Add: $(5-6 i)+(9+9 i)$. Write the result in standard form:
To add two complex numbers, we follow the rule of combining their real parts and their imaginary parts separately. A complex number is written in the standard form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.
Given the two complex numbers:
\(z_1 = 5 - 6i\)
\(z_2 = 9 + 9i\)
We need to calculate their sum:
\(z_1 + z_2 = (5 - 6i) + (9 + 9i)\)
To add these numbers, we group the real parts together and the imaginary parts together:
Real part sum: \(5 + 9 = 14\)
Imaginary part sum: \(-6i + 9i = (-6 + 9)i = 3i\)
By adding these results, we get:
\(14 + 3i\)
Comparing our result \(14 + 3i\) with the given options:
The calculated sum corresponds to option B. Therefore, the final answer is \(\boxed{14 + 3i}\).
Complex numbers are essential in various fields, such as electrical engineering, where they are used to analyze alternating current (AC) circuits. When calculating the total impedance of components connected in series, engineers add complex numbers representing individual impedances (which contain resistance as the real part and reactance as the imaginary part). Just as we added \((5 - 6i) + (9 + 9i)\) to get \(14 + 3i\), an engineer would add impedances to find the total resistance and reactance in a circuit, helping them design systems that operate efficiently and safely.