Summary ® 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. ® Let z = a + ib and z = c + id . Then z + z = ( a + c ) + i ( b + d ) z z = ( ac – bd ) + i ( ad + bc ) ® For any non-zero complex number z = a + ib ( a ≠ , b ≠ ), there exists the complex number b b b , denoted by z or z – , called the multiplicative inverse of z such that ( a + ib ) b b b = + i = ® For any integer k , i k = , i k + = i , i k + = – , i k + = – i ® The conjugate of the complex number z = a + ib , denoted by z , is given by z = a – ib. ® The polar form of the complex number z = x + iy is r (cos θ + i sin θ ), where r = (the modulus of z ) and cos θ = x r , sin θ = 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 . ® A polynomial equation of n degree has n roots. ® The solutions of the quadratic equation ax + bx + c = , where a , b , c ∈ R, a ≠ , b – ac < , are given by x = b ac b i −± COMPLEX NUMBERS AND QUADRATIC EQUATIONS Historical Note The fact that square