Complex numbers: Why is the modulus of z

In summary, the modulus of a complex number, z, is equal to the square root of the sum of the squares of its real and imaginary parts, |z| = √(a^2+b^2). This is because the modulus is meant to measure the "size" of the number, or its distance from 0. Any concept of modulus should satisfy certain properties, including |0| = 0, |x| > 0 for any non-zero number x, and |ax| = |a||x| for a real number a. The suggested alternative, |a+ib| = √(a^2-b^2), does not satisfy these properties and therefore is not a valid measure of modulus
  • #1
Alshia
28
0
Why is the modulus of z, a complex number, |z| = √(a^2+b^2)?

Why is it not |z| = √(a^2+(ib)^2)?
 
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  • #2
Because in that case, the modulus of a+ ai, for any a, would be 0. And the modulus is supposed to measure the "size" of the number- specifically, its distance from 0.

Any concept of "modulus", or, more generally, "norm", should satisfy
1) |0|= 0 and if x is not 0, |x|> 0
2) If a is a real number, |ax|= |a||x| where "|a|" is the usual absolute value of a real number
3) [itex]|a+ b|\le |a|+|b|[/itex]

Your suggestion, |a+ ib|= √(a^2- b^2) would not satisfy those.
 
  • #3
Thank you, [strike]WallsofIvy[/strike] HallsofIvy. :smile:
 
  • #4
Alshia said:
Thank you, [strike]WallsofIvy[/strike] HallsofIvy. :smile:

Well, the halls have walls!
 
  • #5
The modulus is supposed to be the distance between (0,0) and (a,b). You are suggestion does not give the distance.
 

Q1: What is the modulus of a complex number?

The modulus of a complex number z, denoted as |z|, is the distance between the origin and the complex number in the complex plane. It can also be thought of as the absolute value of the complex number, which is calculated by taking the square root of the sum of the squares of the real and imaginary parts of the complex number.

Q2: How is the modulus of a complex number related to its magnitude?

The magnitude of a complex number, also known as its absolute value, is the same as its modulus. Both terms refer to the distance of the complex number from the origin in the complex plane.

Q3: What is the significance of the modulus of a complex number?

The modulus of a complex number provides important information about the complex number, such as its distance from the origin and its magnitude. It can also be used to calculate the argument or phase angle of the complex number, which gives the direction of the complex number in the complex plane.

Q4: How is the modulus of a complex number calculated?

To calculate the modulus of a complex number z = a + bi, where a is the real part and bi is the imaginary part, we use the formula |z| = √(a² + b²). This formula is derived from the Pythagorean theorem, where the modulus acts as the hypotenuse of a right-angled triangle with sides a and b.

Q5: Can the modulus of a complex number be negative?

No, the modulus of a complex number cannot be negative. It is always a positive real number or zero. This is because the modulus represents the distance between the complex number and the origin, and distance cannot be negative.

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