Express $\frac{-2+3 i}{1+2 i}$ in the form $a+b i$, where $a$ and $b$ are real numbers.
Answer (2)
Multiply the numerator and denominator by the conjugate of the denominator: 1 + 2 i − 2 + 3 i = ( 1 + 2 i ) ( 1 − 2 i ) ( − 2 + 3 i ) ( 1 − 2 i ) .
Expand the numerator: ( − 2 + 3 i ) ( 1 − 2 i ) = 4 + 7 i .
Expand the denominator: ( 1 + 2 i ) ( 1 − 2 i ) = 5 .
Simplify the expression: 5 4 + 7 i = 5 4 + 5 7 i . The answer is 5 4 + 5 7 i .
Explanation
Problem Analysis We are asked to express the complex number 1 + 2 i − 2 + 3 i in the form a + bi , where a and b are real numbers. This involves dividing two complex numbers.
Finding the Conjugate To divide complex numbers, we multiply the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of 1 + 2 i is 1 − 2 i .
Multiplying by the Conjugate Multiply the numerator and denominator by the conjugate: 1 + 2 i − 2 + 3 i = ( 1 + 2 i ) ( 1 − 2 i ) ( − 2 + 3 i ) ( 1 − 2 i )
Expanding the Numerator Expand the numerator: ( − 2 + 3 i ) ( 1 − 2 i ) = − 2 + 4 i + 3 i − 6 i 2 = − 2 + 7 i − 6 ( − 1 ) = − 2 + 7 i + 6 = 4 + 7 i
Expanding the Denominator Expand the denominator: ( 1 + 2 i ) ( 1 − 2 i ) = 1 − 2 i + 2 i − 4 i 2 = 1 − 4 ( − 1 ) = 1 + 4 = 5
Simplifying the Expression Therefore, we have: 1 + 2 i − 2 + 3 i = 5 4 + 7 i = 5 4 + 5 7 i
Final Answer Thus, a = 5 4 and b = 5 7 .
Examples
Complex numbers are used in electrical engineering to represent alternating currents. The impedance of a circuit, which is the opposition to the flow of current, can be represented as a complex number. When analyzing circuits, engineers often need to divide complex numbers to find the current or voltage in different parts of the circuit. For example, if you have a circuit with a voltage of − 2 + 3 i volts and an impedance of 1 + 2 i ohms, the current would be 1 + 2 i − 2 + 3 i amps. Expressing this in the form a + bi gives the real and imaginary components of the current.
To express 1 + 2 i − 2 + 3 i in the form a + bi , we multiply by the conjugate of the denominator and simplify. The result is a = 5 4 and b = 5 7 , leading to 5 4 + 5 7 i .
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