/*! 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 30 Write each number in the form a ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 30

Write each number in the form a + bi. See Example 2. a. \(-43\) b. \(\sqrt{-169}\)

Short Answer

Expert verified
a. \(-43 = -43 + 0i\); b. \(\sqrt{-169} = 0 + 13i\).

Step by step solution

01

Express Real Number in Complex Form (Part a)

To express \(-43\) in the form \(a + bi\), we recognize that this number is purely real, with no imaginary part. Thus, \(-43\) can be written as \(-43 + 0i\).
02

Express Imaginary Number (Square Root) in Complex Form (Part b)

Now, consider the square root of a negative number, \(\sqrt{-169}\). Since square roots of negative numbers involve imaginary numbers, we rewrite \(\sqrt{-169}\) as \(\sqrt{169} \times \sqrt{-1}\). This simplifies to \(13i\), since \(\sqrt{169} = 13\) and \(\sqrt{-1} = i\). Therefore, \(\sqrt{-169} = 0 + 13i\).

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.

Real Numbers
Real numbers are the numbers that we use in everyday life. They include whole numbers like 0, 1, 2, negative numbers like -1, -2, and fractions or decimals like 1.5 or 3/4.
Real numbers can be plotted on a number line, making it easy to visualize their position relative to each other.
One unique aspect of real numbers is that they do not include imaginary units such as 'i'. When we express any real number, whether positive or negative, in complex form, it looks like this: \(-43 = -43 + 0i\).
This shows that its imaginary part is zero, hence it lies entirely on the real number line.
  • Composed of both rational numbers (fractions) and irrational numbers (numbers that cannot be expressed as simple fractions, such as 蟺 or 鈭�2).
  • No imaginary component; only the 'a' part in 'a + bi' form is present.
  • Essentially just like the numbers you grew up using with a small twist to include a potential, but zero, imaginary component.
Imaginary Numbers
Imaginary numbers introduce an exciting twist to our otherwise straightforward number system.
These are numbers that, when squared, produce a negative result鈥攊ncluding numbers like \(i\), the widely-used imaginary unit.The imaginary unit \(i\) is defined by the property \(i^2 = -1\).
This means that \(i\) is the square root of \(-1\).
Thus, when dealing with square roots of negative numbers, as we see in equations like \(\sqrt{-169}\), imaginary numbers naturally arise.A number purely imaginary will look like \(bi\) in complex form, where b is nonzero and a is zero.
  • Used to extend the real number system to include solutions to equations that have no real solutions.
  • Allow for the expression of negative square roots which otherwise wouldn鈥檛 make sense in real numbers alone.
  • Introduces the concept of rotational vectors and more advanced mathematics beyond basic algebra.
Square Roots of Negative Numbers
Square roots of negative numbers can seem perplexing because no real number squared gives a negative result.
This is where the concept of imaginary numbers becomes useful.For example, \(\sqrt{-169}\) would ordinarily be unsolvable in terms of real numbers alone.
By using imaginary numbers, however, we reframe the problem: rewrite \(\sqrt{-169}\) as \(\sqrt{169} \times \sqrt{-1}\).
As such, it simplifies to \(13i\), with \(13\) from the \(\sqrt{169}\) and \(i\) because \(\sqrt{-1}\) equals \(i\).This solution demonstrates how every square root of a negative number can be expressed in terms of an imaginary unit.
  • These roots reveal the power of the imaginary unit 'i' as it transforms impossible problems in the real number world into solvable ones in the complex number system.
  • Vital in electrical engineering, quantum physics, and various fields leveraging complex numbers.
  • Allows the unique expression of complex solutions in a mathematically consistent manner.

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 each expression. All variables represent positive real numbers. $$ t^{4 / 3}\left(t^{5 / 3}+t^{-4 / 3}\right) $$

Simplify each expression. All variables represent positive real numbers. See Example 4. $$ -\left(\frac{x^{5}}{32}\right)^{4 / 5} $$

Simplify each radical expression, if possible. Assume all variables are unrestricted. $$ -\sqrt[5]{-243} $$

a. \(\sqrt{16}\) b. \(\sqrt[4]{16}\)

Use a calculator to solve each problem. Round answers to the nearest tenth. Shoelaces. The formula \(S=2[H+L+(p-1) \sqrt{H^{2}+V^{2}}]\) can be used to calculate the correct shoelace length for the criss-cross lacing pattern shown in the illustration, where \(p\) represents the number of pairs of eyelets. Find the correct shoelace length if \(H\) (horizontal distance) \(=50 \mathrm{mm}, L\) (length of end) \(=250 \mathrm{mm},\) and \(V(\) vertical distance \()=20 \mathrm{mm} .\) Round to the nearest tenth.
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

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.