Problem 62 - Algebra A Combined Function

Question

Divide. Write your answers in the form \(a+b i\) $$ \frac{6+2 i}{4-3 i} $$

Step-by-Step Solution

Answer

\( \frac{18}{25} + \frac{26}{25}i \)

Step 1: Identify the Complex Conjugate

To divide complex numbers of the form \( a + bi \) by \( c + di \), we need to multiply the numerator and the denominator by the conjugate of the denominator. For \( 4 - 3i \), the conjugate is \( 4 + 3i \).

Step 2: Set Up the Multiplication

Multiply the numerator and the denominator by the conjugate of the denominator:\[ \frac{6 + 2i}{4 - 3i} \times \frac{4 + 3i}{4 + 3i} \]

Step 3: Simplify the Denominator

Calculate the product of the denominator and its conjugate:\[ (4 - 3i)(4 + 3i) = 4^2 - (3i)^2 = 16 - (-9) = 16 + 9 = 25 \]

Step 4: Simplify the Numerator

Calculate the product of the numerator with the conjugate of the denominator:\[ (6 + 2i)(4 + 3i) = 6 \cdot 4 + 6 \cdot 3i + 2i \cdot 4 + 2i \cdot 3i \]\[ = 24 + 18i + 8i + 6i^2 \]Since \( i^2 = -1 \), substitute:\[ = 24 + 26i - 6 = 18 + 26i \]

Step 5: Divide Real and Imaginary Parts

Now divide each component of the complex number by 25 (the simplified denominator):\[ \frac{18 + 26i}{25} = \frac{18}{25} + \frac{26}{25}i \]

Key Concepts

Complex Conjugate

The complex conjugate is an important tool used to simplify division involving complex numbers. When you have a complex number in the form of \( a + bi \) (where \( i \) is the imaginary unit and \( i^2 = -1 \)), its complex conjugate is \( a - bi \). Simply put, you change the sign of the imaginary part to find the conjugate.

• For example, the complex conjugate of \( 4 - 3i \) is \( 4 + 3i \).
• Complex conjugates are used to "rationalize" the denominator, eliminating any imaginary numbers.
• This process helps in obtaining a result only in terms of real numbers.

By multiplying a complex number by its conjugate, you exploit a property where their product results in a real number: \((a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2\). This is especially useful in dividing complex numbers, as illustrated in the given exercise.

Imaginary Numbers

Imaginary numbers extend our understanding of real numbers by introducing the concept of \( i \), an imaginary unit where \( i^2 = -1 \). These numbers provide a way to deal with equations that have no solutions within the real numbers, such as \( x^2 + 1 = 0 \).

• The real part of a complex number is just that, a real number like 5 or -3.
• The imaginary part is a coefficient multiplied by \( i \), such as \( 2i \) or \( -3i \).

Imaginary numbers, when combined with real numbers, form complex numbers (\( a + bi \)). They are essential in fields that require mathematical precision, such as engineering and physics, especially when dealing with oscillations and waveforms. By understanding imaginary numbers, you unlock the ability to solve more complex arithmetic problems.

Complex Arithmetic

Complex arithmetic involves operations such as addition, subtraction, multiplication, and division on complex numbers. Here’s a quick overview to make things clearer:

• Addition/Subtraction: Simply add or subtract the real parts and the imaginary parts separately. For example, for \( (3 + 4i) + (1 - 2i) \), you combine real numbers (3 + 1) and imaginary numbers (4i - 2i) to get \( 4 + 2i \).
• Multiplication: Use the distributive property, keeping in mind that \( i^2 = -1 \). For instance, \((3 + 4i)(1 - 2i) = 3 \times 1 + 3 \times (-2i) + 4i \times 1 + 4i \times (-2i)\).
• Division: More complex as it typically involves multiplying by the conjugate to simplify into a real number denominator, as shown in the exercise.

Working with complex numbers can seem daunting, but by handling real and imaginary components separately, it's simpler. Practicing these operations enhances your understanding and fluency in complex arithmetic.

Division of Complex Numbers

The division of complex numbers, though complex, is more convenient when using complex conjugates. In the division \( \frac{a + bi}{c + di} \), the main goal is to "clear" the complex number in the denominator. Here's how it’s managed:

• Identify the Complex Conjugate: First, determine the conjugate of the denominator (for example, for \( 4 - 3i \), it's \( 4 + 3i \)).
• Multiply Numerator and Denominator: Both the numerator and the denominator are multiplied by this conjugate to obtain a purely real number in the denominator (for instance, \( (4 - 3i)(4 + 3i) = 16 + 9 = 25 \)). This step changes the division into an ordinary complex multiplication problem.
• Simplify Each Part: After multiplying, calculate separately for real and imaginary numbers so that the result is expressed as a real number plus an imaginary number \( (a/\text{real}) + (b/\text{real})i \).

By using the above method, you turn what seems a perplexing operation into a straightforward arithmetic task. Understanding this concept is vital for higher-level mathematics, where complex numbers frequently appear in equations and formulae.