/*! 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 5 Write the given number in the fo... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 5

Write the given number in the form \(a+i b\). $$ i(5+7 i) $$

Short Answer

Expert verified
The expression simplifies to \(-7 + 5i\).

Step by step solution

01

Distribute the Imaginary Unit

To solve the expression, expand it by distributing the imaginary unit \(i\) across the terms inside the parenthesis. So, multiply \(i\) with each component: \[i(5+7i) = i\times 5 + i \times 7i.\]
02

Simplify Each Term

Simplify the expression from Step 1. Start with \(i\times 5\), which simplifies to \(5i\). Next, simplify \(i\times 7i\). Remember that \(i^2 = -1\), so \(7i^2 = 7(-1) = -7\). This gives us: \[5i + 7i^2 = 5i - 7.\]
03

Write in the Form \(a+ib\)

Now, rewrite the expression \(5i - 7\) by expressing it in the standard form of a complex number, which is \(a + ib\). The real part \(a\) is \(-7\), and the imaginary part \(b\) is \(5\). The final expression is: \[-7 + 5i.\]

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.

Imaginary Unit
The imaginary unit, denoted as \(i\), is a fundamental concept in the realm of complex numbers. It is defined as the square root of
  • -1.
This means that \(i^2 = -1\). The concept may seem a bit abstract because no real number squared gives a negative result. However, it is a powerful tool that extends the number system to include solutions to equations that could not previously be solved. For example, the equation \(x^2 + 1 = 0\) has no real solutions, but within the complex plane, the solutions are \(x = i\) and \(x = -i\). Using \(i\) allows us to handle numbers previously thought of as unsolvable, fostering domains such as engineering, physics, and beyond.
Complex Plane
The complex plane is a unique way of representing complex numbers visually. Think of it as a two-dimensional plane, much like a graph. In this plane:
  • The horizontal axis (real axis) represents the real part of complex numbers.
  • The vertical axis (imaginary axis) represents the imaginary part of complex numbers.
For example, a complex number \(z = a + bi\) would be plotted with \(a\) units along the horizontal axis and \(b\) units along the vertical axis. This visual representation not only helps in understanding complex numbers as coordinates \((a, b)\) but also aids in performing mathematical operations like addition, subtraction, and locating the magnitude of these numbers. The complex plane serves as a valuable tool in various mathematical and applied science fields, facilitating a geometric interpretation of complex numbers.
Standard Form of Complex Numbers
The standard form of a complex number is \(a + ib\), where:
  • \(a\) is the real part.
  • \(b\) is the imaginary part.
  • \(i\) is the imaginary unit.
This form allows for easy identification and operation with complex numbers. For any complex number expressed in this way, operations like addition or multiplication become straightforward. For instance, adding two complex numbers \((a+ib)\) and \((c+id)\) results in
  • \((a+c) + i(b+d)\).
In the exercise, we took the expression \(5i - 7\) and rearranged it using the standard form, giving us \(-7 + 5i\). This explicit format ensures consistency in calculations and clarity when communicating mathematical ideas concerning complex numbers.

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

Use a \(\mathrm{CAS}^{\S}\) to first find \(z^{n}=w\) for the given complex number and the indicated value of \(n\). Then, using the output and the same value of \(n\), determine whether \(w^{1 / n}=\left(z^{n}\right)^{1 / n}=z .\) If not, explain why not. $$ -0.5+0.3 i ; n=5 $$

Sketch the graph of the given equation in the complex plane. $$ \operatorname{Re}(z)=5 $$

Discuss and decide whether the empty set \(\varnothing\) is an open set.

Factor the given quadratic polynomial if the indicated complex number is one root. $$ 4 z^{2}+(-13+18 i) z-5-10 i ; z_{1}=3-4 i $$

Solve the given quadratic equation using the quadratic formula. Then use (5) to factor the polynomial. $$ 3 z^{2}+(2-3 i) z-1-3 i=0 $$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Logic and Functions

Read Explanation

Geometry

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.