( ) = a z

( ) = a z a z  = a z a z  = a z a z  ( a as a r is real for all r ) = a z a z  = a z a z  = P z ( That is P z ( ; this implies that whenever z is a root (i.e. P ( z )= ), its conjugate z is also a root . If one asks whether is a complex number, many students hesitate to say “yes”. As every integer is a rational number, we know that every real number is also a complex number.

So to clearly specify a complex number that is not a real number, that is to specify numbers of form α β + i with β ≠ , we use the term “non-real complex number” . Some authors call such a number an imaginary number . Remark Let z = β with β ≠ . Then z = β .

If α β + i is a root of a polynomial equation P x ( ) = with real coefficients, then by Complex Conjugate Root Theorem, α β − i is also a root of P x ( ) = . Usually the above statement will be stated as complex roots occur in pairs ; but actually it means that non-real complex roots or imaginary roots occur as conjugate pairs , being the coefficients of the polynomial equation are real . Remark From this we see that any odd degree polynomial equation with real coefficients has at least one real root; in fact, the number of real roots of an odd degree polynomial equation with real coefficients is always an odd number. Similarly the number of real roots of an even degree polynomial equation with real coefficients is always an even number.

Example . Find the monic polynomial equation of minimum degree with real coefficients having as a root. Since i is a root of the required polynomial equation with real coefficients, i is also a root. Hence the sum of the roots is and the product of the roots is .

Thus x is the required monic polynomial equation.