Simplify. Write your response in $a+b i$ form.
$(2-11 i)+(1-i)=$
Answer (2)
Combine the real parts: 2 + 1 = 3 .
Combine the imaginary parts: − 11 i − i = − 12 i .
Write the result in a + bi form: 3 − 12 i .
The simplified expression is 3 − 12 i .
Explanation
Understanding the Problem We are asked to simplify the expression ( 2 − 11 i ) + ( 1 − i ) and write the result in the form a + bi , where a and b are real numbers. This involves combining the real parts and the imaginary parts of the complex numbers.
Combining Real and Imaginary Parts First, let's group the real parts and the imaginary parts together: ( 2 − 11 i ) + ( 1 − i ) = ( 2 + 1 ) + ( − 11 i − i ) Now, we perform the addition for the real parts: 2 + 1 = 3 Next, we combine the imaginary parts: − 11 i − i = − 11 i − 1 i = ( − 11 − 1 ) i = − 12 i
Final Result Now, we write the simplified expression in the form a + bi :
3 + ( − 12 i ) = 3 − 12 i
Examples
Complex numbers are used in electrical engineering to analyze alternating current circuits. The voltage and current in a circuit can be represented as complex numbers, and operations with these numbers can be used to calculate the impedance and power in the circuit. For example, if the voltage is 2 − 11 i volts and the impedance is 1 − i ohms, the current can be calculated using Ohm's law.
To simplify the expression ( 2 − 11 i ) + ( 1 − i ) , we combine the real parts to get 3 and the imaginary parts to get − 12 i . The final result in the form a + bi is 3 − 12 i .
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