# What is least positive argument of a comlex number?

sushant singh
66 Points
13 years ago

Dear abinash,...

for this complex number question for arguments...there is a general derivation,,plz go thru it

Given any complex numbers (called coefficients) a0, ..., an, the equation

$a_n z^n + \dots + a_1 z + a_0 = 0, \,$

has at least one complex solution z, provided that at least one of the higher coefficients, a1, ..., an, is nonzero. This is the statement of the fundamental theorem of algebra. Because of this fact, C is called an algebraically closed field. This property does not hold for the field of rational numbers Q (the polynomial x2 − 2 does not have a rational root, since √2 is not a rational number) nor the real numbers R (the polynomial x2 + a does not have a real solution for a > 0, since the square of x is positive for any real number x).

There are various proofs of this theorem, either by analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one root.

Because of this fact, theorems that hold "for any algebraically closed field", apply to C.

Deepanshu
13 Points
2 years ago
the complex number having the least positive argument in the region is the complex number represented by the point of contact of tangent drawn from the origin to the circle.