Problem 81 - Precalculus

Question

Calculate \((1-i)^{n} \cdot(1+i)^{m},\) where \(n\) and \(m\) are positive integers.

Step-by-Step Solution

Answer

\((\sqrt{2})^{n+m} (\cos((m-n)\frac{\pi}{4}) + i\sin((m-n)\frac{\pi}{4}))\).

Step 1: Rewrite Using Euler's Formula

Recall that the complex numbers \((1-i)\) and \((1+i)\) can be written in polar form. The modulus of both numbers is \(\sqrt{1^2 + (-1)^2} = \sqrt{2}\). The argument (angle) of \((1-i)\) is \(-\frac{\pi}{4}\) and for \((1+i)\) is \(\frac{\pi}{4}\). So we can express:\[(1-i) = \sqrt{2}e^{-i\frac{\pi}{4}}\] \[(1+i) = \sqrt{2}e^{i\frac{\pi}{4}}\]

Step 2: Apply Properties of Exponents

Using the properties of exponents, \((x^a)^b = x^{ab}\), we can rewrite the expression:\[(1-i)^n = \left(\sqrt{2}e^{-i\frac{\pi}{4}}\right)^n = (\sqrt{2})^n e^{-in\frac{\pi}{4}}\]\[(1+i)^m = \left(\sqrt{2}e^{i\frac{\pi}{4}}\right)^m = (\sqrt{2})^m e^{im\frac{\pi}{4}}\]

Step 3: Multiply the Expressions

Now, multiply the results from Step 2:\[(1-i)^n \cdot (1+i)^m = (\sqrt{2})^n e^{-in\frac{\pi}{4}} \cdot (\sqrt{2})^m e^{im\frac{\pi}{4}}\]Since exponents with the same bases are added when multiplied, this results in \[(\sqrt{2})^{n+m} \cdot e^{i(-n\frac{\pi}{4} + m\frac{\pi}{4})}\].

Step 4: Simplify the Angle

Simplify the angle in the exponent:\[-n\frac{\pi}{4} + m\frac{\pi}{4} = (m-n)\frac{\pi}{4}\].So the expression becomes:\[(\sqrt{2})^{n+m} \cdot e^{i(m-n)\frac{\pi}{4}}\].

Step 5: Final Expression

The final expression in polar form is:\[(\sqrt{2})^{n+m} (\cos((m-n)\frac{\pi}{4}) + i\sin((m-n)\frac{\pi}{4})).\]This indicates the modulus is \((\sqrt{2})^{n+m}\) and the argument is \((m-n)\frac{\pi}{4}\).

Key Concepts

Euler's Formula

Euler's Formula is a key concept in complex analysis. It provides a deep connection between exponential functions and trigonometric functions, specifically for complex numbers. The formula is expressed as \[ e^{i\theta} = \cos(\theta) + i\sin(\theta)\] where \( \theta \) is the angle in radians. This formula is incredibly useful for transforming complex numbers into a more manageable form.
When applying Euler's Formula, complex numbers can be written as \( re^{i\theta} \), where \( r \) is the modulus (or absolute value) and \( \theta \) is the argument (or angle) of the complex number. This makes it easier to handle operations such as multiplication, division, and powers of complex numbers in exponential form.
In the original exercise, Euler's Formula is used to rewrite the complex numbers \( (1-i) \) and \( (1+i) \) in exponential form with their respective arguments and moduli. This sets the stage for simplifying complex expressions through exponential attributes and highlights the elegance of using polar coordinates.

Polar Form

Polar Form is another powerful representation of complex numbers. Unlike the standard form \( a + bi \), the polar form expresses a complex number as \( r(\cos(\theta) + i\sin(\theta)) \). Using Euler's Formula, this can be simplified to \( re^{i\theta} \), where \( r \) is the distance from the origin (modulus) and \( \theta \) the angle (argument) from the positive x-axis.
The Polar Form simplifies multiplication and division of complex numbers. In multiplication, you simply multiply the moduli and add the arguments:

• If \( z_1 = r_1e^{i\theta_1} \) and \( z_2 = r_2e^{i\theta_2} \), then their product is \( z_1z_2 = (r_1r_2)e^{i(\theta_1+\theta_2)} \).

This reduces the complexity of multiplying complex numbers considerably by transforming it into simple arithmetic involving their magnitudes and angles.
In the exercise, the numbers \((1-i)\) and \((1+i)\) are converted into their polar forms, simplifying calculations involving powers and products of these complex numbers.

Properties of Exponents

The Properties of Exponents are vital tools in simplifying expressions, especially those involving complex numbers expressed using Euler's Formula. The key property that is particularly useful is:

• \((x^a)^b = x^{ab}\), which allows us to simplify nested powers.

In the context of complex numbers, using properties such as \((e^{i\theta})^n = e^{in\theta}\) enables us to raise complex numbers to integer powers easily.
For the original problem, these properties allowed the expression \((1-i)^n\) and \((1+i)^m\) to be rewritten using their polar forms:

• \((1-i)^n = \left(\sqrt{2}e^{-i\frac{\pi}{4}}\right)^n = (\sqrt{2})^n e^{-in\frac{\pi}{4}}\)
• \((1+i)^m = \left(\sqrt{2}e^{i\frac{\pi}{4}}\right)^m = (\sqrt{2})^m e^{im\frac{\pi}{4}}\)

These transformations make complex calculations more straightforward by leveraging the linearity of exponents, particularly when combining exponential terms.

Argument of a Complex Number

The Argument of a Complex Number is the angle \( \theta \) in the polar coordinate representation of the complex number. It indicates the direction of the vector represented by the complex number in the complex plane. The argument is measured in radians, ranging from \(-\pi\) to \(\pi\).
To find the argument of a complex number \( a + bi \), use the arctangent function: \[ \theta = \tan^{-1}\left(\frac{b}{a}\right)\]This gives the angle relative to the positive x-axis in the complex plane, taking into account the signs of \( a \) and \( b \) to place \( \theta \) in the correct quadrant.
In the exercise, finding the arguments of \( 1-i \) and \( 1+i \) yields angles of \(-\frac{\pi}{4}\) and \(\frac{\pi}{4}\) respectively. These angles help in transforming the original expression into a simple polar form for easier computation of higher powers and products of complex numbers.