Problem 22 - Advanced Engineering Mathematics

Question

$$[(1+i)(1-i)]^{2}(1-i)=4-4 i$$

Step-by-Step Solution

Answer

The calculation confirms that \((4 - 4i) = (4 - 4i)\), so the equation is correct.

Step 1: Simplify the Expression Inside the Brackets

First, let's simplify the expression inside the brackets: \[(1+i)(1-i)\]This is a difference of squares, which simplifies to:\[(1)^2 - (i)^2 = 1 - i^2 = 1 - (-1) = 1 + 1 = 2\].The expression reduces to 2.

Step 2: Raise to the Power of 2

Next, take the simplified expression from Step 1 and raise it to the power of 2:\[2^2 = 4\].So the expression after this step is 4.

Step 3: Multiply by (1-i)

Now, take the result from Step 2 and multiply by \((1-i)\):\[4(1-i) = 4 - 4i\].

Step 4: Verify Equality with Given Result

Finally, compare the calculated result \(4 - 4i\) with the given result in the problem \(4 - 4i\). Since both expressions are identical, the original equation holds true.

Key Concepts

Difference of Squares

The expression \( (1+i)(1-i) \) is an example of the difference of squares. This algebraic concept states that the product of two conjugates, \( (a+b) \) and \( (a-b) \), is \( a^2 - b^2 \).
For complex numbers, this translates to simplifying the product by finding the square of the real and imaginary parts separately.
In our example, \( (1+i) \) and \( (1-i) \) are conjugates. Their product simplifies to:

• \( 1^2 = 1 \) for the real part.
• \( i^2 = -1 \) for the imaginary part, because \( i^2 \) is defined as \( -1 \) in complex numbers.

Thus, the difference becomes \( 1 - (-1) = 2 \).
This simplification is essential in making complex algebraic expressions far easier to handle.

Multiplication of Complex Numbers

Multiplying complex numbers involves handling both their real and imaginary components by using distributive properties. For instance, when multiplying the result \( 4 \) by \( (1-i) \), each part of \( (1-i) \) is multiplied by \( 4 \):

• Real part: \( 4 imes 1 = 4 \)
• Imaginary part: \( 4 imes (-i) = -4i \)

The multiplication results in \( 4 - 4i \).
This shows how the multiplication of complex numbers helps transition from a simple algebraic expression to one that reveals both real and imaginary components.

Simplification of Algebraic Expressions

Simplification is the process of reducing an expression to its simplest form. In this exercise, we started by using the difference of squares to resolve \( (1+i)(1-i) \) into \( 2 \).
We then squared this simple result, leading to \( 4 \). This is just basic exponent multiplication: \( 2^2 = 4 \).
Lastly, multiplying \( 4 \) by \( (1-i) \) simplifies the overall expression to its final form.
Each step is crucial for reducing a potentially complex task into understandable parts, ensuring clarity and manageability.

Verification of Mathematical Results

Verification involves confirming that a calculated result matches the expected outcome; it is the final "checkpoint" of problem-solving.
In this exercise, after simplifying and calculating, the expressed result was \( 4 - 4i \).
By comparing this outcome with the given or expected result \( 4 - 4i \), we confirm accuracy.
Verification is essential in mathematics to ensure solutions are correct, logical, and aligned with provided information. This step assures that all previous operations were conducted accurately.