$z_1=6+9 i \quad z_2=-1+2 i$
Answer (1)
Sum the complex numbers: z 1 + z 2 = 5 + 11 i .
Subtract the complex numbers: z 1 − z 2 = 7 + 7 i .
Multiply the complex numbers: z 1 × z 2 = − 24 + 3 i .
Divide the complex numbers: z 2 z 1 = 5 12 − 5 21 i .
z 2 z 1 = 5 12 − 5 21 i
Explanation
Problem Analysis We are given two complex numbers, z 1 = 6 + 9 i and z 2 = − 1 + 2 i . The problem does not specify what to do with these numbers. Let's assume we are asked to find their sum, difference, product, and quotient.
Finding the Sum First, let's find the sum of z 1 and z 2 :
z 1 + z 2 = ( 6 + 9 i ) + ( − 1 + 2 i ) = ( 6 − 1 ) + ( 9 + 2 ) i = 5 + 11 i
Finding the Difference Next, let's find the difference between z 1 and z 2 :
z 1 − z 2 = ( 6 + 9 i ) − ( − 1 + 2 i ) = ( 6 − ( − 1 )) + ( 9 − 2 ) i = 7 + 7 i
Finding the Product Now, let's find the product of z 1 and z 2 :
z 1 × z 2 = ( 6 + 9 i ) × ( − 1 + 2 i ) = 6 × ( − 1 ) + 6 × ( 2 i ) + 9 i × ( − 1 ) + 9 i × ( 2 i ) = − 6 + 12 i − 9 i + 18 i 2 = − 6 + 3 i − 18 = − 24 + 3 i
Finding the Quotient Finally, let's find the quotient of z 1 and z 2 :
z 2 z 1 = − 1 + 2 i 6 + 9 i To simplify this, we multiply the numerator and denominator by the conjugate of the denominator, which is − 1 − 2 i :
− 1 + 2 i 6 + 9 i × − 1 − 2 i − 1 − 2 i = ( − 1 + 2 i ) ( − 1 − 2 i ) ( 6 + 9 i ) ( − 1 − 2 i ) = ( − 1 ) 2 + ( 2 ) 2 6 ( − 1 ) + 6 ( − 2 i ) + 9 i ( − 1 ) + 9 i ( − 2 i ) = 1 + 4 − 6 − 12 i − 9 i − 18 i 2 = 5 − 6 − 21 i + 18 = 5 12 − 21 i = 5 12 − 5 21 i
Final Answer In summary:
Sum: z 1 + z 2 = 5 + 11 i
Difference: z 1 − z 2 = 7 + 7 i
Product: z 1 × z 2 = − 24 + 3 i
Quotient: z 2 z 1 = 5 12 − 5 21 i
Examples
Complex numbers are used in electrical engineering to represent alternating currents and voltages. They simplify circuit analysis by allowing engineers to represent both the magnitude and phase of these quantities. For example, impedance, which is the opposition to current flow in an AC circuit, is represented as a complex number. The real part of the impedance represents resistance, while the imaginary part represents reactance. By using complex numbers, engineers can easily analyze and design complex electrical circuits.
