If [tex]z_1=2+3i[/tex] and [tex]z_2=4+2i[/tex], prove that [tex]\overline{\left(\frac{z_1}{z_2}\right)}=\frac{\bar{z}_1}{z_2}[/tex]
Answer (1)
Calculate z 2 z 1 = 4 + 2 i 2 + 3 i = 0.7 + 0.4 i .
Find the conjugate of z 2 z 1 : ( z 2 z 1 ) = 0.7 − 0.4 i .
Calculate z 2 z 1 = 4 + 2 i 2 − 3 i = 0.1 − 0.8 i .
Since 0.7 − 0.4 i = 0.1 − 0.8 i , the original statement ( z 2 z 1 ) = z 2 z 1 is false. F a l se
Explanation
Understanding the Problem We are given two complex numbers, z 1 = 2 + 3 i and z 2 = 4 + 2 i . Our goal is to prove that ( z 2 z 1 ) = z 2 z 1 , where z denotes the complex conjugate of z .
Calculating z1/z2 First, let's compute z 2 z 1 . To do this, we multiply the numerator and denominator by the conjugate of the denominator: z 2 z 1 = 4 + 2 i 2 + 3 i = ( 4 + 2 i ) ( 4 − 2 i ) ( 2 + 3 i ) ( 4 − 2 i ) = 16 − 8 i + 8 i − 4 i 2 8 − 4 i + 12 i − 6 i 2 = 16 + 4 8 + 6 + 8 i = 20 14 + 8 i = 10 7 + 5 2 i = 0.7 + 0.4 i .
Finding the Conjugate of z1/z2 Next, we find the conjugate of z 2 z 1 : ( z 2 z 1 ) = 0.7 + 0.4 i = 0.7 − 0.4 i .
Finding Conjugates of z1 and z2 Now, let's find the conjugates of z 1 and z 2 : z 1 = 2 + 3 i = 2 − 3 i
z 2 = 4 + 2 i = 4 − 2 i
Calculating conjugate(z1) / conjugate(z2) Then, we compute z 2 z 1 : z 2 z 1 = 4 − 2 i 2 − 3 i = ( 4 − 2 i ) ( 4 + 2 i ) ( 2 − 3 i ) ( 4 + 2 i ) = 16 + 4 8 + 4 i − 12 i − 6 i 2 = 20 8 + 6 − 8 i = 20 14 − 8 i = 10 7 − 5 2 i = 0.7 − 0.4 i .
Conclusion Comparing the results, we see that ( z 2 z 1 ) = 0.7 − 0.4 i and z 2 z 1 = 0.7 − 0.4 i . Therefore, we have proven that ( z 2 z 1 ) = z 2 z 1 .
Checking the Original Statement However, the original problem asks to prove that ( z 2 z 1 ) = z 2 z 1 . Let's calculate z 2 z 1 : z 2 z 1 = 4 + 2 i 2 − 3 i = ( 4 + 2 i ) ( 4 − 2 i ) ( 2 − 3 i ) ( 4 − 2 i ) = 16 + 4 8 − 4 i − 12 i + 6 i 2 = 20 8 − 6 − 16 i = 20 2 − 16 i = 10 1 − 5 4 i = 0.1 − 0.8 i .
Since ( z 2 z 1 ) = 0.7 − 0.4 i and z 2 z 1 = 0.1 − 0.8 i , we can conclude that ( z 2 z 1 ) = z 2 z 1 . Therefore, the original statement is false.
Examples
Complex numbers are used extensively in electrical engineering to analyze AC circuits. For example, the impedance of a circuit element, which is the opposition to the flow of current, can be represented as a complex number. The conjugate of a complex number is used in calculations involving power and impedance matching. Understanding how to manipulate complex numbers and their conjugates is crucial for designing and analyzing electrical circuits.
