Maths > Complex Numbers > 5.0 Representation of complex number
Complex Numbers
1.0 Definition
2.0 Algebraic operations
3.0 Conjugate of complex number
4.0 Properties of conjugate
5.0 Representation of complex number
5.1 Cartesian form (Geometric Representation)
5.2 Principal value of complex number
5.3 Properties of modulus
5.4 Trigonometric/Polar Representation
6.0 Euler's formulae
7.0 Properties of Argument
8.0 De Moivre's Theorem
9.0 Square root of a complex number
10.0 The ${n^{th}}$ root of unity
11.0 Cube roots of unity
12.0 Rotation
13.0 Geometrical properties
14.0 Locus
15.0 Ptolemy's Theorem
5.2 Principal value of complex number
5.2 Principal value of complex number
5.3 Properties of modulus
5.4 Trigonometric/Polar Representation
If the complex number is given as $z=a+ib$, then in order to find the principal value following steps are to be followed:
Step 1. Find $\tan \theta = \left| {\frac{b}{a}} \right|$ and this gives the value of $\theta $ in the first quadrant.
Step 2. Find the quadrant in which $z$ lies by the sign of $a$ and $b$ i.e., $x$ and $y$ coordinates respectively.
Step 3. Argument of $z$ will be $\theta ,\pi - \theta ,\theta - \pi {\text{ and }} - \theta $ according as $z$ lies in the first, second, third or fourth quadrant.
Question 3. Find the principal argument of complex number $z = \sqrt 3 - i$.
Solution: On comparing $z = \sqrt 3 - i$ with $z=a+ib$, we get $a = \sqrt 3 $ and $b=-1$. Therefore, $$\begin{equation} \begin{aligned} \tan \theta = \left| {\frac{b}{a}} \right| = \left| {\frac{{ - 1}}{{\sqrt 3 }}} \right| \\ \Rightarrow \theta = \frac{\pi }{6} \\\end{aligned} \end{equation} $$
Since from the sign of $a$ and $b$, it lies in $IV$ quadrant, therefore, the principal argument is $ - \theta $ i.e., $ - \frac{\pi }{6}$.
