Problem 24 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-34: Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \frac{-3-2 i}{5+2 i} $$

Step-by-Step Solution

Answer

The expression in the form \(a + bi\) is \(\frac{-19}{29} + \frac{-4}{29}i\).

Step 1: Identify the Conjugate

In order to express the division of complex numbers in the form \(a + bi\), we need to eliminate the imaginary part in the denominator. The conjugate of the denominator \(5 + 2i\) is \(5 - 2i\). We will use the conjugate to rationalize the denominator.

Step 2: Multiply the Numerator and Denominator by the Conjugate

We multiply both the numerator and the denominator by the conjugate of the denominator:\[\frac{-3-2i}{5+2i} \cdot \frac{5-2i}{5-2i}\]This results in:\[\frac{(-3-2i)(5-2i)}{(5+2i)(5-2i)}\]

Step 3: Expand the Numerator

Distribute the terms in the numerator:\[(-3)(5) + (-3)(-2i) + (-2i)(5) + (-2i)(-2i)\]This simplifies to:\[-15 + 6i - 10i + 4i^2\]Since \(i^2 = -1\), replace \(4i^2\) with \(-4\) to get:\[-15 + 6i - 10i - 4\]Simplifying gives us:\[-19 - 4i\]

Step 4: Simplify the Denominator

Expand the denominator using the difference of squares formula:\[(5+2i)(5-2i) = 5^2 - (2i)^2\]Which simplifies to:\[25 - 4i^2\]Since \(i^2 = -1\), \(-4i^2 = 4\). So the denominator simplifies to:\[25 + 4 = 29\]

Step 5: Formulate the Fraction and Simplify

Combine the simplified numerator and denominator:\[\frac{-19 - 4i}{29}\]This is equivalent to:\[\frac{-19}{29} + \frac{-4}{29}i\]Thus, in the form \(a + bi\) where \(a = \frac{-19}{29}\) and \(b = \frac{-4}{29}\).

Key Concepts

Imaginary Numbers

Imaginary numbers are an essential concept when dealing with complex numbers. They are used to extend the real number system to solve equations that do not have a solution in the real numbers alone. The most basic imaginary number is represented as the square root of \(-1\), denoted by \(i\). When you see \(i\), it symbolizes \(\sqrt{-1}\). Imaginary numbers become especially important in complex numbers, which take the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.

Let's break this further down:

• The number \(a\) is often called the 'real part' of the complex number.
• \(b i\) represents the 'imaginary part'.
• Together, they form a complex number in the format \(a + bi\).

Understanding imaginary numbers is pivotal because they provide a way to work with problems involving non-real solutions. Imaginary numbers make sure every quadratic equation has a solution, broadening our ability to solve mathematical problems.

Conjugate

In complex numbers, the conjugate is a crucial concept, especially for complex division. The conjugate of a complex number is formed by changing the sign of its imaginary part.

Consider a complex number \(z = a + bi\), where \(a\) and \(b\) are real numbers. The conjugate of \(z\) is denoted as \(\bar{z}\), and it is \(a - bi\).

Why is the conjugate important?

• Conjugates help to eliminate the imaginary component from the denominator, allowing you to rationalize the denominator.
• They are used frequently in complex arithmetic and manipulations where simplification is necessary.
• When a complex number is multiplied by its conjugate, the result is a real number.

In our exercise, \(5 + 2i\) has the conjugate \(5 - 2i\). By multiplying the numerator and the denominator by the conjugate, we simplify the expression and achieve a suitable format for further calculations.

Rationalize the Denominator

Rationalizing the denominator is a strategy used in complex division to remove the imaginary unit \(i\) from the denominator. The process involves multiplying both the numerator and the denominator of a complex fraction by the conjugate of the denominator. This technique helps in expressing complex numbers in a standard form, where both the numerator and denominator are free from imaginary numbers.

Here's why this matters:

• It simplifies division of complex numbers, making them easier to handle in calculations.
• In true mathematical form, denominators should not have imaginary numbers, as it complicates further computation or interpretation.
• Multiplying by the conjugate converts the denominator into a straightforward real number using the difference of squares formula.

For example, in our exercise, \(\frac{-3-2i}{5+2i}\), multiplying by the conjugate \(5-2i\) simplifies the denominator to a real number, effectively 'rationalizing' it.

Complex Division

Complex division involves dividing one complex number by another. This process can seem tricky because it requires a method to eliminate the imaginary numbers in the denominator, thereby converting it into a real number.

The steps for complex division are:

• Identify the conjugate of the denominator.
• Multiply both the numerator and the denominator by this conjugate.
• Simplify the resulting expression to get a complex number in the standard form \(a + bi\).

In our example, \(\frac{-3-2i}{5+2i}\), applying complex division involves \(5 - 2i\) to rationalize the denominator. This leads to operations like expanding polynomials, which allows the resultant expression to fit the form \(a + bi\). For our particular problem, this ensures both the real and imaginary components are properly resolved, with a denominator that is a clear, simple real number.