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Chapter 10: Problem 1

Decide whether each expression is equal to \(1,-1, i, o r-i\) $$ \sqrt{-1} $$

Short Answer

Expert verified
i

Step by step solution

01

Identify the Expression

The given expression is \(\sqrt{-1}\).
02

Recognize the Definition

Recall that in complex numbers, \(\sqrt{-1}\) is defined as \(i\). This is a fundamental property of imaginary numbers.
03

Conclude the Result

Based on the definition, \(\sqrt{-1} = i\).

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Key Concepts

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

Imaginary Numbers
Imaginary numbers are a concept used in mathematics to extend the real number system. They are essential in solving equations that have no real solutions, such as \(x^2 + 1 = 0\).
The imaginary unit, denoted as \(i\), is defined by the property that \(i^2 = -1\). Therefore, imaginary numbers are multiples of \(i\), like \(2i, -3i\), and so forth.
By combining real and imaginary numbers, we get complex numbers, expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers.
  • Imaginary numbers solve problems that real numbers cannot.
  • They are an extension of the real number system.
  • Imaginary unit \(i\) is defined as \(i^2 = -1\).
Square Root of Negative One
The square root of negative one is a unique concept in mathematics. In the realm of real numbers, no number squared gives a negative result. However, in the world of complex numbers, we define \(\sqrt{-1}\) as \(i\).
This allows us to work with equations that involve the square root of negative numbers:
\

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Most popular questions from this chapter

Add or subtract as indicated. Write your answers in the form \(a+b i\) $$ (-1+i)+(2+5 i)+(3+2 i) $$

Find each quotient. $$ \frac{2-3 i}{2+3 i} $$

Find each quotient. $$ \frac{2}{1-i} $$

Find each power of \(i\) $$ i^{89} $$

Find each quotient. $$ \frac{-7+4 i}{3+2 i} $$
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