absolute value, phase, modulus

  

Figure 1: Graph of the absolute value of (z-1)/(z⁶-1).

The radius of a complex number, written in polar form, is zero only when both the real part and the imaginary part of the number are zero, and conversely. It therefore serves as an absolute value, or norm, symbolized in the usual manner as |z|.

$$\surd(x^2+y^2).$$ |x+iy| = (11)

The factorization

x²+y² = (x+iy)(x-iy). (12)

prompts the introduction of the complex conjugatecomplex conjugate $\bar{z}$ of a complex number z, by setting

$$\overline{x+iy}$$ = x - iy. (13)

It thereby follows that $\vert z\vert^2=z\bar{z}$.

The angle $\theta$ in the polar decomposition of a complex number is called its phasephase, or sometimes its amplitudeamplitude, but beware that amplitude does not mean magnitude, unless magnitude refers to the angular distance of the point representing the complex number from the real axis. On the other hand, modulus does mean magnitude.

The absolute value is multiplicative, which can be checked by comparing the algebraic expressions involved; in other words |wz|=|w||z|. It also obeys the triangle inequality, $\vert w+z\vert \le \vert w\vert+\vert z\vert$. Amongst other things, those two properties are sufficient to check the convergence of the sine, cosine, and exponential series for complex numbers, to verify Euler's formula. It would still be premature to interpret the coefficients in the Taylor's series as derivatives, however.