Problem 51 - A Graphical Approach to College Algebra

Question

Add or subtract as indicated. Write each sum or difference in standard form. $$(2-5 i)-(3+4 i)-(-2+i)$$

Step-by-Step Solution

Answer

The sum or difference is \( 1 - 10i \).

Step 1: Distribute the Negative Signs

Start by distributing the negative signs in front of the parentheses: \( (2-5i)-(3+4i)-(-2+i) \). This becomes: \( 2-5i - 3 - 4i + 2 - i \).

Step 2: Combine Like Terms

Group the real parts and the imaginary parts together: Real parts: \( 2 - 3 + 2 \). Imaginary parts: \( -5i - 4i - i \).

Step 3: Simplify the Real Parts

Calculate the sum of the real parts: \( 2 - 3 + 2 = 1 \).

Step 4: Simplify the Imaginary Parts

Calculate the sum of the imaginary parts: \( -5i - 4i - i = -10i \).

Step 5: Write the Final Answer in Standard Form

Combine the simplified real and imaginary parts: \( 1 - 10i \). This is the sum or difference expressed in standard form.

Key Concepts

Standard Form

In the world of complex numbers, the standard form is a way of expressing a complex number as the sum of a real part and an imaginary part. It's usually written in the format: \( a + bi \), where \( a \) represents the real part, and \( bi \) represents the imaginary part.

The idea behind using the standard form is to clearly separate and identify the real and imaginary components of a number. Doing so makes it easier to perform operations like addition, subtraction, multiplication, and division with complex numbers.

For the exercise given, the final expression is \( 1 - 10i \). Here, 1 is the real part, and \( -10i \) is the imaginary part, fitting perfectly into the standard form framework where each component is explicitly highlighted. This clarity is why expressing answers in standard form is a fundamental practice when dealing with complex numbers.

Imaginary Parts

Imaginary parts of complex numbers are an integral aspect of the standard form. They involve the imaginary unit \( i \), which is defined as \( i = \sqrt{-1} \). This means that \( i^2 = -1 \), a key property used in complex arithmetic.

In our example, you initially start with several imaginary components: \(-5i\), \(-4i\), and \(i\). A critical step is to combine these to streamline the expression. Adding these imaginary parts together involves simple arithmetic:

• First, combine: \(-5i - 4i = -9i\).
• Then, add \(-i\) to \(-9i\), resulting in \(-10i\).

This reduction simplifies handling and follows suit in finding the standard form through separating these from any real components. Combining imaginary parts is akin to combining terms like conventional algebra, keeping track of the signs and coefficients along the way.

Real Parts

Real parts in a complex number are simply the components that do not include the imaginary unit \( i \). When manipulating complex numbers, you'll want to treat these just like regular integers.

In the described exercise, you begin with the real numbers 2, -3, and another 2. To simplify:

• Calculate step by step: \(2 - 3 = -1\).
• Then, add \(2\) to \(-1\), yielding \(1\).

Simplifying the real parts involves basic addition or subtraction. Although it's part of a more complex number, treating the real parts separately makes it manageable.

Understanding how to work with real parts is crucial because it directly influences the representation of complex numbers in the standard form \( a + bi \), where \( a \) is the result of simplifying all the real components. This distinctly separates them from the imaginary parts when forming the final answer.