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Chapter 6: Problem 52

Show that \(\overline{\bar{z}}=z\) for every complex number \(z\).

Short Answer

Expert verified
Given a complex number \(z = a + bi\), its conjugate is \(\bar{z} = a - bi\). Calculating the conjugate of \(\bar{z}\) results in \(\overline{\bar{z}} = a - (-bi) = a + bi\), which is equal to the original complex number \(z\). Therefore, \(\overline{\bar{z}} = z\) for every complex number \(z\).

Step by step solution

01

Compute the conjugate of \(z\)

The given complex number \(z\) can be written as \(a + bi\), where \(a\) and \(b\) are real numbers. To find the conjugate of \(z\), we change the sign of the imaginary part, resulting in \(\bar{z} = a - bi\).
02

Compute the conjugate of \(\bar{z}\)

We now have \(\bar{z} = a - bi\). To find its conjugate, which we will call \(\overline{\bar{z}}\), we change the sign of the imaginary part again: \(\overline{\bar{z}} = a - (-bi) = a + bi\).
03

Compare \(\overline{\bar{z}}\) and \(z\)

In step 2, we obtained \(\overline{\bar{z}} = a + bi\), which is the same as our original complex number \(z\). Therefore, we can conclude that \(\overline{\bar{z}} = z\) for every complex number \(z\).

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Key Concepts

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

Complex Conjugate
A complex conjugate is a simple yet fundamental concept when working with complex numbers. If you have a complex number, say \(z = a + bi\), its conjugate is denoted by \(\bar{z}\) and is found by changing the sign of the imaginary part, resulting in \(\bar{z} = a - bi\).
This operation essentially reflects the complex number across the real axis in the complex plane. It's worth noting that the conjugate of the conjugate of a complex number brings you back to the original number: \(\overline{\bar{z}} = z\).
The complex conjugate has several useful properties that help simplify operations in complex arithmetic, making it a crucial tool in mathematics.
Properties of Complex Numbers
Complex numbers have special properties that distinguish them from real numbers.
  • Every complex number can be expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, satisfying \(i^2 = -1\).
  • Complex numbers can be visualized on the complex plane, with the real part \(a\) representing the x-axis and the imaginary part \(b\) on the y-axis.
  • Given their form, complex numbers offer unique operations not just in arithmetic but in geometric interpretations as well.

One essential property related to the article is the conjugate operation: for any complex number \(z\), it holds that \(\overline{\bar{z}} = z\). These properties are foundational for understanding further algebraic operations and transformations in mathematics.
Algebraic Operations with Complex Numbers
Performing algebraic operations with complex numbers extends operations we commonly use with real numbers, but they have their unique aspects.
  • Addition and Subtraction: To add or subtract two complex numbers such as \(z_1 = a + bi\) and \(z_2 = c + di\), simply add or subtract their corresponding parts: \(z_1 + z_2 = (a+c) + (b+d)i\).
  • Multiplication: Multiply complex numbers using distribution, keeping in mind that \(i^2 = -1\). For example, \(z_1 \cdot z_2 = (a+bi)(c+di) = (ac-bd) + (ad+bc)i\).
  • Conjugation: As seen earlier, the conjugation operation involves changing the sign of the imaginary part which plays a vital role in simplifying expressions and finding the magnitude of complex numbers.

These operations help in solving a wide array of mathematical problems and contribute to the broader application of complex numbers in fields like engineering, physics, and applied mathematics.

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Most popular questions from this chapter

Show that addition of complex numbers is associative, meaning that $$ u+(w+z)=(u+w)+z $$ for all complex numbers \(u, w,\) and \(z\).

Write each expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$ (8-4 i)(2-3 i) $$

Write each expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$ (5+7 i)+(4+6 i) $$

Describe the subset of the complex plane consisting of the complex numbers \(z\) such that the real part of \(z^{3}\) is a positive number.

Show that if \(z\) is a complex number, then the real part of \(z\) is in the interval \([-|z|,|z|]\).
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