Contradiction in the definition and properties of the abs

[Jhenrique]

If the abs(a+b) = abs(a) + abs(b), so the abs(z) = abs(x+iy) = abs(x) + abs(iy) = abs(x) + i abs(y). However, the correct wouldn't be abs(z) = √[x²+y²] ?

√[x²+y²] ≠ abs(x) + i abs(y) => abs(z) ≠ abs(z)

It's no make sense. What there is of wrong with those definions?

 

[PeroK]

If the abs(a+b) = abs(a) + abs(b), so the abs(z) = abs(x+iy) = abs(x) + abs(iy) = abs(x) + i abs(y). However, the correct wouldn't be abs(z) = √[x²+y²] ?

√[x²+y²] ≠ abs(x) + i abs(y) => abs(z) ≠ abs(z)

It's no make sense. What there is of wrong with those definions?

$$|a + b| \neq |a| + |b|$$
E.g. a = 2, b = -1

 

[Mark44]

If the abs(a+b) = abs(a) + abs(b)

As PeroK notes, this is incorrect. Since your hypothesis is false, any following work is meaningless.

Also, the | key on the keyboard is usually used to write absolute values, as in |a + b|.

, so the abs(z) = abs(x+iy) = abs(x) + abs(iy) = abs(x) + i abs(y). However, the correct wouldn't be abs(z) = √[x²+y²] ?

√[x²+y²] ≠ abs(x) + i abs(y) => abs(z) ≠ abs(z)

It's no make sense. What there is of wrong with those definions?