The wikipedia article on nth root states the following.
"Every complex number has n different nth roots in the complex plane."
can this also be said of real numbers. I know that using de Moivre's theorem, complex roots of a real number can be found. Would it be correct to say that every real number has n nth roots.
Also, does every nth degree polynomial with real coefficients have n roots (including repeated roots). The wikipedia article on Fundamental Theorem of Algebra, seemed to suggest that the above statement is true for polynomials with complex coefficients.
thanks
"Every complex number has n different nth roots in the complex plane."
can this also be said of real numbers. I know that using de Moivre's theorem, complex roots of a real number can be found. Would it be correct to say that every real number has n nth roots.
Also, does every nth degree polynomial with real coefficients have n roots (including repeated roots). The wikipedia article on Fundamental Theorem of Algebra, seemed to suggest that the above statement is true for polynomials with complex coefficients.
thanks
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Just because a polynomial's coefficients are real does not mean its roots will be.
For example, the polynomial x^2 + 1 has no real roots. It does, however, have 2 complex roots (i and -i).
Also, remember that all real numbers are complex. (not all complex numbers are real, however).
Therefore, the statement that every nth degree polynomial with real coefficients will have n roots implies that the n roots will be complex.
To answer your question, it WOULD be correct to say that every real number has n nth roots, however, you have to allow some or all of those roots to be complex. The reason this is true is because it is true for complex numbers, which is a stronger statement. The reals are a subset of the complex numbers. It would not be correct to say that every real number has n real roots. For example, the number -1 is real, but it has no real square roots, it has complex square roots.
For example, the polynomial x^2 + 1 has no real roots. It does, however, have 2 complex roots (i and -i).
Also, remember that all real numbers are complex. (not all complex numbers are real, however).
Therefore, the statement that every nth degree polynomial with real coefficients will have n roots implies that the n roots will be complex.
To answer your question, it WOULD be correct to say that every real number has n nth roots, however, you have to allow some or all of those roots to be complex. The reason this is true is because it is true for complex numbers, which is a stronger statement. The reals are a subset of the complex numbers. It would not be correct to say that every real number has n real roots. For example, the number -1 is real, but it has no real square roots, it has complex square roots.