Problem 9 Decide whether each expression i... [FREE SOLUTION]

Chapter 10: Problem 9

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

Short Answer

Expert verified

i^2 = -1

Step by step solution

Understanding the Problem

The task is to determine whether the expression $i^2$ is equal to 1, -1, i, or -i. Here, $i$ is the imaginary unit, where $i^2 = -1$.

Recall the Definition

Recall that by definition of the imaginary unit, $i^2 = -1$.

Analyze the Given Expression

Since $i^2$ is given in the question, directly applying the definition, we have $i^2 = -1$.

Conclusion

Therefore, according to the definition, $i^2$ equals $-1$.

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Key Concepts

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

Complex Numbers

Complex numbers are numbers that include a real part and an imaginary part. They are usually expressed in the form of $ a + bi $, where $ a $ is the real part and $ bi $ is the imaginary part. The imaginary unit $ i $ is defined as $ i = \sqrt{-1} $. This means that $ i^2 = -1 $. Complex numbers are essential in many areas of mathematics because they allow for solutions to equations that do not have real solutions, such as quadratic equations with negative discriminants. Understanding complex numbers can greatly expand your mathematical toolkit.

Powers of i

The powers of the imaginary unit $ i $ follow a cyclic pattern due to its definition $ i = \sqrt{-1} $. Here鈥檚 how it works:

$ i^0 = 1 $
$ i^1 = i $
$ i^2 = -1 $
$ i^3 = -i $
$ i^4 = 1 $

Once you reach $ i^4 = 1 $, the cycle repeats. This cyclic behavior is helpful when simplifying expressions involving higher powers of $ i $. For instance, to simplify $ i^6 $, you can think of it as $ i^{(4 + 2)} $, which equals $ i^4 \times i^2 $. Knowing that $ i^4 = 1 $ and $ i^2 = -1 $, you get $ i^6 = 1 \times -1 = -1 $.

Algebraic Expressions

Algebraic expressions involving complex numbers often require you to manipulate and simplify using the properties of $ i $. When working with these expressions:

Combine like terms: Add or subtract the real parts and the imaginary parts separately.
Simplify powers of $ i $: Use the cyclic pattern of powers of $ i $ to reduce higher powers to one of $ i, -1, -i, \text{or} 1 $.
Use distributive property: When multiplying, distribute as you would with binomials, and then simplify using $ i^2 = -1 $.

For example, in the expression $ (2 + 3i)(1 - i) $, distribute to get $ 2(1 - i) + 3i(1 - i) = 2 - 2i + 3i - 3i^2 $. Since $ i^2 = -1 $, this simplifies to $ 2 - 2i + 3i + 3 = 5 + i $. Remember, practicing these steps will make working with algebraic expressions involving $ i $ more intuitive.

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91影视

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

Short Answer

Step by step solution

Understanding the Problem

Recall the Definition

Analyze the Given Expression

Conclusion

Key Concepts

Complex Numbers

Powers of i

Algebraic Expressions

One App. One Place for Learning.

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