Verified Answer :

  • Distribute the negative sign to the terms of the second complex number: \(-3 - 2i - 1 - i\).
  • Group the real parts: \(-3 - 1 = -4\).
  • Group the imaginary parts: \(-2i - i = -3i\).
  • Combine to get the final result: \(\boxed{-4 - 3i}\).

Explanation

  1. Analyzing the Expression

We are tasked with simplifying the expression \((-3 - 2i) - (1 + i)\). Complex numbers are expressed in the standard form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. To subtract complex numbers, we subtract their corresponding real and imaginary parts. For two complex numbers \(z_1 = a + bi\) and \(z_2 = c + di\), the subtraction is defined as \(z_1 - z_2 = (a - c) + (b - d)i\). In our specific expression, \(z_1 = -3 - 2i\) and \(z_2 = 1 + i\), so the real parts are \(-3\) and \(1\), and the imaginary parts are \(-2\) and \(1\).

  1. Grouping and Simplifying Terms

To perform the subtraction, we first distribute the negative sign to each term inside the second set of parentheses. This gives:
\((-3 - 2i) - 1 - i\)
Next, we group the real components and the imaginary components together to simplify the calculation:
\((-3 - 1) + (-2i - i)\)

  1. Calculating the Final Result

Now, we perform the arithmetic for each group:

  • The sum of the real parts is \(-3 - 1 = -4\).
  • The sum of the imaginary parts is \(-2i - 1i = -3i\).

Combining these results, we get the final simplified complex number in the form \(a + bi\), which is \(-4 - 3i\).

Examples

Complex numbers are essential tools in fields like electrical engineering and signal processing. For instance, when analyzing AC (alternating current) circuits, electrical engineers use complex numbers to represent voltage and current as phasors. Simplifying expressions involving complex numbers, like the one we just performed, helps engineers calculate total impedance, analyze circuit resonance, or determine phase shifts in a signal, ensuring that power systems function correctly and efficiently.

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