Problem 93 - Algebra A Combined Function

Question

Write each expression in the form \(a+b i .\) $$ 2+\sqrt{-9} $$

Step-by-Step Solution

Answer

The expression simplifies to \(2 + 3i\).

Step 1: Separate Real and Imaginary Parts

The expression given is \(2 + \sqrt{-9}\). First, we acknowledge that the number we are dealing with involves a square root of a negative number, which is complex. The real part is already \(2\) and we need to address the \(\sqrt{-9}\), which is the imaginary part.

Step 2: Simplify the Imaginary Part

To simplify \(\sqrt{-9}\), recognize that \(-9 = -1 \times 9\), therefore, \(\sqrt{-9} = \sqrt{-1} \times \sqrt{9}\). Since \(\sqrt{-1} = i\) (the imaginary unit), and \(\sqrt{9} = 3\), it follows that \(\sqrt{-9} = 3i\).

Step 3: Combine Real and Imaginary Components

Now that we have the real part as \(2\) and the imaginary part as \(3i\), we can combine these parts into the standard form of a complex number, \(a + bi\). Thus, the expression \(2 + \sqrt{-9}\) simplifies to \(2 + 3i\).

Key Concepts

Imaginary Unit

Complex numbers often involve the imaginary unit, denoted as \( i \). The imaginary unit is defined as \( i = \sqrt{-1} \). This definition is fundamental because it allows us to work with the square roots of negative numbers.
Using \( i \) helps simplify expressions that involve negative square roots. For example, in the expression \( \sqrt{-9} \), we can split it into \( \sqrt{-1} \times \sqrt{9} \). Here, \( \sqrt{-1} \) becomes \( i \), and \( \sqrt{9} \) equals \( 3 \). Thus, \( \sqrt{-9} = 3i \).
The imaginary unit is essential in the field of complex numbers, enabling us to handle "imaginary" aspects seemingly without real-world equivalents. This allows mathematicians and engineers to solve problems otherwise limited by traditional real numbers.

Real Numbers

Real numbers are the numbers you typically encounter in everyday life. They include all positive and negative numbers, zero, integers, fractions, and all points on an infinitely extended number line.
In the context of complex numbers, the real part of a complex number is simply a real number. For example, in the expression \( 2 + \sqrt{-9} \), the real part is 2. This part doesn't involve any \( i \), as it represents a value on the standard number line.
When working with complex numbers, identifying the real part is key to rewriting and simplifying these numbers. Any complex number can be written in the form \( a + bi \), where \( a \) is the real part.

Imaginary Numbers

Imaginary numbers are built on the imaginary unit \( i \). These numbers have a component that involves \( i \), distinguishing them from real numbers.
In our example, the component \( \sqrt{-9} \) resulted in \( 3i \), illustrating the idea of imaginary numbers. Imaginary numbers can be confusing initially, as they don't have a direct real-world parallel. However, they are crucial in solving certain equations and modeling phenomena in physics and engineering.
Imaginary numbers are paired with real numbers to form complex numbers like \( 2 + 3i \). Understanding how to separate and combine these parts is fundamental to working with complex numbers effectively.