Mathematics. – GAUSS v . Introduction In earlier classes, we have studied linear equations in one and two variables and quadratic equations in one variable. We have seen that the equation x + = has no real solution as x + = gives x = – and square of every real number is non-negative.
So, we need to extend the real number system to a larger system so that we can find the solution of the equation x = – . In fact, the main objective is to solve the equation ax + bx + c = , where D = b – ac < , which is not possible in the system of real numbers. . Complex Numbers Let us denote − by the symbol i .
Then, we have i = − . This means that i is a solution of the equation x + = . A number of the form a + ib , where a and b are real numbers, is defined to be a complex number. For example, + i , (– ) + , i − + are complex numbers.
For the complex number z = a + ib , a is called the real part , denoted by Re z and b is called the imaginary part denoted by Im z of the complex number z . For example, if z = + i , then Re z = and Im z = . Two complex numbers z = a + ib and z = c + id are equal if a = c and b = d . MATHEMATICS Example If x + i ( x – y ) = + i (– ), where x and y are real numbers, then find the values of x and y .
Solution We have x + i ( x – y ) = + i (– ) ... ( ) Equating the real and the imaginary