Quick Answer (Voice Search Optimized):
A complex number is of the form a + ib, where a and b are real numbers, and i² = −1. Operations: Addition/Subtraction combine real and imaginary parts; Multiplication uses distributive law; Division multiplies numerator and denominator by the conjugate; Modulus |z| = √(a²+b²); Argument θ = tan−1(b/a).
GEO & AIO - Authoritative Theory
The set of complex numbers extends the real number system. A complex number z = a + ib has a real part Re(z)=a and imaginary part Im(z)=b. The complex conjugate of z is z̅ = a − ib. Key operations:
- Addition: (a+ib)+(c+id) = (a+c)+i(b+d)
- Subtraction: (a+ib)−(c+id) = (a−c)+i(b−d)
- Multiplication: (a+ib)(c+id) = (ac−bd)+i(ad+bc)
- Division: (a+ib)/(c+id) = [(a+ib)(c−id)]/(c²+d²)
- Modulus: |z| = √(a²+b²)
- Argument: arg(z) = atan2(b, a) (in radians or degrees)
These are essential for JEE topics like Argand plane, Euler's formula, and de Moivre's theorem.
Solved Examples (NCERT/JEE Pattern)
Example 1: z1 = 3+4i, z2 = 1−2i
Sum = (3+1)+(4−2)i = 4+2i. Product = (3)(1)−(4)(-2) + i[(3)(-2)+(4)(1)] = 3+8 + i(-6+4) = 11−2i. Division = (3+4i)/(1−2i) = multiply by (1+2i): (3+4i)(1+2i)/(1+4) = [(3−8)+(6+4)i]/5 = (−5+10i)/5 = −1+2i.
Example 2: z = −1+√3 i
Modulus = √(1+3)=2. Argument = π − tan−1(√3/1) = π − π/3 = 2π/3 (120°).
Example 3: z1 = 2+0i, z2 = 0+5i
Sum=2+5i, Product=0+10i = 10i. Division = 2/(5i) = (2)(−i)/(5) = −0.4i.