Problem 9 - College Algebra and Calculus: An Applied Approach

Question

Write the complex number in standard form and find its complex conjugate. $$-3-\sqrt{-12}$$

Step-by-Step Solution

Answer

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

Step 1: Identify the real and complex part

The real part is -3 and the complex part is \(\sqrt{-12}\).

Step 2: Convert complex part into form of bi

We can rewrite \(\sqrt{-12}\) as \(\sqrt{12}*\sqrt{-1}\) which is equal to \(2\sqrt{3}*(-1)\) equals \(2\sqrt{3}i\). So the standard form of the complex number is \(-3 +2\sqrt{3}i\).

Step 3: Find the complex conjugate

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

Key Concepts

Standard Form

Complex numbers are usually expressed in what is known as "standard form," which is written as \(a + bi\). Here, \(a\) is the real part and \(b\) is the coefficient of the imaginary part \(i\). Understanding this form is crucial for dealing with complex numbers, as it allows you to clearly separate the real and imaginary components.
In the exercise, the real part is identified as \(-3\), and the initial complex part is \(\sqrt{-12}\). We convert \(\sqrt{-12}\) into the standard form by expressing it as \(2\sqrt{3}i\), allowing the complex number to be rewritten in standard form as \(-3 + 2\sqrt{3}i\).

• Identify real and imaginary components separately.
• Use \(i = \sqrt{-1}\) to transform imaginary parts.

Grasping standard form helps in analysis and computation with complex numbers. When looking at more advanced topics like polar coordinates or performing operations like addition and multiplication, this clear delineation between real and imaginary parts becomes very useful.

Complex Conjugate

A complex conjugate is a simple yet powerful concept in complex numbers. It involves changing the sign of the imaginary part while keeping the real part the same. For a complex number in the form \(a + bi\), the complex conjugate is \(a - bi\).
This operation is highly useful. For example, multiplying a complex number by its conjugate results in a real number, eliminating the imaginary part. In our example, the complex number \(-3 + 2\sqrt{3}i\) has the complex conjugate \(-3 - 2\sqrt{3}i\).

• Maintains the same real part.
• Switches the sign of the imaginary part.
• Useful for complex arithmetic, particularly division.

Understanding complex conjugates helps in mathematical operations, particularly when you need to simplify fracitons or solve equations involving complex numbers. By recognizing and utilizing complex conjugates, you can simplify complex arithmetic.

Imaginary Unit

The imaginary unit \(i\) is defined as \(i = \sqrt{-1}\). It is the foundation for working with imaginary numbers and, by extension, complex numbers. The presence of \(i\) allows us to work with square roots of negative numbers, something not possible in the realm of real numbers alone.
When solving the given exercise, we convert \(\sqrt{-12}\) into \(2 \sqrt{3} i\) by using the imaginary unit \(i\). This conversion helps in expressing non-imaginary terms as part of the complex number.

• Defined as \(i = \sqrt{-1}\).
• Makes it possible to express square roots of negative numbers.
• Integral in expressing numbers in complex form.

Having a grasp of the imaginary unit \(i\) is crucial for dealing with complex numbers, whether adding, subtracting, or finding the magnitude of these numbers. It forms the basis for all operations involving imaginary components.