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The imaginary unit is a fundamental concept in complex numbers. It is represented by the symbol \(i\). The primary property of the imaginary unit is that \(i^2 = -1\). This means that when you square \(i\), you get \(-1\). This is the basis for understanding more complex algebraic expressions involving imaginary numbers. Imaginary numbers are used to extend the real number system to solve equations that do not have real solutions, such as \(x^2 + 1 = 0\).
Using the imaginary unit, we can express any complex number of the form \(a + bi\), where \(a\) and \(b\) are real numbers.

Complex numbers are numbers that have a real part and an imaginary part. They are usually written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Complex numbers are important because they expand our ability to solve algebraic equations.
Here are some key facts about complex numbers:

• The real part of the complex number \(a + bi\) is \(a\).
• The imaginary part of the complex number \(a + bi\) is \(b\).
• The conjugate of the complex number \(a + bi\) is \(a - bi\).
• The modulus (or absolute value) of a complex number \(a + bi\) is \(\sqrt{a^2 + b^2}\).

Complex numbers can be added, subtracted, multiplied, and divided, just like real numbers, but they follow specific rules that take into account both the real and imaginary parts.

Algebraic expressions involve variables, numbers, and arithmetic operations. When working with imaginary and complex numbers, algebraic expressions can become more intricate.
In the given example, we need to simplify \( (-i)^2 \). To do this, we use the properties of the imaginary unit and follow the correct order of operations:
1. Recognize that \( (-i)^2 = (-i) \times (-i) \).
2. Multiply \( (-i) \times (-i) \):
\[ (-i) \times (-i) = (-1) \times i \times (-1) \times i = (-1) \times (-1) \times (i \times i) = 1 \times (-1) = -1 \]
This clever manipulation reveals that \((-i)^2\) simplifies to \(-1\), demonstrating how algebraic expressions involving imaginary numbers often rely on recognizing and applying key properties.