Problem 73 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The result is \( \frac{14}{17} - \frac{5}{17}i \).

Step 1: Multiply by the Conjugate

To divide complex numbers, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of the denominator \(4 - i\) is \(4 + i\). Therefore, multiply both the numerator \(3 - 2i\) and the denominator \(4 - i\) by \(4 + i\).

Step 2: Apply the Distributive Property

First, distribute \((3 - 2i)\) with \((4 + i)\):\[(3 - 2i)(4 + i) = 3\cdot 4 + 3\cdot i - 2i\cdot 4 - 2i\cdot i = 12 + 3i - 8i - 2i^2\]Recall that \(i^2 = -1\). Therefore, replace \(-2i^2\) with \(2\). Simplify the expression to:\[12 + 2 + 3i - 8i = 14 - 5i\]

Step 3: Simplify the Denominator

Now, multiply the denominator \((4 - i)(4 + i)\):\[(4 - i)(4 + i) = 4\cdot 4 + 4i - 4i - i^2\]This simplifies to:\[16 + 1 = 17\] because \(-i^2 = 1\).

Step 4: Write in Standard Form

Divide the result obtained for the numerator by the result from the denominator:\[\frac{14 - 5i}{17} = \frac{14}{17} - \frac{5}{17}i\]Thus, the division in standard form is:\[\frac{14}{17} - \frac{5}{17}i\]

Key Concepts

Conjugate

The concept of a conjugate is essential when dealing with complex numbers, especially when you're dividing them. A complex number is composed of a real part and an imaginary part, usually written as:

• A real part: a number without any imaginary unit \(i\).
• An imaginary part: a number multiplied by \(i\), the imaginary unit with the property \(i^2 = -1\).

To find the conjugate of a complex number, you switch the sign of its imaginary part.
For instance, if you have a complex number \(a + bi\), its conjugate would be \(a - bi\).
In the context of dividing complex numbers, using the conjugate helps eliminate the imaginary part from the denominator, making calculations much easier.
Multiplying the numerator and denominator by the conjugate of the denominator transforms the problem into a simpler division of real numbers and allows the result to be expressed easily in standard form.

Standard Form

Writing complex numbers in standard form is a straightforward but crucial skill. When you express complex numbers, they should follow the form \(a + bi\), where:

• \(a\) is the real part, and
• \(b\) denotes the coefficient of the imaginary unit \(i\).

This structure makes addition, subtraction, and other arithmetic operations more manageable and clear.
By converting a complex number division result into standard form, you present the solution in a universally recognized format, streamlining further mathematical operations.
The step-by-step division of complex numbers leads up to this standard expression.
In the exercise, the division result was simplified to \(\frac{14}{17} - \frac{5}{17}i\), maintaining its clarity and usability in subsequent calculations.

Distributive Property

The distributive property is a vital arithmetic rule that applies to complex numbers as well as real numbers. It states that for any three numbers (or expressions), \(a\), \(b\), and \(c\), the rule \(a(b + c) = ab + ac\) holds true. This property is straightforward and allows us to break down and simplify expressions, particularly in multiplication.

• For example, applying the distributive property in the exercise allows you to multiply the two complex numbers \((3 - 2i)(4 + i)\).
• The property distributes each term from the first expression with every term from the second.

Through these meticulous steps, we reach the product \(12 + 3i - 8i - 2i^2\), which further simplifies based on the property \(i^2 = -1\).
This simplification is crucial to progressing through division and eventually arriving at a clean, conceivable standard form of a complex number.