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

• m_s_a
In summary, The concept of "squaring" a complex number can vary depending on the field, with mathematicians referring to literal multiplication and physicists sometimes using the complex conjugate. However, it's important to note that (x+iy)(x+iy) is not equal to x^2-y^2, as it is actually x^2-y^2+2ixy. The proper term for the first operation is "modulus squared" and the notation is |z|^2. This was a point of discussion in a conversation about complex numbers and their operations.
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)

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.

Hootenanny said:
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

m_s_a said:
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.

Dick said:
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

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

m_s_a said:
But this is a question in one of the issues
Then I would suggest that,

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

Hootenanny said:
Then I would suggest that,

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

Thank you a lot

1. What are complex numbers?

Complex numbers are numbers that consist of a real part and an imaginary part. The real part is a normal number that we are familiar with, while the imaginary part is a multiple of the imaginary unit, which is denoted by i and is equal to the square root of -1.

2. How do we represent complex numbers?

Complex numbers are often represented in the form of z = x + iy, where x is the real part and y is the imaginary part. This is known as the standard form of a complex number.

3. What is the relationship between complex numbers and the Cartesian plane?

Complex numbers can be visualized in the Cartesian plane, with the real part representing the horizontal axis and the imaginary part representing the vertical axis. The point (x,y) in the Cartesian plane corresponds to the complex number z = x + iy.

4. How do we perform operations on complex numbers?

To perform addition, subtraction, and multiplication on complex numbers, we simply combine the real parts and the imaginary parts separately. For division, we use the conjugate of the denominator to rationalize the expression.

5. What is the geometric interpretation of multiplying complex numbers?

When we multiply two complex numbers, we are essentially scaling and rotating them on the complex plane. The modulus (or magnitude) of the product is equal to the product of the moduli of the two complex numbers, and the argument (or angle) of the product is equal to the sum of the arguments of the two complex numbers.

• Calculus and Beyond Homework Help
Replies
17
Views
1K
• Calculus and Beyond Homework Help
Replies
6
Views
1K
• Calculus and Beyond Homework Help
Replies
27
Views
732
• Calculus and Beyond Homework Help
Replies
19
Views
979
• Calculus and Beyond Homework Help
Replies
13
Views
1K
• Calculus and Beyond Homework Help
Replies
9
Views
1K
• Calculus and Beyond Homework Help
Replies
14
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
605
• Calculus and Beyond Homework Help
Replies
44
Views
4K
• Calculus and Beyond Homework Help
Replies
6
Views
2K