Prove that all the values of √(-1) are 1, , ^2, …, ^n-1. |...

Name: Problem 231, Chapter 2: Abel'S Theorem in Problems and Solutions Based on the Lectures of Professor V.I. Arnold
Uploaded: 2020-05-31T20:41:01-08:00
Duration: 3 min 26 s
Channel: Aamir Mithaiwala
Description: Prove that all the values of √(-1) are 1, , ^2, …, ^n-1.

Key Concepts

Complex Numbers in Polar Form

Complex numbers can be represented in the form r e^(i?), where r is the magnitude and ? is the argument (angle). This form is particularly useful because it allows one to simplify the process of taking powers and roots of complex numbers by operating on the magnitude and the angle separately.

De Moivre's Theorem

De Moivre's Theorem states that (cos ? + i sin ?)^n = cos(n?) + i sin(n?). This theorem is a fundamental tool for raising complex numbers in polar form to integer powers and, conversely, for extracting roots by relating the exponent to the angle, as required in proving formulas for the nth roots of a given number.

nth Roots of a Complex Number

Finding the nth roots of a complex number involves expressing the number in polar form and then taking the nth root of the magnitude while dividing the argument by n. The periodicity of the trigonometric functions implies that one must add multiples of 2? to the original argument, which yields the complete set of n distinct roots. This process explains why the roots are given by a base value raised to successive powers.

Primitive Roots of Unity

A primitive nth root of unity is a complex number that, when raised to the nth power, gives 1, and its successive powers generate all the distinct nth roots of unity. In contexts where a different number such as -1 is considered, a similar idea applies by writing the number in exponential form and identifying a primary root from which all others can be derived by successive multiplication.

Fundamental Theorem of Algebra

The Fundamental Theorem of Algebra guarantees that a polynomial of degree n has exactly n complex roots (counting multiplicities). This theorem underpins the argument that once n distinct roots are found for an expression such as x^n = -1, no further roots exist, thereby completing the proof of the set of values.

Transcript

Please give Ace some feedback