What is a meaning of a complex roots? - Mathematics Stack Exchange 0 In the simplest sense, what does a complex root on the complex plane actually mean? For polynomial equations a root is a x-coordinate of where the curve crosses the x-axis I just find it strange how the roots of complex numbers are symmetrical about the origin of the complex plane, yet no such symmetry exists for polynomial roots
Graphically solving for complex roots -- how to visualize? We can present complex roots to equation on the "complex plane" with one axis for the real part and the other for the imaginary part You can play with, for instance, WolframAlpha, to give it a polynomial equation to solve and get a display of the complex roots If you look up "DeMoivre's Theorem" online, you will find something interesting about the roots of equations $ \ z^n \ = \ c
Complex roots of real polynomials and real roots of their derivatives Applied to quartic equations with two sets of complex conjugate roots, the theorem implies that in general the roots of the quartic are at the vertices of a quadrilateral in the complex plane and the roots of the derivative (real and otherwise) lie inside this quadrilateral
Explaining the nature of complex roots of a quartic Thank you for this I think I will start by demonstrating the distributive and commutative properties of the complex conjugate, then using those to state that if a complex numbers is a root of a polynomial, so is its conjugate, and therefore complex roots must come in conjugate pairs Then I can just list the cases for each order polynomial
Is there an intuitive way of visualising complex roots? The axis of symmetry is the real part of the complex roots; the imaginary part can be found by subtracting the square of the axis (here, $4$) from the intercept ($13-4=9$) and then taking the square root ($3$) This assumes the roots come in conjugate pairs (so the coefficients of your quadratic are real numbers)
complex numbers - What is $\sqrt {i}$? - Mathematics Stack Exchange The square root of i is (1 + i) sqrt (2) [Try it out my multiplying it by itself ] It has no special notation beyond other complex numbers; in my discipline, at least, it comes up about half as often as the square root of 2 does --- that is, it isn't rare, but it arises only because of our prejudice for things which can be expressed using small integers
How to prove that a polynomial of degree $n$ has at most $n$ roots? 5 Just a clarification here The Fundamental Theorem of Algebra says that a polynomial of degree n will have exactly n roots (counting multiplicity) This is not the same as saying it has at most n roots To get from "at most" to "exactly" you need a way to show that a polynomial of degree n has at least one root Then you can proceed by induction
Do complex roots always come in pairs? - Mathematics Stack Exchange Following this line of reasoning, it shouldn't be necessary for complex roots of a complex polynomial to come in pairs However, this premise was used in solving one of my school assignments Do complex roots always have to come in pairs, regardless of the field in which the polynomial was defined?
Counting the Number of Real Roots of A Polynomial The best way to solve this is to use Sturm's theorem This gives an algorithm for computing the number of distinct real roots of any polynomial The Wikipedia page is quite good, but I'll outline the method here