/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Find the indicated sums and diff... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 8

Find the indicated sums and differences of complex numbers. $$(-1+6 i)+(5-4 i)$$

Short Answer

Expert verified
The sum is \(4 + 2i\).

Step by step solution

01

Identify the Real Parts

The real part of \(-1+6i\) is \(-1\) and the real part of \(5-4i\) is \(5\).
02

Identify the Imaginary Parts

The imaginary part of \(-1+6i\) is \(6i\) and the imaginary part of \(5-4i\) is \(-4i\).
03

Add the Real Parts

Add the real parts together: \(-1 + 5 = 4\).
04

Add the Imaginary Parts

Add the imaginary parts together: \(6i - 4i = 2i\).
05

Combine the Results

Combine the results from Step 3 and Step 4 to get the final sum: \(4 + 2i\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Adding Complex Numbers
When adding complex numbers, we treat both parts - real and imaginary - separately. A complex number is usually written in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. To add two complex numbers, such as \((-1 + 6i)\) and \((5 - 4i)\), follow these steps:
  • First, identify and add the real parts.
  • Then, identify and add the imaginary parts.

This method helps streamline the process and makes it easier to avoid mistakes.
Imaginary Part
The imaginary part of a complex number is the component that involves the imaginary unit \(i\). In our example, for the complex number \(-1 + 6i\), the imaginary part is \(6i\). For \(5 - 4i\), it is \(-4i\). Imaginary units are critical in performing operations with complex numbers. When adding imaginary parts together, remember to only combine the coefficients of \(i\). For instance, in step 4 of our solution, we added \(6i - 4i\) to get \(2i\).
  • Always treat \(i\) as a separate entity and just focus on the numerical coefficients when combining.
Real Part
The real part of a complex number is the component that does not involve the imaginary unit \(i\). For example, in \(-1 + 6i\), the real part is \(-1\), and in \(5 - 4i\), it is \(5\). To add real parts, simply treat them as regular numbers. In step 3 of our step-by-step solution, we added the real parts: \(-1 + 5\), resulting in \(4\).
  • This step examines only the numeric value without the imaginary unit.
  • Focus on these values when performing operations to ensure accurate addition.
Combining Results in Complex Numbers
After separately adding the real and imaginary parts of complex numbers, the final step is to combine these results into one expression. From our example, we had found the real part to be \(4\) and the imaginary part to be \(2i\). Combining these results gives the final complex number \(4 + 2i\).
  • This combined result represents the sum of the initial complex numbers.
  • Ensure you correctly combine the parts for accurate results.
By following these steps and understanding each part individually, the process of adding complex numbers becomes simpler and more systematic.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Simplify. $$\frac{\sqrt{h k}}{\sqrt{h}+3 \sqrt{k}}$$

Solve each problem by writing an equation and solving it. Find the exact answer and simplify it using the rules for radicals. If the sides of a rectangle are 30 feet and 40 feet in length, find the length of the diagonal of the rectangle.

Solve each equation. $$(4 y-5)^{7}=0$$

Solve each problem. Find the exact area of a rectangle that has a length of \(\sqrt{6}\) feet and a width of \(\sqrt{3}\) feet.

Simplify. $$\frac{\sqrt{x}}{\sqrt{x}-3}+\frac{5}{\sqrt{x}}$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Geometry

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.