Problem 107 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The expression is \(2 + 3i\).

Step 1: Understand the Problem

The expression given is \(2 + \sqrt{-9}\). We need to express it in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, defined as \(\sqrt{-1}\).

Step 2: Simplify the Square Root of Negative Number

The square root of \(-9\) can be written as \(\sqrt{9} \cdot \sqrt{-1}\). Since \(\sqrt{9} = 3\) and \(\sqrt{-1} = i\), we have \(\sqrt{-9} = 3i\).

Step 3: Combine Terms

Now, substitute \(\sqrt{-9} = 3i\) back into the original expression to get \(2 + 3i\).

Step 4: Express in the Form \(a + bi\)

In the expression \(2 + 3i\), the real part \(a\) is 2 and the imaginary part \(b\) is 3. Therefore, the expression \(2 + \sqrt{-9}\) is in the form \(2 + 3i\).

Key Concepts

Imaginary Unit

The imaginary unit is a fundamental concept in complex numbers, represented by the symbol \(i\). It is defined as the square root of \(-1\).
The equation \(i^2 = -1\) is the foundation of working with complex numbers, allowing us to handle otherwise undefined operations like the square root of negative numbers.
When you encounter the imaginary unit, think of it as a tool that allows us to work with a broader range of numbers and solve equations that don't have solutions within the realm of real numbers.

Square Root of Negative Number

Understanding the square root of a negative number can initially be confusing since square roots are typically associated with positive values.
However, when you take the square root of a negative number, you utilize the imaginary unit \(i\).
For example, to find \(\sqrt{-9}\), you rewrite it as \(\sqrt{9} \cdot \sqrt{-1}\).
This simplifies to \(3i\) because \(\sqrt{9} = 3\) and \(\sqrt{-1} = i\).
This approach converts the problem from dealing strictly with negative numbers into one that involves imaginary numbers, expanding our ability to solve problems.

Real and Imaginary Parts

A complex number is generally expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.
In the expression \(2 + 3i\), \(2\) is the real part, and \(3i\) is the imaginary part.
This form is quite useful because it clearly separates the different components of a complex number.
When performing operations like addition or multiplication with complex numbers, always keep track of these parts separately to simplify your calculations effectively.
Recognizing real and imaginary parts can also simplify complex mathematical problems and help better understand their solutions.