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Complex numbers might sound complicated, but they are just numbers with two parts: a real part and an imaginary part. When you want to add or subtract complex numbers, you deal with these parts separately.

For addition, you take the real parts of both numbers and add them together. Do the same with the imaginary parts. It's like adding two separate pairs of numbers. Subtraction works similarly; you subtract the real parts and the imaginary parts separately. You always handle them in the same order to maintain consistency.

Once you combine the results of these operations, you arrive at the final complex number.

Recognizing the real and imaginary components of a complex number is essential in these operations. Each complex number is written in the form of a real number plus an imaginary number (e.g., \( a + bi \)). Here, \( a \) represents the real part, and \( bi \) denotes the imaginary part.

Consider the expression \( 7 - 6i \). The real part, here, is simply \( 7 \). The imaginary part is \( -6i \), which includes both the imaginary component and its sign. Similarly, in the second number, \( 3 - 4i \), \( 3 \) is the real part, and \( -4i \) is the imaginary part.

This step is an important starting point before performing any operations with complex numbers; it sets the stage for accurate addition or subtraction.

Let’s walk through the step-by-step solution to add \( (7-6i) \) and \( (3-4i) \).

• **Step 1**: Identify the parts. We have two complex numbers, \( 7-6i \) and \( 3-4i \). For the first, the real part is \( 7 \) and imaginary part is \( -6i \). For the second, the real part is \( 3 \) and imaginary part is \( -4i \).


• **Step 2**: Add the real parts. Simply take \( 7 \) and \( 3 \), and get \( 7 + 3 = 10 \).


• **Step 3**: Add the imaginary parts. Here, combine \( -6i \) and \( -4i \) to get \( -6i - 4i = -10i \).


• **Step 4**: Combine your results. Join the real part \( 10 \) with the imaginary part \( -10i \) to form the final result of \( 10 - 10i \).

Understanding each step is key in simplifying and mastering operations with complex numbers. Each part of this walkthrough builds on the previous, ensuring clarity and a thorough grasp of the solution process.