Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Write the complex number in standard form and find its complex conjugate. $$-6 i+i^{2}$$

Step-by-Step Solution

Answer

The complex number in standard form is \(-1 - 6i\) and its complex conjugate is \(-1 + 6i\).

Step 1: Simplify the expression

Firstly, we know that \(i^{2} = -1\). Thus, by substituting this into the given expression, we get \(-6i + (-1)\). This simplifies to \(-1 - 6i\).

Step 2: Write in standard form

The standard form of a complex number is \(a+bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. After simplifying, the complex number is already in standard form as \(-1 - 6i\), where \(a = -1\) and \(b = -6\).

Step 3: Find the complex conjugate

The complex conjugate of a complex number is obtained by changing the sign of its imaginary part. If our complex number is \(-1 - 6i\), then its complex conjugate will be \(-1 + 6i\).

Key Concepts

Standard Form

Complex numbers can often seem intimidating, but once you understand their structure, they become easy to work with. A complex number is traditionally expressed in what's called the "standard form": \(a + bi\). Here:

• \(a\) is the real part of the complex number.
• \(b\) is the factor of the imaginary part, which is attached to the imaginary unit \(i\).

For example, if we consider the expression \(-1 - 6i\), it's already in standard form. In this case:

• The real part \(a\) is \(-1\).
• The imaginary part \(b\) is \(-6\).

The standard form allows us to easily navigate operations like addition, subtraction, and visualization on the complex plane.
Breaking down more complicated expressions into standard form can often simplify calculations, helping you easily manipulate and understand complex numbers.

Complex Conjugate

A powerful tool in the realm of complex numbers is the "complex conjugate," which helps in simplification and division. The complex conjugate of a number is found by changing the sign of its imaginary part.
For a complex number like \(-1 - 6i\), the complex conjugate is simply \(-1 + 6i\).
This process is straightforward:

• Take the original complex number.
• Flip the sign of the imaginary part.

Creating the conjugate helps in various mathematical operations, especially when dividing complex numbers.
Multiplying a complex number by its conjugate can be used to eliminate the imaginary component, leaving us with a real number—an invaluable step in many calculations. In our case, multiplying \(-1 - 6i\) with its conjugate \(-1 + 6i\), yields a real number.

Imaginary Unit

At the heart of complex numbers lies the imaginary unit, denoted by \(i\). This small letter holds a very particular property: \(i^2 = -1\).
You might wonder why this property matters. Well, it allows us to extend numbers beyond the real number line in a logical way.

• \(i\) introduces a dimension perpendicular to the real numbers.
• It allows representation of numbers that are considered "imaginary," hence opening new ways of calculation and applications.

Imaginary units are crucial in expressing numbers that involve square roots of negative numbers, which aren't otherwise possible within the scope of real numbers.
In calculations, like expressing \(-6i + i^2\), understanding \(i^2 = -1\) is key to simplifying expressions to their real and imaginary parts. Knowing this property, you can tackle complex expressions with confidence, as seen in converting the expression to \(-1 - 6i\), aiding understanding and calculation efficiency.