COMPLEX NUMBERS AND QUADRATIC EQUATIONS
Frequently Asked Questions
A complex number is a number of the form \(z = a + ib\), where \(a, b \in \mathbb{R}\) and \(i = \sqrt{-1}\).
For \(z = a + ib\), the real part is \(\Re(z)=a\) and the imaginary part is \(\Im(z)=b\).
The imaginary unit \(i\) is defined by \(i^2 = -1\).
If \(b=0\), the complex number is purely real; if \(a=0\), it is purely imaginary.
The modulus of \(z=a+ib\) is \(|z|=\sqrt{a^2+b^2}\).
The argument \(\theta\) of \(z=a+ib\) satisfies \(\tan\theta=\frac{b}{a}\), taking the correct quadrant into account.
The principal argument \(\arg z\) lies in the interval \((-\pi,\pi]\).
The conjugate of \(z=a+ib\) is \(\bar z=a-ib\).
It is represented as a point \((a,b)\) or a vector in the Argand plane.
It is a plane in which the x-axis represents real parts and the y-axis represents imaginary parts.
The polar form is \(z=r(\cos\theta+i\sin\theta)\), where \(r=|z|\).
Euler’s form is \(z=re^{i\theta}\).
It represents the distance of the point from the origin.
Conjugation represents reflection across the real axis.
It follows the parallelogram law of vector addition.