Problem 32 - Precalculus : Building Concepts and Connections

Question

Find the complex conjugate of each number. $$-2 i+4$$

Step-by-Step Solution

Answer

The complex conjugate of the number \(-2i + 4\) is \(4 + 2i\).

Step 1: Identify the terms of the complex number

The complex number presented in this exercise is \(-2i+4\), where \(4\) is the real part and \(-2i\) is the imaginary part.

Step 2: Apply the concept of complex conjugate

The complex conjugate of a number \(a + bi\) is \(a - bi\). Therefore, if we apply this to our complex number, we change the sign of the imaginary part, forming the conjugate as \(4 + 2i\).

Key Concepts

Complex Conjugate

In the world of complex numbers, the complex conjugate is an essential concept. A complex number is usually denoted as \(a + bi\), where \(a\) represents the real part, and \(bi\) is the imaginary part. The complex conjugate of \(a + bi\) is \(a - bi\). To find it, we simply change the sign of the imaginary part.

For example, consider the complex number \(-2i + 4\). Its real part is \(4\), and its imaginary part is \(-2i\). To find the complex conjugate, you keep the real part the same and change the sign of the imaginary part, turning \(-2i\) into \(2i\). So, the complex conjugate is \(4 + 2i\).

The importance of the complex conjugate comes in various applications, such as simplifying division of complex numbers and solving polynomial equations.

• Helps in rationalizing denominators when dividing complex numbers.
• Used in finding the magnitude (or modulus) of a complex number as \(z \times \overline{z} = a^2 + b^2\).
• In electrical engineering and physics, it helps work with phasors.

Real Part

The real part of a complex number is straightforward: it is the component without the imaginary unit \(i\). Consider the complex number \(a + bi\); here \(a\) is the real part. It behaves like a regular real number.

For the given example, \(-2i + 4\), the real part is \(4\). It functions just like any other real number you encounter in basic arithmetic. When analyzing complex numbers, identifying the real part can be crucial, especially when performing operations like addition, subtraction, or finding the complex conjugate.

• Real parts are used in pairing complex numbers with purely real numbers in certain operations.
• In polar form, the real part contributes to determining the magnitude and argument of the complex number.
• In graphical representations, it is plotted along the horizontal axis.

Imaginary Part

The imaginary part of a complex number consists of the coefficient \(b\) alongside the imaginary unit \(i\). In a generic complex number \(a + bi\), \(bi\) represents the imaginary component. Despite its name, it has real-world applications, especially in engineering and physics.

In our example \(-2i + 4\), the imaginary part is \(-2i\). To isolate the imaginary part, remove the \(i\), and you are left with \(-2\). When working with complex numbers, always remember that the imaginary part involves the unit \(i\), which signifies \(\sqrt{-1}\).

• The imaginary part is critical in frequency domain analysis in electrical engineering.
• It plays a role in calculating the phase or angle in polar coordinates.
• On the complex plane, the imaginary part is plotted along the vertical axis.