Answered

Subtract $(5-2 i)-(1+8 i)$.


  1. Which property allows you to write the expression as $5-2 i-1-8 i$?
  • Commutative
  • Associative
  • Identity
  • Distributive


  1. What is the real part of the difference?


  1. What is the imaginary part of the difference?

Verified Answer :

  • The expression \((5 - 2i) - (1 + 8i)\) becomes \(5 - 2i - 1 - 8i\) using the distributive property.
  • Combining the real parts \(5 - 1\) gives \(4\).
  • Combining the imaginary parts \(-2i - 8i\) gives \(-10i\).
  • Property used: Distributive.
  • Real part: \(\boxed{4}\).
  • Imaginary part: \(\boxed{-10} i\).

Explanation

  1. Removing Parentheses and Identifying the Property

To subtract the complex numbers \((5 - 2i) - (1 + 8i)\), we first remove the parentheses. The expression \(a - (b + c)\) is equivalent to \(a - b - c\) due to the distributive property of multiplication (by -1) over addition. Thus, we have:
\[(5 - 2i) - (1 + 8i) = 5 - 2i - 1 - 8i.\]
This operation is justified by the distributive property.

  1. Simplifying and Finding the Parts

Next, we group the real parts and the imaginary parts separately to simplify the expression:
\[(5 - 1) + (-2i - 8i) = 4 + (-2 - 8)i.\]
Calculating these, we find:
\[4 - 10i.\]
The real part of the resulting complex number is \(4\), and the imaginary part (the coefficient of \(i\)) is \(-10\).

Examples

Complex numbers are widely used in engineering and physics, especially in alternating current (AC) circuit analysis and control systems. For example, when calculating the total impedance of a circuit, you might subtract voltage vectors or current values represented as complex numbers to find the potential difference or phase shift between two components. Understanding how to correctly subtract these values is fundamental for accurate circuit design and performance analysis.

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