If [tex]i=\sqrt{-1}[/tex], then [tex]i^2=[/tex] ?
Answer (2)
We are given that i = − 1 .
We square both sides of the equation to find i 2 = ( − 1 ) 2 .
Simplifying, we get i 2 = − 1 .
Therefore, i 2 = − 1 .
Explanation
Understanding the Problem We are given that i = − 1 . We need to find the value of i 2 .
Squaring Both Sides To find i 2 , we simply square both sides of the equation i = − 1 . This gives us i 2 = ( − 1 ) 2 .
Simplifying Since squaring a square root cancels out, we have i 2 = − 1 .
Final Answer Therefore, the value of i 2 is − 1 .
Examples
Complex numbers, involving the imaginary unit i , are crucial in electrical engineering for analyzing AC circuits. They help represent impedance, which is the opposition to current flow. The relationship i 2 = − 1 allows engineers to perform calculations and understand the behavior of these circuits effectively. For example, when calculating power dissipation or voltage drops in AC circuits, complex numbers simplify the analysis and provide accurate results.
The value of i 2 is − 1 , derived from the definition of i as the square root of − 1 . This understanding is fundamental in mathematics, especially in complex number theory. The relationship i 2 = − 1 has significant applications in engineering and other fields.
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