Absolute value of complex numbers

In summary, when |a| = |b|, then either a = b or a = b*, where a and b are complex numbers and b* is the complex conjugate of b. This statement can also be rewritten or interpreted as a / b lying on the unit circle, a and b lying on a circle in the complex plane, the distance of a and b to the origin being the same, or other conditions involving the real and imaginary parts of a and b.
  • #1
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When is it true that is |a|=|b|, then either a=b or a=b*, where a and b are complex numbers and b* is the complex conjugate of b?
 
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  • #2
-1 is not the complex conjugate of 1.
 
  • #3
What I'm asking is, are there any conditions on a and b such that the above statement is true?
 
  • #4
Yes, but they are all trivial rewritings of |a| = |b|.
If it were as simple as "a = b or a = b*", for example, we wouldn't need the bar notation, would we :smile:

Some of those rewritings / interpretations are:
* a / b lies on the unit circle
* a and b lie on a circle in the complex plane
* the distance of a and b to the origin is the same
* Re(a)^2 - Re(b)^2 = Im(b)^2 - Im(a)^2
* There exists [itex]r > 0, \theta_1, \theta_2 \in [0, 2\pi)[/itex] such that [itex]a = r e^{i\theta 1}[/itex], [itex]b = r e^{i\theta_2}[/itex] (and r = |a| = |b|)
* There exists [itex]\theta \in [0, 2\pi)[/itex] such that [itex]a = e^{i \theta} b[/itex].
 

1. What is the absolute value of a complex number?

The absolute value of a complex number is a measure of its distance from the origin on the complex plane. It is also known as the modulus or magnitude of the complex number and is always a non-negative real number.

2. How is the absolute value of a complex number calculated?

The absolute value of a complex number is calculated by taking the square root of the sum of the squares of its real and imaginary parts. In other words, if a complex number is expressed as z = a + bi, where a and b are real numbers, then the absolute value of z is given by |z| = √(a² + b²).

3. What is the difference between the absolute value of a real number and a complex number?

The absolute value of a real number is simply the distance of that number from zero on the number line, while the absolute value of a complex number is the distance of that number from the origin on the complex plane. This means that while the absolute value of a real number is always positive, the absolute value of a complex number can be positive, zero, or even negative.

4. Can the absolute value of a complex number be imaginary?

No, the absolute value of a complex number is always a real number. This is because it represents a distance, which is a real quantity, and cannot have an imaginary component.

5. How is the absolute value of a complex number used in mathematics?

The absolute value of a complex number is used in various mathematical operations and concepts such as finding the magnitude of a vector, solving equations involving complex numbers, and calculating complex roots of polynomials. It is also a key concept in understanding the geometry of the complex plane.

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