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Chapter 1: Problem 73

Evaluate the given expression for \(z=3-4 i\) and \(w=5+2 i\) $$\bar{z}+\bar{w}$$

Short Answer

Expert verified
The expression evaluates to \(8 + 2i\).

Step by step solution

01

Find the Conjugate of z

The conjugate of a complex number is obtained by changing the sign of its imaginary part. For the complex number \(z = 3 - 4i\), the conjugate \(\bar{z}\) is \(3 + 4i\).
02

Find the Conjugate of w

Similarly, the conjugate of a complex number \(w = 5 + 2i\) is found by changing the sign of its imaginary part. Thus, the conjugate \(\bar{w}\) is \(5 - 2i\).
03

Add the Conjugates

Now, add the conjugates \(\bar{z}\) and \(\bar{w}\) calculated in the previous steps: \[\bar{z} + \bar{w} = (3 + 4i) + (5 - 2i)\].Separate the real and imaginary parts:\[=(3+5) + (4i - 2i) \].
04

Simplify the Expression

Simplify the expression by adding the real parts and the imaginary parts:\[\bar{z} + \bar{w} = 8 + 2i \].

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Key Concepts

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

Imaginary Numbers Explained
Imaginary numbers are a fundamental concept in mathematics. They are numbers written in the form of a real number multiplied by the imaginary unit. This imaginary unit is denoted as "i" and is defined as the square root of -1.
  • The essence of imaginary numbers began with the necessity to solve equations where no real number root existed, like the equation \(x^2 + 1 = 0\).
  • An imaginary number is typically written as \(bi\) where \(b\) is a real number and \(i\) is the imaginary unit.
Imaginary numbers, when combined with real numbers, form complex numbers. These are incredibly useful, not just in mathematics, but also in fields like engineering and physics. In the given exercise, both \(z = 3 - 4i\) and \(w = 5 + 2i\) have imaginary components of \(-4i\) and \(2i\) respectively.
Understanding Complex Conjugates
A complex conjugate is closely tied to complex numbers. For any complex number, its conjugate is found by simply changing the sign of its imaginary component. Let's break this down:
  • For a complex number \(z = a + bi\), its complex conjugate is \(\bar{z} = a - bi\).
  • This operation is useful in many mathematical applications, such as simplifying division of complex numbers and finding the magnitude of a complex number.
In the context of our exercise, knowing the conjugates of \(z = 3 - 4i\) and \(w = 5 + 2i\) allows us to combine these values easily. By determining \(\bar{z} = 3 + 4i\) and \(\bar{w} = 5 - 2i\), we shift the imaginary part's sign to facilitate further operations.
Adding Complex Numbers
The addition of complex numbers is a straightforward process. When adding two complex numbers, we separately sum up their real and imaginary parts:
  • For two complex numbers \(z = a + bi\) and \(w = c + di\), their sum is \((a+c) + (b+d)i\).
  • This means you treat the real and imaginary parts independently.
In our exercise, once we found the conjugates of \(z\) and \(w\), we simply added them together. So, \(\bar{z} + \bar{w} = (3 + 4i) + (5 - 2i)\) becomes \((3+5) + (4-2)i\), resulting in \(8 + 2i\). This addition helps consolidate two complex numbers into one manageable expression.

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Most popular questions from this chapter

Use scientific notation, the Laws of Exponents, and a calculator to perform the indicated operations. State your answer rounded to the number of significant digits indicated by the given data. $$\frac{\left(3.542 \times 10^{-6}\right)^{9}}{\left(5.05 \times 10^{4}\right)^{12}}$$

Decimal Notation Write each number in decimal notation. (a) \(3.19 \times 10^{5}\) (b) \(2.721 \times 10^{8}\) (c) \(2.670 \times 10^{-8}\) (d) \(9.999 \times 10^{-9}\)

The gravitational force \(F\) exerted by the earth on an object having a mass of \(100 \mathrm{kg}\) is given by the equation $$F=\frac{4,000,000}{d^{2}}$$ where \(d\) is the distance (in \(\mathrm{km}\) ) of the object from the center of the earth, and the force \(F\) is measured in newtons (N). For what distances will the gravitational force exerted by the earth on this object be between \(0.0004 \mathrm{N}\) and \(0.01 \mathrm{N} ?\)

Manufacturer's Profit If a manufacturer sells \(x\) units of a certain product, revenue \(R\) and cost \(C\) (in dollars) are given by $$ \begin{array}{l} R=20 x \\ C=2000+8 x+0.0025 x^{2} \end{array} $$ Use the fact that profit \(=\) revenue \(-\) cost to determine how many units the manufacturer should sell to enjoy a profit of at least \(\$ 2400\).

Complete the squares in the general equation \(x^{2}+a x+y^{2}+b y+c=0\) and simplify the result as much as possible. Under what conditions on the coefficients \(a, b,\) and \(c\) does this equation represent a circle? A single point? The empty set? In the case in which the equation does represent a circle, find its center and radius.
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