Problem 77 - Precalculus Mathematics for Calculus

Question

Recall that the symbol \(\bar{z}\) represents the complex conjugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. \(z+\bar{z}\) is a real number.

Step-by-Step Solution

Answer

\(z + \bar{z} = 2a\) is a real number, as the imaginary parts cancel out.

Step 1: Identify the Complex Conjugate

The complex conjugate of a complex number \(z = a + bi\) is \(\bar{z} = a - bi\). This flips the sign of the imaginary part but keeps the real part the same.

Step 2: Express the Sum of z and its Conjugate

Compute \(z + \bar{z}\) by adding \(z = a + bi\) and \(\bar{z} = a - bi\). This gives \(z + \bar{z} = (a + bi) + (a - bi)\).

Step 3: Simplify the Expression

Combine like terms: \(z + \bar{z} = a + a + bi - bi\). The imaginary terms \(+bi\) and \(-bi\) cancel each other out, resulting in \(z + \bar{z} = 2a\).

Step 4: Conclude with Real Number

Notice that the result \(2a\) is a real number, since it consists purely of the real coefficient \(a\), with no imaginary part. Therefore, \(z + \bar{z}\) is indeed a real number.

Key Concepts

Complex Conjugate

In the world of complex numbers, each complex number is typically in the form of \(z = a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \(i^2 = -1\). The complex conjugate, denoted as \(\bar{z}\), is found by changing the sign of the imaginary part. So, the complex conjugate of \(z = a + bi\) is \(\bar{z} = a - bi\).

This operation has important mathematical properties which are useful in various computations:

• When you add a complex number and its conjugate, the imaginary parts cancel out.
• Multiplying a complex number by its conjugate results in a real number.
• Conjugation helps in division of complex numbers, as it simplifies the process of obtaining the modulus.

Using complex conjugates helps keep computations safe from non-real disturbances, making machinery and algorithms stable.

Real Numbers

Real numbers are just what they sound like - numbers that can be found on the number line. These include whole numbers, fractions, and decimals. In complex number algebra, the real part is crucial because it interacts with both the real and imaginary parts.

Here's a fun fact for dealing with complex numbers:

• If you add a number to its complex conjugate, the result is always a real number.
• For example, if you have \(z = 3 + 4i\), the complex conjugate \(\bar{z}\) would be \(3 - 4i\). Adding these results in a purely real number: \(z + \bar{z} = (3 + 4i) + (3 - 4i) = 6\).

Real numbers in complex equations remind us that even within the imaginary realm of mathematics, real, tangible numbers still play a pivotal role.

Imaginary Part

The imaginary part of a complex number is what sets it apart from real numbers. For any complex number \(z = a + bi\), the term \(bi\) represents the imaginary part, where \(b\) is a real number and \(i\) is the imaginary unit.

Here’s why the imaginary part is so interesting:

• The imaginary unit \(i\) is defined by the equation \(i^2 = -1\), which introduces a new dimension in mathematics that doesn't exist in real numbers.
• Although the imaginary part seems to be illusory, it is very real in many mathematical applications including engineering and physics.
• Adding or subtracting the imaginary parts of complex numbers is the key operation in maintaining the balance and determining their effects on a system.

In our exercise, when the complex number \(z\) and its conjugate \(\bar{z}\) were added, the imaginary parts \(+bi\) and \(-bi\) canceled each other, showing the power of imaginary numbers in balancing equations.