Problem 15 - Advanced Engineering Mathematics

Question

$$\frac{2-4 i}{3+5 i} \cdot \frac{3-5 i}{3-5 i}=\frac{-14-22 i}{34}=-\frac{7}{17}-\frac{11}{17} i$$

Step-by-Step Solution

Answer

The expression simplifies to \(-\frac{7}{17} - \frac{11}{17} i\).

Step 1: Identify the initial expression

The initial expression given is \( \frac{2-4i}{3+5i} \). Our goal is to simplify this expression by removing the imaginary unit from the denominator.

Step 2: Multiply by the conjugate

To eliminate the imaginary part in the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(3+5i\) is \(3-5i\). So, we consider \( \frac{2-4i}{3+5i} \times \frac{3-5i}{3-5i}\).

Step 3: Distribute the multiplication in the numerator

Expand \((2-4i)(3-5i)\) using the distributive property:\[(2 \cdot 3) + (2 \cdot (-5i)) + (-4i \cdot 3) + (-4i \cdot (-5i)) = 6 - 10i - 12i + 20i^2\]. Since \(i^2 = -1\), replace \(20i^2\) with \(-20\), leading to: \[6 - 10i - 12i - 20 = -14 - 22i\].

Step 4: Distribute the multiplication in the denominator

Expand \((3+5i)(3-5i)\) using the formula for the difference of squares:\[3^2 - (5i)^2 = 9 - 25i^2\]. Since \(i^2 = -1\), replace \(-25i^2\) with \(25\), leading to: \[9 + 25 = 34\].

Step 5: Write down the simplified form

After simplifying both the numerator and the denominator, the expression becomes \(\frac{-14 - 22i}{34}\).

Step 6: Simplify the fraction

Divide both terms in the numerator by the denominator to give:\(-\frac{14}{34} - \frac{22}{34}i = -\frac{7}{17} - \frac{11}{17}i\).

Key Concepts

Imaginary Unit

The imaginary unit, represented as \( i \), is an essential component in complex numbers. It is defined by the property \( i^2 = -1 \). This unique property allows \( i \) to represent the square root of \(-1\), a number that does not exist within the realm of real numbers. Complex numbers are expressed in the form \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part.

In the context of the exercise, the imaginary unit \( i \) appears in both the numerator \((2-4i)\) and denominator \((3+5i)\) of the initial expression. Simplifying these expressions often requires using the property of \( i^2 \) as demonstrated in the solution steps, where \( i^2 \) is replaced with \(-1\). This substitution helps in obtaining real numbers from complex expressions, crucial for further simplification, like in the provided multiplication steps.

Complex Conjugate

A complex conjugate is used to transform a complex expression into a real number. The complex conjugate of a number, say \( a + bi \), is \( a - bi \). By multiplying a complex number by its conjugate, the result is always a real number, as the imaginary parts cancel out.

For instance, in the given problem, the conjugate of the denominator \( 3 + 5i \) is \( 3 - 5i \). By multiplying both the numerator and the denominator by this conjugate, you eliminate the imaginary part in the denominator. This is shown when transforming the denominator using the formula for the difference of squares:

• Calculation: \((3+5i)(3-5i) = 3^2 - (5i)^2\)
• Substitution: \(9 + 25 = 34\)

This process results in a real number \(34\) as the new denominator, simplifying further calculations.

Simplifying Fractions

To simplify fractions that include complex numbers, you must ensure the expression is in its simplest form. This involves both reducing the numeric fractions and simplifying the complex components.

After determining the numerator \(-14 - 22i\) and the denominator \(34\), the fraction can be simplified by dividing each component of the numerator by the denominator:

• Real part: \(-\frac{14}{34} = -\frac{7}{17}\)
• Imaginary part: \(-\frac{22}{34} = -\frac{11}{17}\)

This results in the expression \(-\frac{7}{17} - \frac{11}{17}i\).

Simplifying in such steps ensures each part of the complex fraction is properly reduced, leading to the most concise and comprehensible form for presenting a solution involving complex numbers.