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Can you please explain me modulus and arguement of a complex number

Can you please explain me modulus and arguement of a complex number

Grade:11

1 Answers

vikas askiitian expert
509 Points
10 years ago

Modulus and Argument

Thinking in terms of the Argand diagram we can specify the position of the complex number z = x + jy on the plane by giving the polar coordinates of the point (x, y).

 

Figure 10.3: The modulus - argument representation of z.
\psfrag{r}{$r$}\psfrag{theta}{$\theta$}\psfrag{z=x+jy}{$z = x + jy$}\includegraphics[height=2.5cm ] {xfig/polar.eps}

The polar coordinate r is the distance from O to P and is called the modulus of the complex number z and written as | z|.

r = | z| = $\displaystyle \sqrt{{x^2+y^2}}$ = $\displaystyle \sqrt{{z\bar{z}}}$

The polar coordinate $ \theta$ is called an argument of z. If we take $ \theta$ in the range - $ \pi$ < $ \theta$$ \le$$ \pi$ then we call it the (principal) argument of z and we denote it by arg(z). Note that any argument of z differs from arg(z) by an integer multiple of 2$ \pi$ (working in radians) or of 360o (working in degrees)10.2.

Since x = r cos$ \theta$ and y = r sin$ \theta$ we can write z in terms of its modulus and argument as

$\displaystyle \boxed{z = r(\cos\theta + j\sin\theta) } \qquad r\ge 0,\quad -\pi < \theta \le \pi. $
This is called writing z in polar form or modulus - argument form. Any non-zero complex number can be written in this form. The point 0 is a slightly special case, it has r = 0 but the angle $ \theta$ is not defined.

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