Add and Subtract Complex Numbers - Algebra-2

1. Fundamental Concepts

The addition and subtraction of complex numbers essentially involve performing corresponding addition and subtraction operations on the real and imaginary parts of the complex numbers separately.
Since the standard form of a complex number is (where a is the real part and b is the imaginary part), when adding or subtracting, like terms (real parts with real parts, imaginary parts with imaginary parts) need to be combined for calculation.

Addition Rule: For two complex numbers and , their sum is: That is, the sum of the real parts serves as the real part of the result, and the sum of the coefficients of the imaginary parts serves as the coefficient of the imaginary part of the result.
Subtraction Rule: For two complex numbers and , their difference is: That is, the difference of the real parts serves as the real part of the result, and the difference of the coefficients of the imaginary parts serves as the coefficient of the imaginary part of the result.
Closure Property of Operations: The result of adding or subtracting two complex numbers is still a complex number.

Easy difficulty: Calculate
Solution: Add the real parts , add the coefficients of the imaginary parts .
The result is .
Medium difficulty: Calculate
Solution: Subtract the real parts , subtract the coefficients of the imaginary parts .
The result is .
Hard difficulty: Calculate
Solution: Calculate step by step, first add then subtract.
Step 1: ;
Step 2: .

Step 1: Remove parentheses: If there are parentheses in the expression, first remove them according to the sign rules (when subtracting a complex number, both the real part and the imaginary part inside the parentheses must change their signs).
Step 2: Separate real and imaginary parts: Gather all real part terms together and all coefficient terms of the imaginary parts together (pay attention to retaining their respective signs).
Step 3: Calculate separately: Perform addition and subtraction operations on the real part terms, and do the same for the coefficient terms of the imaginary parts.
Step 4: Arrange the result: Combine the calculated real part and imaginary part into the standard form (if the coefficient of the imaginary part is 0, it can be directly written as a real number).