Question

In Problems \(87-90, z=3-4 i\) and \(w=8+3 i .\) Write each expression in the standard form \(a+b i .\) $$ z+\bar{z} $$

Short answer

The expression in standard form is 6.

Step-by-step solution

Identify the values of z and its conjugate

Given the complex number \(z = 3 - 4i\), identify its conjugate. The conjugate of \(z\) is \(\bar{z} = 3 + 4i\).

Set up the expression

Write the expression \(z + \bar{z}\) with the identified values of \(z\) and \(\bar{z}\): \(z = 3 - 4i\) and \(\bar{z} = 3 + 4i\).

Add the expressions

Combine the real parts and the imaginary parts separately: \[(3 - 4i) + (3 + 4i) = (3 + 3) + (-4i + 4i)\]

Simplify the expression

Simplify the result: \[6 + 0i = 6\]

Key concepts

complex conjugate

The complex conjugate of a complex number is found by changing the sign of the imaginary part. For example, if we have a complex number in the form of \(z = a + bi\), its conjugate, denoted as \(\bar{z}\), will be \(a - bi\).

Let's take the given problem where \(z = 3 - 4i\). The conjugate of \(z\) is \(\bar{z} = 3 + 4i\). Notice how only the sign of the imaginary part has changed from \(-4i\) to \(4i\) while the real part remains the same.

Understanding how to identify the conjugate is essential for various operations in complex number arithmetic.

standard form

Standard form for a complex number is written as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit satisfying \(i^2 = -1\).

In the given problem, we see that both \(z = 3 - 4i\) and \(\bar{z} = 3 + 4i\) are already in standard form, where \(a = 3\) is the real part and \(b = -4\) or \(b = 4\) is the imaginary part.

Writing a complex number in standard form helps to clearly separate the real and imaginary components, making computations more straightforward.

real part

The real part of a complex number is the component that is not multiplied by \(i\). In our examples, \(z = 3 - 4i\) and \(\bar{z} = 3 + 4i\), the real part is \(3\).

When performing arithmetic operations involving complex numbers, adding or subtracting the real parts is crucial. In the provided example, when we add \(z + \bar{z}\), we add their real parts separately: \((3 + 3) = 6\).

Recognizing the real part will simplify processes such as addition, subtraction, and finding magnitudes of complex numbers.

imaginary part

The imaginary part of a complex number is the component that is multiplied by \(i\). For \(z = 3 - 4i\) and \(\bar{z} = 3 + 4i\), the imaginary parts are \(-4i\) and \(4i\), respectively.

When adding complex numbers, their imaginary parts should also be combined. For our specific example \(z + \bar{z}\), adding the imaginary parts \((-4i + 4i)\) results in \(0i\).

This demonstrates that when you add a complex number to its conjugate, the imaginary parts always cancel out, leaving only the real part of the result. Thus, understanding the imaginary part is important for simplifying and accurately performing complex number operations.