Combination of one Modulus and one argument can make a complex number?

Question

In my book it is written in bold letters that

Combination of one Modulus and one argument can make one complex number

but it is matter of sorrow that they did not give any example to make understand what they have said . If any one could help me with an example ?

Answer 1

You can express your complex number $$ z=x+iy$$ in polar form $$ z=re^{i\theta} =r(cos(\theta) + i\sin(\theta))$$ where $$r = \sqrt {a^2+b^2}$$ is called the modulus and $ \theta $ the angle from positive $x$-axis to the line segment joining the origin to $z$ is called the argument.

For example for $$ z= \sqrt 3 +i =2e^{i(\pi /6)} $$ The modulus is$ 2$ and the argument is $\pi /6 $

$$ For z= 1 +i = \sqrt 2e^{i(\pi /4)} $$

The modulus is$ \sqrt 2$ and the argument is $\pi /4 $

Answer 2

It is probably referring to the polar and exponential form

$$z=|z|e^{i\theta}=|z|(\cos \theta + i \sin \theta)$$

even if $e^{i\theta}$ and $(\cos \theta + i \sin \theta)$ are also complex number and not argument, in the sense that $z$ is completely defined by

• one modulus $|z|$ and
• one argument $\theta + 2k\pi$