Problem 74 - Algebra and Trigonometry

Question

Recall that the symbol \(\overline{z}\) represents the complex con- jugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. \(z=\overline{z}\) if and only if \(z\) is real

Step-by-Step Solution

Answer

\(z = \overline{z}\) if and only if \(z\) is real because this implies the imaginary part is zero.

Step 1: Understanding the Complex Conjugate

The complex conjugate of a complex number \(z = a + bi\) is given by \(\overline{z} = a - bi\). This means that the real part \(a\) remains the same, while the imaginary part \(b\) changes sign.

Step 2: Setting the Equality Condition

For \(z = \overline{z}\) to hold true, we equate \(z\) and its complex conjugate, resulting in the equation \(a + bi = a - bi\).

Step 3: Equate Real and Imaginary Parts

Since complex numbers are equal if and only if their corresponding real and imaginary parts are equal, we equate the imaginary parts: \(bi = -bi\). Simplifying gives us \(2bi = 0\) which reduces to \(b = 0\).

Step 4: Conclusion on Real Number

Since \(b = 0\), the imaginary part of \(z\) is zero, making \(z = a\), which is purely real. Thus, \(z\) is real if \(z = \overline{z}\).

Step 5: Proving the Converse

If \(z\) is real, then \(z = a\) where \(bi = 0\). Hence, the conjugate of \(z\) is also \(a\), i.e., \(\overline{z} = z\). This proves the converse, completing the 'if and only if' nature of the statement.

Key Concepts

Complex Numbers

Complex numbers are numbers that take the form of \(z = a + bi\), where \(a\) and \(b\) are real numbers. The term \(a\) is known as the real part, while \(bi\) represents the imaginary part. Complex numbers extend the concept of one-dimensional numbers (the real numbers) into two dimensions by adding an imaginary number component, \(i\), which is defined as \(\sqrt{-1}\).

One way to visualize complex numbers is to imagine them as points or vectors on a plane, called the complex plane. The horizontal axis represents the real part, and the vertical axis represents the imaginary part.

Complex numbers can be added, subtracted, multiplied, and divided, similar to real numbers, with certain rules and properties that guide these operations. A particularly important property is the complex conjugate, which helps simplify complex number calculations. Conjugates are reflections across the real axis in the complex plane.

Real Numbers

Real numbers are the backbone of the number system and are represented as any number on the number line. These include both rational numbers (like 2, \(\frac{3}{4}\), and -5) and irrational numbers (like \(\pi\) and \(\sqrt{2}\)). Real numbers do not have an imaginary component, meaning they can be written simply as \(a + 0i\).

When we consider a real number in the context of complex numbers, it simply means that the imaginary part is zero. For instance, if a complex number is \(z = a + bi\), for it to be a real number, the imaginary coefficient \(b\) must be zero (\(b = 0\)).

This concept is crucial when understanding the conditions \(z = \overline{z}\). For such conditions to be met, a complex number must essentially become a real number, verifying that its imaginary part is zero. Thus, understanding real numbers simplifies the grasp of complex arithmetic and aids in proofs like the one in the original exercise.

Imaginary Numbers

Imaginary numbers are a fascinating part of mathematics, anchored by the symbol \(i\), which is defined as \(\sqrt{-1}\). Imaginary numbers allow us to work with the square roots of negative numbers and are denoted in the form \(bi\), where \(b\) is a real number.

When a complex number is expressed as \(z = a + bi\), the term \(bi\) stands for the imaginary part. This coordinate can be represented on a complex plane, where the imaginary number is plotted along the vertical axis. Unlike real numbers, imaginary numbers cannot be found on the standard number line, which only supports real numbers.

Imaginary numbers are fundamental when performing operations on complex numbers. They play a critical role in understanding the complex conjugate; for a number \(z\) to equal its complex conjugate (\(\overline{z}\)), its imaginary part must cancel out, leaving no imaginary component. Hence, the exploration of imaginary numbers not only broadens one's mathematical toolkit but also enhances comprehension of more abstract mathematical concepts.