Problem 27 - Precalculus : Building Concepts and Connections

Question

Find the real and imaginary parts of the complex number. $$1+\sqrt{-5}$$

Step-by-Step Solution

Answer

The complex number \(1+ \sqrt{-5}\) has a real part of \(1\) and an imaginary part of \(i\sqrt{5}\).

Step 1: Identifying the real part of the complex number

The real part of the complex number is the component without the square root of a negative number. Here, it is \(1\).

Step 2: Simplifying the square root of a negative number

The square root of a negative number is imaginary. We know that \(\sqrt{-1} = i\). Hence, we can rewrite the \(\sqrt{-5}\) as \(i\sqrt{5}\).

Step 3: Assembling the components

Combine the real and imaginary parts identified in Steps 1 and 2. The complex number \(1+\sqrt{-5}\) is represented as \(1 + i\sqrt{5}\).

Key Concepts

Real and Imaginary Parts of Complex Numbers

Understanding the real and imaginary components of a complex number is vital to grasp the foundation of complex number algebra. Complex numbers include both a real part and an imaginary part. Essentially, the real part is the portion without the imaginary unit 'i', which is the square root of -1. In the case of the complex number 1 + \(\sqrt{-5}\), the real part is simply 1.

The imaginary part is typically associated with the imaginary unit 'i' and represents the square root of a negative number. In our example, the imaginary part comes from the term \(\sqrt{-5}\). The key to understanding complex numbers is recognizing these two parts and how they combine to form the entire number.

Simplifying Square Roots of Negative Numbers

When you encounter a square root of a negative number, it signifies the presence of an imaginary number. By definition, the imaginary unit 'i' is equal to \(\sqrt{-1}\), so simplifying square roots of negative numbers involves factoring out this 'i'. To simplify \(\sqrt{-5}\), observe that it is the same as \(\sqrt{-1}\) times \(\sqrt{5}\), which simplifies to \(i\sqrt{5}\).

Simplifying such expressions is crucial for working with complex numbers because it allows you to clearly identify and separate the real and imaginary parts. This clarity is not only important for basic arithmetic operations with complex numbers but also for more advanced applications in mathematics and engineering.

Complex Number Representation

A complex number can be represented in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, multiplied by the imaginary unit 'i'. Our example, after simplification, yields \(1 + i\sqrt{5}\). This format is standard for complex numbers and allows for straightforward addition, subtraction, and even multiplication or division with other complex numbers.

It is important to be comfortable with this representation, as it is the cornerstone of complex number arithmetic. With practice and understanding of real and imaginary components, as well as simplification techniques, working with complex numbers becomes an intuitive process. This familiarity is also essential for delving into deeper topics like complex conjugates, absolute values, and polar representation of complex numbers.