Use complete sentences to describe why [tex]$\sqrt{-1} \neq-\sqrt{1}$[/tex].
Answer (2)
Define − 1 as i .
Evaluate − 1 as -1.
Explain that i is an imaginary number, while -1 is a real number.
State that imaginary numbers and real numbers are distinct, therefore − 1 = − 1 .
Explanation
Understanding the Problem The question asks us to explain why the square root of -1 is not equal to the negative square root of 1.
Defining the Square Root of -1 The square root of -1, denoted as − 1 , is defined as the imaginary unit, represented by the symbol i . In mathematical terms, we have: − 1 = i
Evaluating the Negative Square Root of 1 On the other hand, we are given − 1 . The square root of 1 is 1, so we have: − 1 = − 1
Comparing Imaginary and Real Numbers Now, we need to explain why i is not equal to -1. The number i is an imaginary number, which means it is a multiple of the imaginary unit. The number -1 is a real number. Imaginary numbers and real numbers are fundamentally different types of numbers.
Conclusion Since i is an imaginary number and -1 is a real number, they cannot be equal. Therefore, we can conclude that: − 1 = − 1
Examples
Imaginary numbers, like − 1 = i , might seem abstract, but they're crucial in electrical engineering for analyzing alternating current circuits. Real numbers, like − 1 = − 1 , are used for everyday measurements such as length or temperature. Understanding the difference between them is essential for solving problems in various fields of science and engineering.
The square root of -1 is defined as the imaginary unit i , while the negative square root of 1 equals -1. As i is an imaginary number and -1 is a real number, they cannot be equal. Thus, we conclude that − 1 = − 1 .
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