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Chapter 8: Problem 56

For Exercises 49-64, write each quotient in standard form. $$ \frac{8}{1+6 i} $$

Short Answer

Expert verified
\( \frac{8}{37} - \frac{48}{37}i \)

Step by step solution

01

Identify the Quotient and Complex Denominator

You have the fraction \( \frac{8}{1+6i} \) where the denominator is the complex number \( 1 + 6i \). To write this in standard form, we must eliminate the imaginary part from the denominator.
02

Multi-denominator by the Conjugate

The standard form of a complex number is \( a + bi \). To achieve this, multiply the numerator and the denominator of \( \frac{8}{1 + 6i} \) by the conjugate of the denominator \( 1 - 6i \).
03

Multiply Numerator and Denominator

Perform the multiplication:1. Multiply the numerator: \( 8(1 - 6i) = 8 \cdot 1 - 8 \cdot 6i = 8 - 48i \).2. Multiply the denominator: \( (1 + 6i)(1 - 6i) = 1^2 - (6i)^2 = 1 - 36(-1) = 1 + 36 = 37 \).
04

Write the Result in Standard Form

Now, the fraction becomes \( \frac{8 - 48i}{37} \). This can be separated into the real and imaginary parts: - Real part: \( \frac{8}{37} \)- Imaginary part: \( \frac{-48}{37}i \)Thus, the standard form is \( \frac{8}{37} - \frac{48}{37}i \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form of Complex Numbers
Complex numbers offer a fascinating blend of real and imaginary values. The standard form of a complex number is expressed as \( a + bi \), where:
  • \( a \) represents the real part.
  • \( b \) represents the imaginary part.
  • \( i \) is the imaginary unit, equivalent to \( \sqrt{-1} \).
To convert a complex number into its standard form, it's crucial to eliminate any imaginary components from the denominator. This is often done using the conjugate of the denominator. In our exercise, \( \frac{8}{1+6i} \) was converted into its standard form, \( \frac{8}{37} - \frac{48}{37}i \), by multiplying both the numerator and denominator by the conjugate \( 1 - 6i \). Through simplification, the fraction becomes a clear representation: one part real, one part imaginary.
Complex Conjugate
The complex conjugate is a simple yet powerful concept used in handling complex numbers. If you have a complex number in the form \( a + bi \), its complex conjugate is \( a - bi \). Reversing the sign of the imaginary part, transforms a complex number into its conjugate.
  • For example, the conjugate of \( 1 + 6i \) is \( 1 - 6i \).
  • The product of a complex number and its conjugate is always a real number.
Multiplying by the conjugate is key in converting complex fractions to standard form, as it neutralizes the imaginary component in the denominator. This technique ensures the denominator becomes a real number, making it easier to separate real and imaginary parts of the complex quotient.
Imaginary Numbers
Imaginary numbers are numbers that, when squared, yield a negative result. They can be puzzling at first but are integral to the concept of complex numbers. The imaginary unit \( i \) serves as the base element, defined by \( i = \sqrt{-1} \). This neat little property allows us to incorporate otherwise impossible calculations.
  • Imaginary numbers are written as \( bi \), where \( b \) is a real number coefficient.
  • \( i^2 = -1 \), reinforcing the distinctive nature of imaginary numbers.
In our example \( \frac{8}{1+6i} \), the imaginary part was \( 6i \). By multiplying by the complex conjugate, the pesky imaginary unit was removed from the denominator. Simplifying complex equations often involves this crucial step to reframe expressions in the more intuitive real-number domain.

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