Problem 4 Decide whether each expression i... [FREE SOLUTION]

91影视

Chapter 8: Problem 4

Decide whether each expression is equal to $1,-1, i,$ or $-i$ $$ -i^{2} $$

Short Answer

Expert verified

Step by step solution

Recall the definition of i

Remember that the imaginary unit, denoted as $i$, is defined by the property $i^2 = -1$.

Evaluate $i^2$

Using the definition $i^2 = -1$, substitute $i^2$ with $-1$.

Substitute $-1$ in the expression

Rewrite the original expression $-i^2$ as $-(-1)$.

Simplify the expression

Simplify $-(-1)$ which results in $1$.

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, represented as $i$, is a fundamental concept in complex numbers.
It is defined by the equation $i^2 = -1$.
This definition helps to extend the real numbers to the complex number system.
We can think of $i$ as a solution to the equation $x^2 = -1$, which has no solution within the real numbers.
Understanding $i$ is crucial for working with complex numbers.

complex number operations

Complex numbers involve operations that combine real and imaginary components.
A complex number has the form $a + bi$, where $a$ is the real part, and $bi$ is the imaginary part.
Here are some key operations:

**Addition**: $(a + bi) + (c + di) = (a + c) + (b + d)i$
**Subtraction**: $(a + bi) - (c + di) = (a - c) + (b - d)i$
**Multiplication**: $(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i$ (using $i^2 = -1$)
**Division**: $\frac{a + bi}{c + di}$ involves multiplying the numerator and denominator by the conjugate of the denominator to simplify.

Practice these operations to become comfortable with complex numbers.

evaluating expressions

Evaluating expressions involving complex numbers often requires substituting and simplifying using properties of $i$.
Let鈥檚 break down the given expression: $-i^2$

First, recall that $i^2 = -1$. This means you can replace $i^2$ with $-1$.
The expression $-i^2$ then becomes $-(-1)$.
Simplify the double negative: $-(-1) = 1$.

Therefore, the expression $-i^2$ evaluates to $1$.
This step-by-step approach can be applied to similar complex number problems. Remember to always substitute $i^2$ with $-1$ and follow through with simplification.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Decide whether each expression is equal to \(1,-1, i,\) or \(-i\) $$ -i^{2} $$

Short Answer

Step by step solution

Recall the definition of i

Evaluate \(i^2\)

Substitute \(-1\) in the expression

Simplify the expression

Key Concepts

imaginary unit

complex number operations

evaluating expressions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Logic and Functions

Calculus

Theoretical and Mathematical Physics

Decision Maths

Mechanics Maths

Study anywhere. Anytime. Across all devices.