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Chapter 10: Problem 43

Perform each indicated operation. Write the result in the form \(a+b i\) $$ (6-3 i)-(4-2 i) $$

Short Answer

Expert verified
The result is \(2 - i\).

Step by step solution

01

Identify and Organize Parts

Start by identifying the real and imaginary parts of each complex number. For \((6-3i)\), the real part is 6, and the imaginary part is -3i. For \((4-2i)\), the real part is 4, and the imaginary part is -2i.
02

Subtract the Real Parts

Subtract the real part of the second complex number from the real part of the first one. This gives: \(6 - 4 = 2\).
03

Subtract the Imaginary Parts

Subtract the imaginary part of the second complex number from the imaginary part of the first one. This gives: \(-3i - (-2i) = -3i + 2i = -i\).
04

Combine Results into Standard Form

Combine the results from Steps 2 and 3 into the standard form \(a + bi\). Here, \(a = 2\) and \(b = -1\), so the complex number is \(2 - i\).

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Key Concepts

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

Real and Imaginary Parts
Complex numbers are composed of two essential parts: the real part and the imaginary part. To better understand this, let's analyze the complex number \((6 - 3i)\). Here, the number 6 represents the **real part**, while \(\-3i\)\ is the **imaginary part**.
  • The real part is just a regular number without any imaginary component, like the 6 in our example.
  • The imaginary part is paired with \(i\), which is the imaginary unit defined as \(i = \sqrt{-1}\).
In any complex number, you will always find these two components working together to form expressions like \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Understanding the real and imaginary parts is crucial, as they will guide you in various operations, such as subtraction or addition, to achieve the standard form of a complex number.
Complex Number Subtraction
Subtracting complex numbers involves separating and subtracting their real and imaginary parts. Given two complex numbers, let鈥檚 say \((a + bi)\) and \((c + di)\), subtraction is performed as follows:
  • **Subtract the real parts**: This is done by subtracting the real component of the second number from the real component of the first. In our example, we have 6 and 4, so \(6 - 4 = 2\).
  • **Subtract the imaginary parts**: Similarly, subtract the second number's imaginary part from the first. For \(-3i\) and \(-2i\), this is \(-3i - (-2i) = -3i + 2i = -i\).
These calculations give you the new real and imaginary parts. It's like performing standard subtraction but done separately for each part of the complex numbers. Mastering this step leads to the correct and simplified form.
Standard Form of Complex Numbers
Once you've completed operations on the real and imaginary parts, it's important to express the result in the standard form, which is \(a + bi\). After our subtraction, we combined the real and imaginary results:
  • Real part: 2
  • Imaginary part: \(-i\)
The standard form we get is \(2 - i\). This form is critical as it provides a clear and structured way to express complex numbers, making it easy to read and interpret. Always remember:
  • Write the result as \(a + bi\), even if \(b\) is negative, as \(2 - i\) is the same as \(2 + (-1)i\).
Consistency in using the standard form ensures clarity, especially when solving more complex problems involving quadratic equations, polar coordinates, or electrical engineering applications.

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