Problem 20 - Advanced Engineering Mathematics

Question

$$\frac{4+3 i}{3+4 i} \cdot \frac{3-4 i}{3-4 i}=\frac{24-7 i}{25}=\frac{24}{25}-\frac{7}{25} i$$

Step-by-Step Solution

Answer

The simplified result is \( \frac{24}{25} - \frac{7}{25}i \).

Step 1: Understand the Exercise

The exercise requires simplifying the complex fraction \( \frac{4+3i}{3+4i} \) by multiplying it with \( \frac{3-4i}{3-4i} \), which is the conjugate of the denominator.

Step 2: Multiply the Fractions

Multiply the given fraction by the conjugate. The product is \( \frac{(4+3i)(3-4i)}{(3+4i)(3-4i)} \). This uses the formula \((a+bi)(c-di) = (ac+bd) + (bc-ad)i\) for complex multiplication.

Step 3: Simplify the Numerator

Calculate the numerator: \( (4+3i)(3-4i) = 4 \cdot 3 + 4 \cdot (-4i) + 3i \cdot 3 + 3i \cdot (-4i) = 12 - 16i + 9i - 12i^2 \). Since \( i^2 = -1 \), this becomes \( 12 - 16i + 9i + 12 = 24 - 7i \).

Step 4: Simplify the Denominator

Calculate the denominator: \( (3+4i)(3-4i) = 3^2 - (4i)^2 = 9 - (-16) = 9 + 16 = 25 \). The denominator is a real number because it is the difference of squares.

Step 5: Combine and Simplify

Combine the results from Step 3 and Step 4. The fraction is \( \frac{24-7i}{25} \). Divide each component separately: \( \frac{24}{25} - \frac{7}{25}i \).

Step 6: Result Verification

Verify the result by comparing \( \frac{24-7i}{25} \) and \( \frac{24}{25} - \frac{7}{25}i \). They are equivalent, confirming the solution.

Key Concepts

Complex Conjugate

A complex conjugate is a pair of complex numbers that have the same real part but an opposite imaginary part. If a complex number is written as \( a + bi \), its conjugate would be \( a - bi \). This concept is essential when simplifying complex fractions because it helps in making the denominator a real number.

• Conjugates look similar but have a different sign for the imaginary component.
• Multiplying a complex number by its conjugate eliminates the imaginary part in the denominator.

The reason this works is because the product of a complex number and its conjugate is always a real number. This product is given by the formula \((a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2\). Notice how the imaginary parts cancel each other out.
In the problem, multiplying \(3+4i\) by its conjugate \(3-4i\) changed the denominator to \(25\), a real number.

Complex Multiplication

Complex multiplication involves two complex numbers and follows similar basic principles to the multiplication of algebraic expressions. For two complex numbers \(x = a + bi\) and \(y = c + di\), their product is described using the distributive property:

• Multiply the real parts: \(ac\).
• Multiply the imaginary parts: \(bd\).
• Combine the product of the real part of one and the imaginary part of the other to form the imaginary part: \((bc + ad)i\).

Thus, the multiplication results in \((a + bi)(c + di) = (ac - bd) + (ad + bc)i\).
In the exercise, multiplying \(4+3i\) by the conjugate \(3-4i\) yields \(24 - 7i\), using the formula described. This carefully combines both real and imaginary components uniquely to form another complex number.

Simplification of Complex Fractions

Simplifying a complex fraction requires manipulating both the numerator and the denominator to achieve a standard complex format. When simplifying, typically the complex number in the denominator is removed, turning it into a real number. This is achieved by multiplying by the conjugate:

• The given complex fraction is multiplied by a fraction equivalent to 1, such as \(\frac{3-4i}{3-4i}\).
• The multiplication is computed separately for both the numerator and the denominator.
• The denominator becomes a real number, simplifying the entire fraction.

Finally, you separate the real and imaginary parts, expressing them as two distinct fractions/as two separate terms.
In the original exercise, the simplified form of the fraction \(\frac{4+3i}{3+4i}\) becomes \(\frac{24-7i}{25}\), further separated into \(\frac{24}{25} - \frac{7}{25}i\). This transformation simplifies calculations in subsequent mathematical operations.