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Chapter 6: Problem 95

Is it true that the product of a complex number and its conjugate is a real number? Explain.

Short Answer

Expert verified
Yes, the product of a complex number and its conjugate is always a real number.

Step by step solution

01

Define a Complex Number

A complex number can be written in the form of \(z = a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit such that \(i^2 = -1\).
02

Find the Conjugate

The conjugate of the complex number \(z = a + bi\) is denoted by \(\bar{z}\) and it is found by changing the sign of the imaginary part: \(\bar{z} = a - bi\).
03

Multiply the Complex Number by Its Conjugate

To find the product of the complex number \(z\) and its conjugate \(\bar{z}\), multiply them together: \(z \cdot \bar{z} = (a + bi)(a - bi)\).
04

Expand the Product

Use the distributive property to expand the product: \((a + bi)(a - bi) = a^2 - abi + abi - b^2i^2\).
05

Simplify the Expression

Notice that \(-abi\) and \(+abi\) cancel out, and recall that \(i^2 = -1\): \(a^2 - b^2i^2 = a^2 - b^2(-1) = a^2 + b^2\).
06

Conclusion

The expression \(a^2 + b^2\) is a sum of squares of real numbers \(a\) and \(b\), which is a real number. Therefore, the product of a complex number and its conjugate is always a real number.

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Key Concepts

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

complex number
A complex number is a number that has both a real part and an imaginary part. It is usually written in the form \(z = a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \(i^2 = -1\). Because of this combination, complex numbers are a richer set of numbers than real numbers, allowing us to solve equations that do not have solutions within the set of real numbers.

Some key points about complex numbers are:
  • Real part: The term \(a\) in \(z\) is the real part of the complex number.
  • Imaginary part: The term \(bi\) in \(z\) is the imaginary part of the complex number.
  • The imaginary unit \(i\) helps in expressing roots of negative numbers, which are not possible with real numbers alone.
Understanding complex numbers is essential because they extend the concept of one-dimensional real numbers to two-dimensional complex numbers in the complex plane.
conjugate
The conjugate of a complex number is a significant concept when working with complex numbers. If you have a complex number \(z = a + bi\), its conjugate is denoted by \(\bar{z}\) and is defined as \(a - bi\). Essentially, you change the sign of the imaginary part.

Here's how the conjugate helps:
  • The product of a complex number and its conjugate always results in a real number. For example, multiplying \(z = a + bi\) and \(\bar{z} = a - bi\) gives us a result containing no imaginary part.
  • Finding the conjugate can simplify calculations, particularly when dividing complex numbers or finding the modulus (magnitude) of a complex number.
Conjugates play a crucial role, not just in mathematical operations, but also in various fields such as signal processing and quantum physics.
real number
A real number is a value representing a quantity along a continuous number line. Real numbers include rational numbers (such as 7 and -0.3) and irrational numbers (such as \(\frac{\backslash\backslashsqrt{2}}\) and \(\backslash\backslashpi\)).

When dealing with complex numbers, the real number aspect comes into play when:
  • You are identifying the real part of a complex number \(z = a + bi\).
  • You find that the product of a complex number and its conjugate is always a real number. This was shown by expanding the product \((a + bi)(a - bi) = a^2 + b^2\).
Real numbers are the foundation of most numerical concepts we encounter, but in complex numbers, they enhance our ability to describe more complex phenomena. When a complex number is multiplied by its conjugate, the imaginary components cancel out, leaving only the real part, providing an excellent demonstration of the interplay between real and complex numbers.

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Most popular questions from this chapter

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