Problem 20 - Advanced Engineering Mathematics

Question

Write the given number in the form \(a+i b\). $$ \frac{(4+5 i)+2 i^{3}}{(2+i)^{2}} $$

Step-by-Step Solution

Answer

The expression simplifies to \(-1 - \frac{7}{25}i\).

Step 1: Apply the power of i

Recall that \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), and \( i^4 = 1 \). Applying this to \(2i^3\), we have \(i^3 = -i\), so \(2i^3 = 2(-i) = -2i\). Substitute this into the equation to simplify the expression: \((4+5i) - 2i\).

Step 2: Simplify the numerator

Combine the imaginary parts in the numerator: \(4 + 5i - 2i = 4 + 3i\). The expression becomes \( \frac{4 + 3i}{(2+i)^2} \).

Step 3: Expand the squared denominator

Compute \((2+i)^2 = (2+i)(2+i) = 4 + 4i + i^2\). Since \(i^2 = -1\), we get \(4i - 1 = 3 + 4i\). So, the denominator is \(3 + 4i\).

Step 4: Multiply by the conjugate

To simplify the complex fraction, multiply the numerator and denominator by the conjugate of the denominator, \(3 - 4i\).

Step 5: Calculate the denominator multiplied by its conjugate

The multiplication of \((3+4i)(3-4i)\) gives \(9 + 16 = 25\). Therefore, the denominator becomes \(25\).

Step 6: Calculate the numerator times the conjugate of the denominator

Multiply \((4 + 3i)\) by \((3 - 4i)\):\\((4 + 3i)(3 - 4i) = 12 - 16i + 9i + 12 = 0 + 25i\)\which simplifies to \(-25 - 7i\).

Step 7: Complete the fraction

Now divide \(-25 - 7i\) by the denominator \(25\):\\( \frac{-25}{25} + \frac{-7i}{25} = -1 - \frac{7}{25} i \).

Step 8: Final form

Therefore, in the form \(a + ib\), the simplified expression is \(-1 - \frac{7}{25} i\).

Key Concepts

Imaginary Unit

Complex numbers introduce an intriguing concept known as the imaginary unit. The imaginary unit is denoted by \(i\) and is defined as the square root of \(-1\), meaning \(i^2 = -1\). This fundamental property leads to various powers of \(i\), which are frequently used in calculations involving complex numbers. Here are the key powers:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

Notice that every four powers, the cycle of values repeats. Understanding and applying these cycles is crucial when you're simplifying expressions that involve powers of \(i\). For example, in the exercise you have \(2i^{3}\). Since \(i^3 = -i\), it results in \(-2i\). This substitution reduces the expression into a more manageable form for further operations.

Complex Conjugate

In complex numbers, the concept of a complex conjugate is pivotal for simplifying expressions, especially rational expressions with complex numbers. For a given complex number \(a + bi\), its complex conjugate is \(a - bi\). The product of a complex number with its conjugate results in a real number:

• \((a + bi)(a - bi) = a^2 + b^2\)

This property is particularly useful in eliminating the imaginary part from the denominator of a fraction. For instance, if the denominator is \(3 + 4i\), its conjugate is \(3 - 4i\).
When you multiply \(3 + 4i\) by \(3 - 4i\), you get \(9 + 16 = 25\), which is a real number. Thus, this tactic ensures the denominator of a complex fraction is a pure number, simplifying the division process.

Complex Fraction Simplification

Simplifying complex fractions involves transforming them into a standard form \(a + ib\). Here, it’s essential to manage both the numerator and denominator effectively. The process often starts by evaluating or simplifying the numerator and then focusing on the denominator, especially its complex terms.

• Firstly, simplify the numerator as much as possible.
• Then, focus on the denominator by multiplying both the numerator and denominator by the complex conjugate of the denominator. This step turns a potentially tricky division by a complex number into a straightforward division by a real number.
• Subsequently, carry out the multiplication across the numerator and reduce the terms.
• Finally, divide each term in the simplified numerator by the real number denominator.

In our exercise, after breaking down the fraction to \(\frac{4 + 3i}{3 + 4i}\), the next step was to multiply by the conjugate \(3 - 4i\).
This simplification turns the problem into computing simpler multiplicative operations. Eventually, arriving at the form \(-1 - \frac{7}{25}i\), showcasing how each stage steadily transforms the expression towards its simplest form of \(a + ib\).