Complex Numbers and Quadratic Equations

• Need for complex numbers, especially √1, to be motivated by inability to solve some of the quadratic equations

• Algebraic properties of complex numbers

• Argand plane and polar representation of complex numbers

• Statement of Fundamental Theorem of Algebra

• Solution of quadratic equations in the complex number system

• Square root of a complex number

SUMMARY

1. A number of the form a + ib, where a and b are real numbers, is called a complex number, a is called the real part and b is called the imaginary part of the complex number.

2. Let z1 = a + ib and z2 = c + id. Then (i) z1 + z2 = (a + c) + i (b + d) (ii) z1 z2 = (ac – bd) + i (ad + bc)

2. The conjugate of the complex number z = a + ib, denoted by z̄ , is given by z̄ = a – ib.

3. The polar form of the complex number z = x + iy is r (cosθ + isinθ), where r = √(x2 + y2) (the modulus of z) and cosθ = x/r , sinθ = y/r . (θ is known as the argument of z. The value of θ, such that – π < θ ≤ π, is called the principal argument of z.

4. A polynomial equation of n degree has n roots.

5. The solutions of the quadratic equation ax2 + bx + c = 0, where a, b, c ∈ R.