Complex Numbers - Algebra-2

1. Fundamental Concepts

Imaginary unit i: Defined as , which means . It is used to represent the square root of negative numbers.
Square root of a negative real number: For any positive real number n, (e.g., ).

1: Identify the real part and the imaginary part of the complex number .
Answer: The real part is 7, and the imaginary part is -4 (where ).
2: Express in the form of an imaginary number.
Answer: .
3: Determine whether the following numbers are complex numbers and classify them: 0, , .
Answer: All are complex numbers. Among them, is a real number; is a pure imaginary number; is an imaginary number.

Distinguishing the real part and the imaginary part of a complex number: For a complex number , directly extract a (the real part) and b (the imaginary part). Note that b can be positive, negative, or zero.
Calculating the square root of a negative number: Use the formula () to convert the square root of a negative number into the product of the imaginary unit and the square root of a positive number.
Classifying complex numbers: Judge whether a complex number is a real number, an imaginary number, or a pure imaginary number based on whether the real part and the imaginary part are zero (a pure imaginary number has a zero real part and a non-zero imaginary part; a real number has a zero imaginary part; others are imaginary numbers).