Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Perform operations and write the result in standard form. $$5 i-(2-i)$$

Step-by-Step Solution

Answer

-2 + 6i in standard form.

Step 1: Understand the Expression

We are given the expression \(5i - (2 - i)\). Our goal is to simplify it and write it in standard form.

Step 2: Remove Parentheses

The first thing we do is remove the parentheses by distributing the negative sign to the terms within the parentheses:\[5i - (2 - i) = 5i - 2 + i\]

Step 3: Combine Like Terms

Next, we combine the like terms. In this case, we combine the imaginary terms and the real terms separately:Real part: \(-2\) Imaginary part: \(5i + i = 6i\)Thus, the expression becomes \(-2 + 6i\).

Step 4: Write in Standard Form

The standard form of a complex number is \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary part. We already have it in this form as \(-2 + 6i\).

Key Concepts

Imaginary Unit

Understanding the imaginary unit is a crucial part of working with complex numbers. The imaginary unit is denoted by the symbol \(i\). It is defined as the square root of -1, such that \(i^2 = -1\). This definition allows us to work with numbers that are not real, which we call imaginary numbers.
Imaginary numbers are used in conjunction with real numbers to form complex numbers. A complex number is typically written in the form \(a + bi\), where:

• \(a\) is the real part
• \(b\) is the imaginary part

Thus, any number involving \(i\) is part of this broader category of complex numbers. Understanding \(i\) and its properties helps simplify expressions, especially in the context of solving algebraic problems where negative square roots are involved.

Standard Form

The standard form of a complex number is expressed as \(a + bi\). Here, \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. This notation is essential because it organizes complex numbers in a consistent way, making operations like addition and subtraction straightforward. For example, the expression \(-2 + 6i\) ensures we know exactly what the real and imaginary parts are.
The key features of standard form include:

• The real part \(a\) might be any real number, including integers or fractions.
• The imaginary part \(bi\) is the product of a real number \(b\) and the imaginary unit \(i\).

Using standard form also aids in graphing complex numbers on a complex plane, where the horizontal axis represents the real part, and the vertical axis represents the imaginary part. This visualization helps in understanding complex number operations better.

Combining Like Terms

Combining like terms involves grouping terms in an expression that have the same variables to simplify the expression. In the context of complex numbers, this means separating and combining the real parts and the imaginary parts strictly.
For the expression \(5i - (2 - i)\), we:

• First distribute the negative sign: \(5i - 2 + i\)
• Then combine like terms to simplify it:

- Combine the imaginary parts: \(5i + i = 6i\)- The real number \(-2\) remains unpaired
This process results in the simplified expression: \(-2 + 6i\). By combining like terms, we ensure the expression is in its simplest form, fulfilling the requirement to express it in standard form as well. This step is vital because it reduces expressions to a cleaner, more understandable form, easing further calculations or interpretations.