Problem 71 - Intermediate Algebra

Question

Divide. Write all answers in the form a \(+b i.\) $$ \frac{11 i}{4-7 i} $$

Step-by-Step Solution

Answer

\(-\frac{77}{65} + \frac{44}{65}i\)

Step 1: Multiply by the Conjugate

To eliminate the imaginary part from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of the denominator \(4 - 7i\) is \(4 + 7i\). So, multiply the expression by \(\frac{4 + 7i}{4 + 7i}\).

Step 2: Use the Distributive Property on the Numerator

Multiply \(11i\) by \(4 + 7i\). Use the distributive property: \[11i \times (4 + 7i) = 11i \times 4 + 11i \times 7i = 44i + 77i^2\]Since \(i^2 = -1\), replace \(i^2\) with \(-1\):\[77i^2 = 77(-1) = -77\]So, the numerator becomes: \[-77 + 44i\].

Step 3: Use the Distributive Property on the Denominator

Multiply \(4 - 7i\) by \(4 + 7i\). Use the formula for a complex number's conjugate:\[(a - bi)(a + bi) = a^2 + b^2\]Here, \(a = 4\) and \(b = 7\), so:\[(4 - 7i)(4 + 7i) = 4^2 + 7^2 = 16 + 49 = 65\]

Step 4: Write the Quotient in Standard Form

Combine the results from Steps 2 and 3 to form the quotient:\[\frac{-77 + 44i}{65}\]Separate into real and imaginary components:\[-\frac{77}{65} + \frac{44}{65}i\]

Step 5: Simplify the Expression (if possible)

Check if \(-\frac{77}{65}\) and \(\frac{44}{65}\) can be simplified. Since 77, 44, and 65 do not have common factors, they are already in simplest form:\[-\frac{77}{65} + \frac{44}{65}i\].

Key Concepts

Imaginary Unit

The imaginary unit is a cornerstone of the complex numbers system and is represented by the symbol \(i\). It is defined as the square root of \(-1\), leading to the mathematical relationship \(i^2 = -1\). This unique feature allows complex numbers to handle problems involving negative square roots, which are not possible with real numbers alone.

Complex numbers combine real and imaginary components, typically written in the form \(a + bi\), where \(a\) is the real part, and \(bi\) denotes the imaginary part. Here, \(b\) is the coefficient and can be any real number, showcasing the flexibility of complex numbers in mathematical applications.

In the given exercise, we use \(i\) to express the number in a division problem between a pure imaginary number \(11i\) and a complex number \(4 - 7i\). Understanding \(i\) and its properties is integral to performing operations involving complex numbers, as observed in this division example.

Conjugate of Complex Numbers

The conjugate of a complex number is essential when simplifying complex fractions, especially in division tasks. For any complex number \(a + bi\), its conjugate is \(a - bi\).

Using the conjugate eliminates the imaginary part from the denominator when multiplying both the numerator and the denominator by the conjugate. This transformation adheres to the rule:

• If the denominator is \(4 - 7i\), its conjugate is \(4 + 7i\).
• Multiplying a complex number by its conjugate results in a real number, as shown by the equation \((a - bi)(a + bi) = a^2 + b^2\).

In the exercise, multiplying the given complex fraction by \(\frac{4 + 7i}{4 + 7i}\) achieves this, allowing us to simplify the expression to a real denominator, making further calculations straightforward.

Distributive Property

The distributive property is an algebraic rule that applies to both real and complex numbers, essential for calculating products of various terms. It states that for any numbers or expressions \(a, b,\) and \(c\), the property can be written as \(a(b + c) = ab + ac\).

Applying this property in the multiplication of complex numbers helps distribute terms effectively:

• In the exercise, we distribute \(11i\) over \(4 + 7i\), yielding \(44i + 77i^2\).
• The imaginary unit's property \(i^2 = -1\) results in simplifying \(77i^2\) as \(-77\).

This step-by-step distribution process helps locate each term's contribution to the final outcome of a complex expression, facilitating straightforward simplification when performed correctly.

Standard Form of Complex Numbers

Expressing complex numbers in standard form is crucial for readability and further mathematical operations. The standard form is \(a + bi\), where \(a\) and \(b\) are real numbers representing the real and imaginary parts, respectively.

After performing the division in the exercise, you combine the simplified terms to write the quotient as a standard form expression:

• The simplified result is \(\frac{-77 + 44i}{65}\).
• Separate into real and imaginary components: \(-\frac{77}{65} + \frac{44}{65}i\).

This separation makes it easy to identify and understand the two distinct parts of the complex number, ensuring consistent interpretation across various contexts in mathematics. Using this approach allows for simple integration with other mathematical processes and operations, reinforcing the usefulness of the standard form.