🌀 Complex Numbers Argand Plane Visualizer
Complex NumbersWhat's happening? z = a + bi is a point on the Argand plane. |z| = √(a²+b²) is the distance from origin. Rotating by θ multiplies by e^(iθ).
Complex numbers: 3-5 questions every year.
JEE tests modulus, argument, De Moivre, cube roots.
Key formulas
Complex Numbers Formulas
Modulus
|z| = sqrt(a^2+b^2)
Distance from origin
Argument
arg(z) = atan2(b,a)
Angle with real axis
Polar
z = r(cos+i sin) = r*e^(i*theta)
Euler form
De Moivre
z^n = r^n(cos(n*theta)+i*sin(n*theta))
Powers and roots
Step-by-step problem solving
How to solve maths problems in JEE
- Convert to polar: r, theta
- Multiply: multiply r, add theta
- Divide: divide r, subtract theta
- Powers/roots: De Moivre
Common JEE mistakes
What students get wrong
Argument quadrant
atan2 gives correct quadrant, atan doesn't
Conjugate
z_bar = a-bi, |z|^2 = z*z_bar
Cube roots unity
1, omega, omega^2. 1+omega+omega^2=0
|z1+z2| != |z1|+|z2|
Triangle inequality: <=
FAQs
Complex Numbers Questions
Argand plane?
x-axis = real, y-axis = imaginary. z = point (a,b)
Cube roots?
Polar form, cube root r, theta/3 + 2pi*k/3
Cube roots of unity?
1, omega, omega^2. Sum = 0, product = 1
Multiplication geometrically?
Scale by |z|, rotate by arg(z)