Problem 33 - Precalculus : Building Concepts and Connections

Question

Find the complex conjugate of each number. $$3+\sqrt{2}$$

Step-by-Step Solution

Answer

The complex conjugate of the number \(3 + \sqrt{2} i\) is \(3 - \sqrt{2} i\).

Step 1: Identify the complex number

The given number is \(3+ \sqrt{2} i\). Here, \(a=3\) is the real part and \(b= \sqrt{2}\) is the imaginary part

Step 2: Applying complex conjugate formula

The complex conjugate of a complex number \(a+bi\) is \(a-bi\). So, replace \(a=3\) and \(b=\sqrt{2}\) which yields the complex conjugate as \(3 - \sqrt{2} i\).

Key Concepts

Complex Numbers

A complex number is a fundamental concept in mathematics that extends the idea of a one-dimensional number line to a two-dimensional plane, commonly referred to as the complex plane. It consists of two parts: a real part and an imaginary part.
Complex numbers are usually expressed in the form \(a + bi\), where \(a\) is the real component, and \(bi\) is the imaginary component. In this expression, \(i\) is the imaginary unit, defined as the square root of \(-1\).

• The real part, \(a\), can be any real number.
• The imaginary part, \(b\), can also be any real number, but is multiplied by \(i\).

Understanding complex numbers is essential for solving equations that don't have real solutions. They are crucial in various fields including engineering, physics, and advanced mathematics to represent phenomena that cannot be captured by real numbers alone.

Imaginary Part

The imaginary part of a complex number plays a significant role in its structure. In the standard representation \(a + bi\), \(bi\) denotes this part. The inclusion of the imaginary unit \(i\) allows us to work with square roots of negative numbers.
In the complex number \(3 + \sqrt{2}i\), \(\sqrt{2}i\) is the imaginary component:

• \(b\) is the coefficient of \(i\), which is \(\sqrt{2}\) in this case.
• This coefficient is a real number.

The imaginary part does not exist independently in reality (hence the term imaginary), but it provides a complete framework alongside the real part to solve complex problems. Key to complex operations like finding roots and in electrical engineering for AC circuit analysis, the imaginary component helps visualize a whole new dimension of mathematical possibilities.

Real Part

The real part of a complex number is the component that exists on the real number line, making it straightforward to comprehend. In a complex number \(a + bi\), the term \(a\) represents the real portion.In our example, \(3 + \sqrt{2}i\), the real part is simply \(3\):

• This is often straightforward as it operates like any other real number.
• The real part is crucial when plotting complex numbers on the complex plane, as it determines the horizontal position on the real axis.

Understanding the real part is vital for operations like addition, subtraction, and magnification of complex numbers. It forms one-half of the components needed to form complex numbers. Insights into the real part give us the ability to navigate the complex number system confidently and to bridge the gap between real-life applications and theoretical mathematics.