- Identify the real and imaginary parts of the given complex numbers \(z_1 = 21 - 9i\) and \(z_2 = 24 + 7i\).
- Add the numbers by combining the real components \((21+24)\) and imaginary components \((-9+7)\) to get \(\boxed{45-2i}\).
- Subtract the numbers by subtracting the real components \((21-24)\) and imaginary components \((-9-7)\) to get \(\boxed{-3-16i}\).
Explanation
- Data and Problem Analysis
We are given two complex numbers: \(z_1 = 21 - 9i\) and \(z_2 = 24 + 7i\). Our objective is to perform both addition and subtraction operations on these numbers and express the final results in the standard form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Complex numbers are added or subtracted by combining their corresponding real and imaginary components separately.
- Performing Addition
To calculate the sum \((21 - 9i) + (24 + 7i)\), we group the real parts together and the imaginary parts together:
\[ (21 - 9i) + (24 + 7i) = (21 + 24) + (-9 + 7)i \]
Calculating the sum of the real parts: \(21 + 24 = 45\).
Calculating the sum of the imaginary parts: \(-9 + 7 = -2\).
Thus, the sum is \(45 - 2i\).
- Performing Subtraction
To calculate the difference \((21 - 9i) - (24 + 7i)\), we first distribute the subtraction sign to each term of the second complex number:
\[ (21 - 9i) - (24 + 7i) = 21 - 9i - 24 - 7i \]
Next, we group the real parts and the imaginary parts:
\[ (21 - 24) + (-9 - 7)i \]
Calculating the difference of the real parts: \(21 - 24 = -3\).
Calculating the difference of the imaginary parts: \(-9 - 7 = -16\).
Thus, the difference is \(-3 - 16i\).
- Final Conclusion
We have successfully performed the operations:
The sum of the two complex numbers is \(45 - 2i\).
The difference of the two complex numbers is \(-3 - 16i\).
These are the final answers in standard form.
Examples
In engineering, particularly in electrical circuits, alternating current (AC) is often represented by complex numbers (phasors). The real part represents the resistance, and the imaginary part represents the reactance. When components are connected in series, we add their complex impedances; when analyzing the difference in potential across different points in a circuit, we subtract these values. Mastering complex number arithmetic is essential for calculating total impedance and voltage drops in such systems.