SOLUTION: Any help would be appreciated with this. If s and t are two complex numbers, prove the following. a) |s+t|^2=|s|^2+|t|^2+(st*+s*t) |s|^2+|t|^2+(st*+s*t) =ss*+tt*+st*+s*

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Any help would be appreciated with this. If s and t are two complex numbers, prove the following. a) |s+t|^2=|s|^2+|t|^2+(st*+s*t) |s|^2+|t|^2+(st*+s*t) =ss*+tt*+st*+s*      Log On


   

Question 979594: Any help would be appreciated with this.
If s and t are two complex numbers, prove the following.
a) |s+t|^2=|s|^2+|t|^2+(st*+s*t)

|s|^2+|t|^2+(st*+s*t)
=ss*+tt*+st*+s*t
Using s=a+bi and t=c+di
=a^2+b^2+c^2+d^2+2ac+2bd

|s+t|^2 = a^2+2abi-b2+2ac+2bci+2adi-2bd+c^2+2cdi-d^2
Which seems quite wrong.
Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!
If s and t are two complex numbers, prove the following.
|s+t|^2 = a^2+2abi-b2+2ac+2bci+2adi-2bd+c^2+2cdi-d^2
You did that wrong.  add s=a+bi and t=c+di before you find the magnitude
or absolute value:

|s+t|² = |(a+bi)+(c+di)|² = |(a+c)+(b+d)i| = (a+c)²+(b+d)² = 

a²+2ac+c²+b²+2bd+d² 

Then when you do the other side:

|s|²+|t|²+(st*+s*t) = a²+b²+c²+d²+(a+bi)(c-di)+(a-bi)(c+di) = 

a²+b²+c²+d²+ac-adi+bci+bd+ac+adi-bci+bd = a²+b²+c²+d²+2ac+2bd

and they are equal.

Edwin