar form.

• This method of multiplication simplifies calculations immensely compared to the traditional rectangular form.
• In practice, this yields a new complex number with a magnitude of \(2\sqrt{2}\) and an argument of \(-\frac{\pi}{2}\), as shown in the given solution.

Complex Division

Complex division, analogous to multiplication in polar form, requires dividing the magnitudes and subtracting the arguments of the two complex numbers.

For two complex numbers \(z_1 = r_1(\cos \theta_1 + i\sin \theta_1)\) and \(z_2 = r_2(\cos \theta_2 + i\sin \theta_2)\), the quotient \(\frac{z_1}{z_2}\) has a magnitude \(\frac{r_1}{r_2}\) and an argument \(\theta_1 - \theta_2\). This contrasts the multiplication process where angles are added.

• This approach helps avoiding complex algebraic manipulations involving imaginary units.
• The simplicity of operation in polar form makes division straight forward, especially for expressing results accurately.
• The step-by-step solution illustrates the process, finding the result is a complex number with magnitude \(\sqrt{2}\) and argument \(0\).