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Chapter 8: Problem 45

For Exercises \(41-48\), for each complex number \(z\), write the complex conjugate \(\bar{z}\), and find \(z \bar{z}\). $$ z=6+4 i $$

Short Answer

Expert verified
The complex conjugate is \( \bar{z} = 6 - 4i \); \( z \bar{z} = 52 \).

Step by step solution

01

Identify the Complex Number

The given complex number is \( z = 6 + 4i \).
02

Formulate the Complex Conjugate

The conjugate of a complex number \( z = a + bi \) is \( \bar{z} = a - bi \). For \( z = 6 + 4i \), the conjugate is \( \bar{z} = 6 - 4i \).
03

Calculate the Product of z and its Conjugate

To find \( z \bar{z} \), multiply \( z = 6 + 4i \) by \( \bar{z} = 6 - 4i \). Use the formula: \[ (a + bi)(a - bi) = a^2 + b^2 \]Substitute \( a = 6 \) and \( b = 4 \):\[ (6 + 4i)(6 - 4i) = 6^2 + 4^2 = 36 + 16 = 52 \].

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Key Concepts

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

Complex Conjugate
In the world of complex numbers, a complex conjugate plays an important role in simplifying expressions and solving equations. It is derived from a given complex number by changing the sign of the imaginary part. For a complex number in the form of \( z = a + bi \), the conjugate \( \bar{z} \) is \( a - bi \). This simple alteration is particularly useful in a variety of mathematical calculations.

When working with complex conjugates, there are some simple rules:
  • The conjugate of \( z = a + bi \) is \( \bar{z} = a - bi \).
  • It has the same real part and an opposite imaginary part.
  • Complex conjugation is commutative: \( \overline{\overline{z}} = z \).
The complex conjugate helps eliminate the imaginary component when multiplied with the original complex number, leaving just a real number. This characteristic facilitates problem solving in fields ranging from algebra to control systems engineering.
Multiplying Complex Numbers
Multiplying complex numbers involves both the real and imaginary parts. The process can seem tricky at first, but with practice, it becomes a useful skill. For two complex numbers, \( z_1 = a + bi \) and \( z_2 = c + di \), the multiplication \( z_1 \times z_2 \) follows these steps:

  • Multiply the real parts: \( ac \).
  • Multiply the real part of the first by the imaginary part of the second: \( adi \).
  • Multiply the imaginary part of the first by the real part of the second: \( bci \).
  • Multiply the imaginary parts: \( bdi^2 \).
Putting it all together, we use distributive properties to get \( ac + bdi^2 + (ad + bc)i \). Here, keep in mind that \( i^2 = -1 \). This equation becomes \( ac - bd + (ad + bc)i \).

This calculation turns complex multiplication into a manageable series of steps consisting of simple multiplications and additions.
Imaginary Unit
The imaginary unit \( i \) is a fundamental aspect of complex numbers. By definition, \( i \) is the square root of \(-1\), allowing us to extend real numbers into the complex plane. This might seem abstract at first, but \( i \) allows mathematicians and engineers to express complex numbers and solve a broader range of equations than with just real numbers alone.

Understanding the properties of \( i \) can simplify calculations with complex numbers:
  • \( i^1 = i \)
  • \( i^2 = -1 \)
  • \( i^3 = -i \)
  • \( i^4 = 1 \)
The pattern repeats every four powers. This repetitive cycle makes it easier to handle powers of \( i \) in complex calculations since the value will always be one of four possibilities. Remember, the imaginary unit \( i \) is what makes complex numbers a versatile and powerful tool in both mathematics and physics.

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Most popular questions from this chapter

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