Understanding the Relationship between Absolute Value and Complex Numbers

In summary, |z|^2 = z*z because the absolute value of a complex number, denoted as |z|, is equal to the modulus of z which is defined as the square root of the sum of the squares of the real and imaginary parts of z. This can be seen geometrically as the distance of the complex number from the origin on the Argand Plane. However, |z|^2 is not equal to z^2, as the latter is the product of z with its complex conjugate.
  • #1
jaejoon89
195
0
Why is |z|^2 = z*z?

z = a + ib
z*z = (a - ib)(a + ib) = a^2 + b^2
z^2 = (a + ib)^2 = a^2 + 2iab - b^2

So it must have something to do with the absolute value, but I don't understand what or why.
 
Physics news on Phys.org
  • #2
|z| does not define the "Absolute Value" of a complex number. The notation |z| refers to the modulus of z, which is by definition

|z| = sqrt(a^2 + b^2)

Geometrically it gives the distance of the complex number from the origin on the Argand Plane.

And quite obviously |z|^2 is NOT EQUAL TO z^2
 
  • #3
|z|^2 is conjugate(z)*z=a^2+b^2. It's not equal to z^2.
 
  • #4
although the modulus of z when z is of the form a + 0i (i.e. it is only in the reals), then wouldn't that be essentially like an absolute value?
 
  • #5
Sure. Modulus of z is |z| is sqrt(a^2+b^2). It's still not the same as z^2. What's the question again?
 

What are complex numbers?

Complex numbers are numbers that consist of a real part and an imaginary part. They are written in the form a + bi, where a is the real part and bi is the imaginary part, with i being the square root of -1. Complex numbers are used to represent quantities that cannot be expressed with only real numbers, such as the solution to the equation x^2 + 1 = 0.

What are the properties of complex numbers?

The main properties of complex numbers are addition, subtraction, multiplication, and division. Addition and subtraction work the same way as with real numbers, where the real parts and imaginary parts are added or subtracted separately. Multiplication is done by distributing and using the property that i^2 = -1. Division is done by multiplying the numerator and denominator by the complex conjugate of the denominator, which is found by changing the sign of the imaginary part.

How are complex numbers represented on a graph?

Complex numbers can be represented on a graph called the complex plane. The real part is plotted on the horizontal axis, while the imaginary part is plotted on the vertical axis. This allows for complex numbers to be visualized and used in geometric applications.

What is the conjugate of a complex number?

The conjugate of a complex number is found by changing the sign of the imaginary part. For example, the conjugate of 3 + 4i is 3 - 4i. Conjugates are useful in simplifying complex number expressions and in finding the inverse of a complex number.

What is the absolute value of a complex number?

The absolute value, or modulus, of a complex number is its distance from the origin on the complex plane. It is found by taking the square root of the sum of the squares of the real and imaginary parts. This can also be thought of as the magnitude or length of the complex number.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
2K
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
9
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Precalculus Mathematics Homework Help
Replies
3
Views
469
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
8
Views
343
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
Back
Top