Problem 13 - Algebra and Trigonometry

Question

Find the real and imaginary parts of the complex number. $$ \sqrt{3}+\sqrt{-4} $$

Step-by-Step Solution

Answer

The real part is \( \sqrt{3} \) and the imaginary part is 2.

Step 1: Simplify the Imaginary Part

The given complex number is \( \sqrt{3} + \sqrt{-4} \). Recognize that \( \sqrt{-4} \) involves the imaginary unit \( i \), since \( \sqrt{-1} = i \). Thus, \( \sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i \). The complex number can now be expressed as \( \sqrt{3} + 2i \).

Step 2: Identify the Real Part

In the expression \( \sqrt{3} + 2i \), the term \( \sqrt{3} \) is the real part of the complex number because it does not involve the imaginary unit \( i \).

Step 3: Identify the Imaginary Part

In the expression \( \sqrt{3} + 2i \), the term \( 2i \) is the imaginary part of the complex number. Here, the coefficient of \( i \), which is \( 2 \), is considered the imaginary part.

Key Concepts

Real Part

When discussing complex numbers, we often refer to their components: the real part and the imaginary part. For a complex number expressed in the form of \(a + bi\), the "real part" is denoted by \(a\). It is the portion of the number without any association with the imaginary unit \(i\). In the problem you encountered, the expression \(\sqrt{3} + 2i\) represents a complex number made up of two distinct parts. The real part here is \(\sqrt{3}\).

Let's break down why \(\sqrt{3}\) is real:

• The term does not have the imaginary unit \(i\) multiplying it.
• It is a square root of a positive number, ensuring it is indeed a real number.

Understanding the real part helps clarify what portion of the complex number corresponds to traditional, non-imaginary quantities. This clarity is crucial for further operations, such as addition or multiplication with other complex numbers.

Imaginary Part

In any complex number, the imaginary part is identified by its association with the imaginary unit \(i\). In the standard form \(a + bi\), the part \(bi\) is regarded as the imaginary part, where \(b\) is a real number and \(i\) squares to \(-1\).

Taking the example of \(\sqrt{3} + 2i\), the imaginary part is \(2i\). More specifically, the coefficient \(2\) is what defines the magnitude of the imaginary part:

• The term involves the imaginary unit \(i\).
• The coefficient before \(i\) is a real number, determining the imaginary part's size.

This concept plays a key role when evaluating how complex numbers interact, especially in operations like addition and subtraction, where imaginary components are handled just like real parts but kept separate during computation.

Imaginary Unit

The imaginary unit, denoted as \(i\), forms the core of complex number arithmetic. By definition, \(i^2 = -1\). This property distinguishes imaginary numbers from real numbers, allowing mathematicians to handle square roots of negative numbers effectively.

In the given problem, when addressing \(\sqrt{-4}\), the imaginary unit comes into play:

• Recognize \(\sqrt{-4}\) as \(\sqrt{4} \times \sqrt{-1}\).
• This results in \(2 \cdot i\), turning the imaginary portion of the complex number into \(2i\).

Understanding the role of \(i\) is essential for making sense of negative square roots and expanding mathematical operations into the realm of complex numbers. It transforms an otherwise undefined operation into something that can be calculated and understood as part of a larger, complex structure.