# Complex numbers: Why is the modulus of z

1. May 26, 2012

### Alshia

Why is the modulus of z, a complex number, |z| = √(a^2+b^2)?

Why is it not |z| = √(a^2+(ib)^2)?

2. May 26, 2012

### HallsofIvy

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) $|a+ b|\le |a|+|b|$

Your suggestion, |a+ ib|= √(a^2- b^2) would not satisfy those.

3. May 26, 2012

### Alshia

Thank you, [strike]WallsofIvy[/strike] HallsofIvy.

4. May 27, 2012

### HallsofIvy

Well, the halls have walls!

5. May 27, 2012

### micromass

The modulus is supposed to be the distance between (0,0) and (a,b). You are suggestion does not give the distance.