Question

Prove the following properties:

Re(z) = `(z + bar(z))/2` and Im(z) = `(z - bar(z))/(2"i")`

Answer 1

Let z = x + iy

`bar(z)` = x – iy

`(z + bar(z))/2 = (x + "i"y + x - "i"y)/2`

= `(2x)/2`

= x

= Re(z)

Hence proved.

`(z - bar(x))/(2"i") = ((x + "i"y) - (x - "i"y))/(2"i")`

= `(x + "i"y - x + "i"y)/(2"i")`

= `(2"i"y)/(2"i")`

= y

= Im(z)

Hence proved.