Question

Use complete sentences to describe why

√−1 ≠ √−1
a) The square root of -1 is imaginary, while -\sqrt{1} equals -1, so they are not equal.
b) Both expressions are equal.
c) The square root of -1 equals -1, while -\sqrt{1} is imaginary.
d) These expressions are equivalent in the complex number system.

Answer 1

The square root of -1 is represented as the imaginary unit i, while -√1 is -1, a real number; thus they are not equal. The correct answer is a) since these expressions represent different values, one imaginary and one real.

Step-by-step explanation:

The question involves understanding the concept of imaginary numbers in the complex number system. The expression √−1 refers to the imaginary unit, commonly represented by the symbol i. This imaginary unit denotes the square root of -1, which is not a real number.

On the other hand, -√1 is a different expression where the square root of 1, which is 1, is multiplied by -1, giving us -1, a real number. Therefore, the correct answer is a), as the two expressions are not equal: √−1 is imaginary, and -√1 equals -1, which is real.

In the complex number system, the imaginary unit is crucial for representing solutions to equations that do not have real solutions, and it is used in various fields within mathematics and engineering.