ncert books for class maths cbsc · Part

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 root of a negative number does not exist in the real number system was recognised by the Greeks.

But the credit goes to the Indian mathematician Mahavira ( ) who first stated this difficulty clearly. “He mentions in his work ‘ Ganitasara Sangraha ’ as in the nature of things a negative (quantity) is not a square (quantity)’, it has, therefore, no square root”. Bhaskara , another Indian mathematician, also writes in his work Bijaganita , written in . “There is no square root of a negative quantity, for it is not a square.” Cardan ( ) considered the problem of solving x + y = , xy = .

He obtained x = + and y = – as the solution of it, which was discarded by him by saying that these numbers are ‘useless’. Albert Girard (about ) accepted square root of negative numbers and said that this will enable us to get as many roots as the degree of the polynomial equation. Euler was the first to introduce the symbol i for − and W.R. Hamilton (about ) regarded the complex number a + ib as an ordered pair of real numbers ( a , b ) thus giving it a purely mathematical definition and avoiding use of the so called ‘ imaginary numbers ’.