Simplify: $(4-3 i)+(1-6 i)+(4+5 i)$
To simplify the expression \((4-3i)+(1-6i)+(4+5i)\), we follow the rules for adding complex numbers. A complex number is represented in the form \(a+bi\), where \(a\) is the real part and \(b\) is the imaginary part. To add two or more complex numbers, we sum their real parts together and their imaginary parts together independently.
For the given expression, the real parts are \(4\), \(1\), and \(4\). The sum of these real parts is:
\[4 + 1 + 4 = 9\]
The imaginary parts are \(-3\), \(-6\), and \(5\). The sum of these imaginary parts is:
\[-3 + (-6) + 5 = -9 + 5 = -4\]
Combining these, we get the result in the form \(a+bi\):
\[9 + (-4)i = 9 - 4i\]
Comparing our result \(9-4i\) with the given choices:
The calculated sum \(9-4i\) matches option B. Therefore, the simplified form of the given expression is \(9-4i\), which corresponds to option B. \(\boxed{B}\)
Complex numbers are fundamental in electrical engineering, particularly in analyzing alternating current (AC) circuits. In these circuits, voltage and current are often represented as complex quantities because they possess both a magnitude (amplitude) and a phase. When combining several signals or circuit components in series, engineers add their complex representations just like we added these complex numbers: summing the real components (resistive parts) and the imaginary components (reactive parts) to find the total effective impedance or voltage.