Modulus ($|z|$)

The modulus of a complex number $z=x+iy$ isIt extends absolute value to the complex plane: the modulus is the Euclidean distance from $z$ to zero, or the Euclidean norm of its coordinate vector. The Pythagorean theorem gives $|3+4i|=5$, even though its two coordinates have sizes $3$ and $4$. The modulus is always nonnegative, and is zero exactly at zero.

Two essential rules areThe second is the triangle inequality. For $z=3+4i$ and $w=1+i$, the product has modulus $\sqrt{50}=5\sqrt{2}\approx 7.071068$, while the sum has modulus $\sqrt{41}\approx 6.403124<5+\sqrt{2}\approx 6.414214$. The strict inequality records that the two displacement vectors point in different directions.

The Euclidean distance between two complex numbers $z,w$ is $|z-w|$. Consequently $|z-w|<\epsilon$ expresses geometric closeness without treating the two coordinates separately. This makes the modulus a natural measure of error in complex analysis; the triangle inequality bounds accumulated error, while the product rule tracks how multiplication changes its size. Equal moduli do not imply equal complex numbers: $5$, $-5$, and $3+4i$ all lie on the same circle around zero. The modulus records size and deliberately forgets direction; the argument supplies the missing information.