Problem 24 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ (-4+2 i)+(7+35 i) $$

Step-by-Step Solution

Answer

The expression in standard form is \(3 + 37i\).

Step 1: Identify the Terms

Break down each complex number in the expression. We have two complex numbers: 1. \(-4 + 2i\)2. \(7 + 35i\)

Step 2: Combine Like Terms

Add the real parts and the imaginary parts separately. Real part: \(-4 + 7 = 3\)Imaginary part: \(2i + 35i = 37i\)

Step 3: Write in Standard Form

The standard form for a complex number is \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. So, the expression simplifies to: \(3 + 37i\)

Key Concepts

Standard Form

Complex numbers are written in a specific format known as the standard form. In this form, a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, which is defined as \(\sqrt{-1}\).
Writing a complex number in standard form allows you to easily identify the real and imaginary components.

• "\(a\)" represents the real part of the complex number.
• "\(bi\)" represents the imaginary part of the complex number, where \(b\) is a real number.

Using this form simplifies the manipulation of complex numbers, especially in algebra and calculus. When asked to write a complex number like \((-4 + 2i) + (7 + 35i)\) in standard form, your goal is to rearrange it into the format of \(a + bi\). This step generally involves operations like addition, subtraction, multiplication, or division of complex numbers.

Real and Imaginary Parts

In a complex number, such as \(a + bi\), understanding the distinction between the real and imaginary parts is crucial.
The real part is the component without the imaginary unit \(i\). In our example, this would be the numbers \(-4\) and \(7\).

• The combination of these real parts in the expression is calculated as \(-4 + 7 = 3\).

Meanwhile, the imaginary part includes the coefficient of the imaginary unit \(i\). Consider our numbers \(2i\) and \(35i\), which we sum together by adding their coefficients:

• The imaginary parts combine as \(2i + 35i = 37i\).

Grasping these components helps in processing complex number operations smoothly, as it lays the groundwork for subsequent algebraic manipulations.

Addition of Complex Numbers

Adding complex numbers involves combining their real parts and their imaginary parts separately. Let's look at the expression \((-4 + 2i) + (7 + 35i)\) as an example.
This addition occurs in two main steps:

• First, you add the real numbers: \(-4 + 7\), resulting in \(3\).
• Second, you add the imaginary numbers: \(2i + 35i\), resulting in \(37i\).

Once these calculations are done, the complex number is now in its simplest form, \(3 + 37i\).
This consistent method helps in maintaining clarity, especially when dealing with more complicated expressions involving multiple complex numbers. Remember the standard form \(a + bi\), as it assures that operations on complex numbers remain straightforward and easily interpretable.