Problem 16 - Advanced Engineering Mathematics

Question

Write the given number in the form \(a+i b\). $$ \frac{10-5 i}{6+2 i} $$

Step-by-Step Solution

Answer

\( \frac{5}{4} - \frac{5i}{4} \).

Step 1: Identify the Task

The task is to simplify the given complex fraction \( \frac{10-5i}{6+2i} \) to the form \( a + bi \), where \( a \) and \( b \) are real numbers.

Step 2: Multiply by the Conjugate

To simplify a complex fraction, multiply the numerator and denominator by the conjugate of the denominator. Here, the conjugate of \( 6+2i \) is \( 6-2i \). The expression becomes: \[ \frac{(10-5i)(6-2i)}{(6+2i)(6-2i)} \]

Step 3: Simplify the Denominator

Use the formula for the product of a complex number and its conjugate: \[ (a+bi)(a-bi) = a^2 + b^2 \]. Thus, the denominator: \[ (6+2i)(6-2i) = 6^2 + (2)^2 = 36 + 4 = 40 \]

Step 4: Simplify the Numerator

Expand the numerator using the distributive property: \[ (10-5i)(6-2i) = 10(6) + 10(-2i) - 5i(6) + 5i(2i) \] Calculate each term: \[ = 60 - 20i - 30i - 10 = 60 - 50i - 10 \] Combine like terms: \[ = 50 - 50i \]

Step 5: Divide Both Parts by the Real Denominator

Divide the real and imaginary parts of the numerator by the real denominator: \[ \frac{50}{40} - \frac{50i}{40} = \frac{5}{4} - \frac{5i}{4} \] Thus, we have \( a = \frac{5}{4} \) and \( b = -\frac{5}{4} \).

Key Concepts

Conjugate of a Complex Number

A complex number is written in the form of a + bi, where a is the real part and b is the imaginary part. The conjugate of a complex number involves changing the sign of the imaginary component. For instance, if we have a complex number c = a + bi, its conjugate is c* = a - bi.
Understanding the concept of conjugates is crucial when working with complex numbers, particularly in simplifying complex fractions. Multiplying a complex number by its conjugate is a technique to eliminate the imaginary unit 'i' from the denominator. This is because the product of a complex number and its conjugate is always a real number. The formula for this is: (a + bi)(a - bi) = a^2 + b^2.
In the case of the fraction \( \frac{10 - 5i}{6 + 2i} \), the conjugate of the denominator \(6 + 2i \) is \( 6 - 2i \). By multiplying both the numerator and the denominator by this conjugate, we facilitate the simplification process.

Simplifying Complex Fractions

Simplifying complex fractions involves a straightforward process, often centered on removing imaginary units from the denominator by using the conjugate method. Here's how it works:

• Identify the complex number in the denominator you need to simplify, e.g., \( 6 + 2i \).
• Find its conjugate, which is \( 6 - 2i \).
• Multiply both the numerator \( (10 - 5i) \) and the denominator \( (6 + 2i) \) by this conjugate.

This yields the expression \( \frac{(10-5i)(6-2i)}{(6+2i)(6-2i)} \). The denominator simplifies to \( 40 \) using the formula for conjugates, while the numerator expands to \( 50 - 50i \).
The final step is to divide both the real and imaginary parts of the expanded numerator by the simplified denominator, yielding the simplified form \( \frac{5}{4} - \frac{5i}{4} \).

Real and Imaginary Parts of Complex Numbers

In mathematics, complex numbers are composed of two distinct parts: a real part and an imaginary part. These are expressed as \( a + bi \), where \( a \) is the real component, and \( bi \) is the imaginary component. Both parts are essential for identifying the position of the number on the complex plane.

• The real part (\( a \)) corresponds to the horizontal axis.
• The imaginary part (\( b \)) corresponds to the vertical axis.

When working to express a complex number such as \( \frac{10 - 5i}{6 + 2i} \) in its simplest form, both the real and imaginary parts must be easily identifiable. After simplifying, we arrive at \( \frac{5}{4} - \frac{5i}{4} \), where \( \frac{5}{4} \) is the real part and \( -\frac{5i}{4} \) is the imaginary part.
Understanding these components helps one visualize and work with complex numbers, particularly in fields like engineering and physics, where complex numbers often model real-world phenomena.