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When working with complex numbers, it is crucial to simplify expressions to make them easier to understand and solve. Simplifying expressions involves combining like terms, following the standard arithmetic operations. In this context, we're dealing with subtraction of complex numbers.

Let's break down the given expression \((-5 + 3i) - (6 - i)\). First, identify and separate the real and imaginary components. This separation helps in methodically simplifying the expression in two parts:

• Real components: \(-5\) and \(6\)
• Imaginary components: \(+3i\) and \(-i\)

After separating, apply the subtraction operation individually to the real and imaginary components. This two-step process allows us to simplify the expression effectively.

Once calculations are performed, the results are combined to form a single, simplified complex number.

Complex numbers are built upon two parts: real and imaginary components. The real component is a normal number that you are used to dealing with in everyday mathematics. In this example, the real parts are \(-5\) and \(6\).

The imaginary component, however, includes the term \(i\), which represents the square root of \(-1\). The imaginary components in our given expression are \(3i\) and \(-i\).

Understanding that the imaginary part behaves similarly to variables like \(x\) or \(y\) helps in manipulating and simplifying these terms. Remember:

• Real components deal with numbers without \(i\).
• Imaginary components contain the term \(i\), where \(i^2 = -1\).

By handling these components separately, the complex arithmetic becomes simpler and more organized.

To subtract complex numbers effectively, one needs to follow systematic steps for both the real and imaginary parts. The expression \((-5 + 3i) - (6 - i)\) involves subtracting the respective components:

• Real Parts: Subtract the second real component from the first \((-5) - 6 = -11\).
• Imaginary Parts: Subtract the second imaginary component from the first \(3i - (-i) = 3i + i = 4i\).

Performing the subtraction on each component separately ensures clear and accurate results.

Finally, combine both the simplified real and imaginary results to provide the solution \(-11 + 4i\). This represents the new complex number after subtraction. Mastering these steps will enhance your confidence when dealing with complex number operations.