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In mathematics, arithmetic operations are fundamental. They include addition, subtraction, multiplication, and division. These operations apply to all numbers, including complex numbers. In a complex number, you combine both real numbers and imaginary numbers. This is important because each part can be manipulated through arithmetic.

For example, in the exercise, subtraction was used. Subtraction involves distributing the negative sign across the terms in the parentheses. This is similar to subtracting regular numbers, but now both real and imaginary numbers are involved.

To solve any arithmetic operation with complex numbers, it’s key to separate and deal with the real and imaginary parts individually.

The imaginary unit, denoted as \(i\), is a fascinating part of complex numbers. It is defined by the property that \(i^2 = -1\). This means \(i\) is not a real number but a concept to help work with numbers not on the real number line.

The imaginary unit allows for the existence of complex numbers, represented as \(a+bi\), where \(a\) and \(b\) are real numbers. Here, \(bi\) represents the imaginary part.

Understanding the imaginary unit is crucial as it introduces the ability to compute values that wouldn’t be possible otherwise, like the square roots of negative numbers. Without the imaginary unit, equations with negative discriminants would have no solution in the set of real numbers. The ability to add, subtract, and multiply with \(i\) broadens the scope of arithmetic to a wider variety of applications.

Complex numbers, like those encountered in the exercise, have two components: the real part and the imaginary part. These parts can be separated as the first step in solving equations. The real part of a complex number corresponds to the term not multiplied by \(i\), and the imaginary part is the term that multiplies \(i\).

In a complex number written as \(a+bi\):

• \(a\) is the real part
• \(bi\) is the imaginary part

For calculation purposes, especially in addition and subtraction, separating these parts allows for simpler and clearer arithmetic. You deal with each part separately, without mixing the real numbers with the imaginary ones, ensuring clarity and accuracy in the result.

The process of addition and subtraction with complex numbers is straightforward once you understand the concept of separating real and imaginary parts. To add or subtract two complex numbers, you simply align and manipulate the real parts together and the imaginary parts together.

For example, in the problem provided, you have two complex expressions:

• \(-4 + i\)
• \(2 - 5i\)

Adding or subtracting these two involves grouping:

• Real parts: \(-4 - 2\)
• Imaginary parts: \(i + 5i\)

The combination of these results gives the final answer, ensuring each component is carefully computed. This combines both techniques and understanding of complex number arithmetic, reinforcing foundational math skills.