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If the discriminant is positive, the number of non-real roots is a multiple of 4. That is, there is a nonnegative integer k ≤ n /4 such that there are 2 k pairs of complex conjugate roots and n − 4 k real roots.
Furthermore, if the Jacobian determinant at p is positive, then f preserves orientation near p; if it is negative, f reverses orientation. The absolute value of the Jacobian determinant at p gives us the factor by which the function f expands or shrinks volumes near p; this is why it occurs in the general substitution rule.
For operators in a finite factor, one may define a positive real-valued determinant called the Fuglede−Kadison determinant using the canonical trace. In fact, corresponding to every tracial state on a von Neumann algebra there is a notion of Fuglede−Kadison determinant.
In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. ... If the discriminant is positive, ...
If the three roots are real and distinct, the discriminant is a product of positive reals, that is > If only one root, say r 1 , is real, then r 2 and r 3 are complex conjugates, which implies that r 2 – r 3 is a purely imaginary number , and thus that ( r 2 – r 3 ) 2 is real and negative.
The discriminant of K is 49 = 7 2. Accordingly, the volume of the fundamental domain is 7 and K is only ramified at 7. In mathematics, the discriminant of an algebraic number field is a numerical invariant that, loosely speaking, measures the size of the (ring of integers of the) algebraic number field.
In number theory, Carmichael's theorem, named after the American mathematician R. D. Carmichael, states that, for any nondegenerate Lucas sequence of the first kind U n (P, Q) with relatively prime parameters P, Q and positive discriminant, an element U n with n ≠ 1, 2, 6 has at least one prime divisor that does not divide any earlier one except the 12th Fibonacci number F(12) = U 12 (1, − ...
If the discriminant is positive, then there are two distinct roots +, both of which are real numbers. For quadratic equations with rational coefficients, if the discriminant is a square number , then the roots are rational—in other cases they may be quadratic irrationals .