Complex Numbers - Understanding and Working with z = x + iy and z^2 = x^2 + y^2

[m_s_a]

hi,








let
z=x+iy

z^2=z.zpar=(x+iy)(x-iy)=x^2+y^2
or
z^2=(x+iy)(x+iy)=(x^2-y^2)

 

[Hootenanny]

It very much depends on your field. Generally in mathematics when one says the 'square' of a complex number one means literally multiplication by itself as in your latter example. However, physicists working in QM often refer to the multiplication of a complex number by it's complex conjugate as 'squaring' it, as for your former example.

 

[m_s_a]

It very much depends on your field. Generally in mathematics when one says the 'square' of a complex number one means literally multiplication by itself as in your latter example. However, physicists working in QM often refer to the multiplication of a complex number by it's complex conjugate as 'squaring' it, as for your former example.

Thank you for you on your response
And on the new information for me

 

[Dick]

hi,








let
z=x+iy

z^2=z.zpar=(x+iy)(x-iy)=x^2+y^2
or
z^2=(x+iy)(x+iy)=(x^2-y^2)

Take note that (x+iy)(x+iy) is NOT equal to x^2-y^2. It's x^2-y^2+2ixy. Your first 'z^2' is the modulus (size) of the complex number squared. The second is the complex function z*z. They are quite different. A physicist who refers to the first operation as 'squaring' is being pretty sloppy. The proper term is 'modulus squared' and the proper notation is |z|^2.

 

[Hootenanny]

Take note that (x+iy)(x+iy) is NOT equal to x^2-y^2. It's x^2-y^2+2ixy.

Nice catch Dick, didn't even see it

 

[m_s_a]

Thank you for you on the note
And thank you on the information that you presented
But this is a question in one of the issues
Thanks

 

[Hootenanny]

But this is a question in one of the issues

Then I would suggest that,

$$z^2 = x^2 +2ixy - y^2$$

 

[m_s_a]

Then I would suggest that,

$$z^2 = x^2 +2ixy - y^2$$

Thank you a lot