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Chapter 8: Problem 48

For Exercises \(41-48\), for each complex number \(z\), write the complex conjugate \(\bar{z}\), and find \(z \bar{z}\). $$ z=-3-9 i $$

Short Answer

Expert verified
The complex conjugate is \(-3 + 9i\) and \(z \bar{z} = 90\).

Step by step solution

01

Identify the complex number

The given complex number is \( z = -3 - 9i \). It has a real part \( a = -3 \) and an imaginary part \( b = -9 \).
02

Write the complex conjugate

The complex conjugate \( \bar{z} \) of a complex number \( z = a + bi \) is given by \( \bar{z} = a - bi \). Therefore, for \( z = -3 - 9i \), the complex conjugate is \( \bar{z} = -3 + 9i \).
03

Calculate the product of the complex number and its conjugate

The product \( z \bar{z} \) is calculated by multiplying the complex number with its conjugate: \((-3 - 9i)(-3 + 9i)\). Use the difference of squares formula: \((a + b)(a - b) = a^2 - b^2\). Here, \( a = -3 \) and \( b = 9i \).
04

Apply the difference of squares formula

Substituting into the difference of squares formula, we get: \((-3)^2 - (9i)^2 = 9 - (81i^2)\). Since \( i^2 = -1 \), the expression becomes \( 9 - 81(-1) = 9 + 81 = 90 \).

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Key Concepts

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

Complex Conjugate
In mathematics, every complex number has a counterpart known as its _complex conjugate_. For a complex number expressed in the form \(z = a + bi\), where \(a\) and \(b\) are real numbers, the complex conjugate is denoted as \(\bar{z}\) and is given by \(\bar{z} = a - bi\). Essentially, to find the complex conjugate, you simply change the sign of the imaginary part.
  • The given complex number is \(-3 - 9i\).
  • Its complex conjugate is \(-3 + 9i\).
This operation is crucial when you need to simplify expressions or solve equations involving complex numbers. By multiplying a complex number by its conjugate, you eliminate the imaginary part, leaving a real number.
Difference of Squares
The _difference of squares_ is a formula that greatly simplifies the multiplication of expressions of a specific form. It can be stated as:\[(a + b)(a - b) = a^2 - b^2\]In the realm of complex numbers, this formula is particularly useful. When you multiply a complex number by its conjugate, you apply this principle by using the pattern:
  • \(a = -3\) and \(b = 9i\) for our example.
  • Thus, \((-3 - 9i)(-3 + 9i)\) becomes \((-3)^2 - (9i)^2\).
After resolving, you'll end up with a real number because the \(i^2\) term becomes \(-1\), thus turning the subtraction into addition, \((9 + 81) = 90\).
The difference of squares not only simplifies calculations but also allows us to eliminate complex parts, bolstering our ability to handle complex equations effectively.
Imaginary Unit
The _imaginary unit_, denoted as \(i\), is the cornerstone of complex numbers. The imaginary unit is defined as \(i = \sqrt{-1}\), meaning that when squared, \(i\) yields \(-1\):\[i^2 = -1\]
This characteristic is instrumental in manipulating and simplifying problems involving complex numbers.
  • In our solved exercise, we encounter an \(i^2\) term when managing the expression \((9i)^2\).
  • According to the fundamental property \(i^2 = -1\), \((9i)^2\) breaks down to \(-81\).
When you multiply the complex number by its conjugate, these properties of \(i\) ensure the result is a purely real number, simplifying calculations and making complex numbers more tangible to work with.

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Most popular questions from this chapter

In Exercises 21-36, each set of parametric equations defines a plane curve. Find an equation in rectangular form that also corresponds to the plane curve. $$ x=\frac{t}{2}, y=2 \tan t $$

Find the polar equation that is equivalent to a vertical line, \(x=a\).

In Exercises 45-68, graph each equation. In Exercises 63-68, convert the equation from polar to rectangular form first and identify the resulting equation as a line, parabola, or circle. $$ r^{2} \sin ^{2} \theta+2 r \cos \theta=3 $$

In Exercises 21-36, each set of parametric equations defines a plane curve. Find an equation in rectangular form that also corresponds to the plane curve. $$ x=2 \sin ^{2} t, y=2 \cos ^{2} t $$

In Exercises 21-36, each set of parametric equations defines a plane curve. Find an equation in rectangular form that also corresponds to the plane curve. $$ x=t, y=\sqrt{t^{2}+1} $$
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