Problem 94 - Algebra for College Students

Question

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{5+i}{2+9 i}$$

Step-by-Step Solution

Answer

The quotient is \( \frac{19}{85} - \frac{43}{85}i \) in standard form.

Step 1: Identify the Expression and Standard Form

We need to find the quotient \( \frac{5+i}{2+9i} \) and express it in the standard form of a complex number, which is \( a+bi \), where \( a \) and \( b \) are real numbers.

Step 2: Multiply by the Conjugate

Multiply both the numerator and denominator by the conjugate of the denominator, which is \( 2-9i \). The expression becomes: \[ \frac{(5+i)(2-9i)}{(2+9i)(2-9i)} \].

Step 3: Simplify the Denominator

The denominator \( (2+9i)(2-9i) \) simplifies using the difference of squares: \( (2)^2 - (9i)^2 = 4 - 81(-1) = 4 + 81 = 85 \).

Step 4: Expand the Numerator

Expand \( (5+i)(2-9i) \) using the distributive property: \[ 5 \cdot 2 + 5 \cdot (-9i) + i \cdot 2 + i \cdot (-9i) \] = \( 10 - 45i + 2i - 9i^2 \).

Step 5: Simplify the Numerator

Simplify the expression from the previous step: \( 10 - 45i + 2i + 9 \) (since \( i^2 = -1 \), so \( -9i^2 = 9 \)). This simplifies to \( 19 - 43i \).

Step 6: Divide Numerator by Denominator

Divide the simplified numerator \( 19 - 43i \) by the denominator 85: \[ \frac{19}{85} - \frac{43}{85}i \].

Step 7: Express in Standard Form

The quotient in standard form is \( \frac{19}{85} - \frac{43}{85}i \).

Key Concepts

Complex Conjugate

The complex conjugate of a complex number is essential for simplifying expressions involving division. To find the complex conjugate, simply change the sign of the imaginary part of the number.
For example, the complex conjugate of \(2+9i\) is \(2-9i\). This technique is crucial when rationalizing the denominator of a complex fraction.
By multiplying both the numerator and the denominator by the conjugate of the denominator, you eliminate the imaginary unit \(i\) from the denominator, often resulting in a real number.

• Always remember: the complex conjugate of \(a+bi\) is \(a-bi\).
• This step is necessary to simplify the division of complex numbers.
• The result will often be much simpler after utilizing the conjugate.

Standard Form

Every complex number can be expressed in the standard form, which is \(a+bi\). In this form, \(a\) represents the real part, and \(bi\) represents the imaginary part.
This format is useful because it separates the real and imaginary components, making it easy to perform arithmetic operations like addition and subtraction.

• When simplifying complex quotients, the goal is often to end up with a \(a+bi\) form.
• This ensures the expression is easy to understand and consistent with mathematical conventions.

In the original exercise, after simplification, the quotient \( \frac{5+i}{2+9i} \) ends up in the form \( \frac{19}{85} - \frac{43}{85}i \), showcasing both real and imaginary parts.

Imaginary Unit

The imaginary unit, symbolized as \(i\), is a crucial concept in the realm of complex numbers. By definition, \(i\) satisfies the equation \(i^2 = -1\).
Complex numbers build on this unit and are often expressed in terms of \(i\), combining real numbers with imaginary ones. Understanding \(i\) is vital since it is frequently encountered in both simple and advanced mathematics tasks.

In expressions involving \(i\), any occurrence of \(i^2\) should be replaced with \(-1\). For instance, in the step of expanding the numerator \((5+i)(2-9i)\), the term \(-9i^2\) simplifies to \(9\) because \(i^2 = -1\).

• Use \(i\) to express the square roots of negative numbers.
• Always replace \(i^2\) with \(-1\) during calculations to simplify expressions.

Simplifying Expressions

Simplifying expressions involving complex numbers involves several crucial steps. Often, this begins with ensuring the expression is in a form that's easy to work with, such as using the complex conjugate to rationalize denominators.
Once in the desired form, expressions can be expanded and sometimes simplified further using the imaginary unit's properties.

In the exercise given, simplifying involved careful multiplication and replacement of \(i^2\). After expanding \((5+i)(2-9i)\), the term \(-9i^2\) was rewritten as \(9\) since \(i^2 = -1\).
Subsequent steps often require dividing and rewriting in a clean \(a+bi\) format, as seen when the expression \( \frac{19 - 43i}{85} \) transforms into \( \frac{19}{85} - \frac{43}{85}i \).

• Recognize patterns like the distributive property to manage complex multiplications.
• Always utilize the basic properties of \(i\), especially \(i^2 = -1\).
• Ensure final expressions clearly show separation of real and imaginary parts.

Successful simplification provides clarity and readiness for further mathematical operations or interpretations.