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Chapter 3: Problem 20

Perform the indicated operation and express the result as a simplified complex number. \((2-3 i)-(3+2 i)\)

Short Answer

Expert verified
The result is \(-1 - 5i\).

Step by step solution

01

Identify the Operation

Examine the problem to determine that we need to perform a subtraction between two complex numbers: \((2 - 3i)\) and \((3 + 2i)\).
02

Subtract the Real Parts

Separate the real parts from both complex numbers. Subtract the real part of the second complex number from the real part of the first: \[ 2 - 3 = -1 \]
03

Subtract the Imaginary Parts

Similarly, separate the imaginary parts with the imaginary unit \(i\). Subtract the imaginary part of the second complex number from the imaginary part of the first: \[ -3i - 2i = -5i \]
04

Combine the Results

Combine the results from the subtraction of the real and imaginary parts to form the final complex number: \[ -1 - 5i \]
05

Conclusion

The simplified complex number after performing the subtraction operation is \(-1 - 5i\). This form represents the combination of the subtracted real and imaginary components.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Subtraction of Complex Numbers
Complex numbers are written in the form of \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Subtraction of complex numbers involves subtracting both the real and imaginary components separately. Consider two complex numbers: \((2 - 3i)\) and \((3 + 2i)\). It is crucial to handle these two parts independently for subtraction to ensure accuracy. You begin by:
  • Separating the numbers based on their real and imaginary components.
  • Subtracting the real parts: \(2 - 3\).
  • Subtracting the imaginary parts: \(-3i - 2i\).
By paying close attention to these steps, especially when negative signs are involved, you execute the operation correctly. This process results in a new complex number derived from the distinct subtraction of each part.
Real and Imaginary Parts
Understanding the distinct roles of real and imaginary parts in complex numbers is essential in performing operations like subtraction. The real part of a complex number is the constant term without the imaginary unit \(i\), while the imaginary part is multiplied by \(i\).
  • In \((2 - 3i)\), the real part is 2 and the imaginary part is \(-3i\).
  • In \((3 + 2i)\), the real part is 3 and the imaginary part is \(2i\).
Knowing how to identify and manage these parts separately is crucial for mathematical operations. Each part is manipulated on its own, meaning that the interaction of real with real and imaginary with imaginary ensures proper processing of complex numbers. This separation and calculated interaction simplify the arithmetic process.
Simplified Complex Number
After performing operations on complex numbers, the final goal is often to express the result as a simplified complex number. A simplified complex number is a standard form \(a + bi\) that clearly distinguishes both the real and imaginary components. In the example given:
  • The real parts subtraction gave us \(-1\).
  • The imaginary parts subtraction resulted in \(-5i\).
When combined, these results form the simplified complex number \(-1 - 5i\). This straightforward form is standard practice in mathematics, allowing easy comparison and further operations. The simplicity of this format aids in understanding and manipulating complex numbers in more advanced problems.

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