OLYMPIAD > COMPLEX NUMBERS
Complex numbers are built around the idea of the square root of a negative number. Of course, such a number does not exist on the real plane and so one must
The Mandelbrot using complex numbersventure into the complex plane to explore its nature. The first reference of the square root of a negative number occured during the 1st century AD when Greek mathematician Heron of Alexandria considered the volume of an impossible frustum of a pyramid.
Complex numbers later showed its existence in the 16th century when two mathematicians, Tartaglia and Cardano, were caught in an endless dispute of who was the first to solve a cubic equation. Their search for a solution to a cubic equation brought them to complex plane where square roots of negative numbers started to show. More work was spent on investigating these mysterious roots and it was Rene Descartes who in 1637 first coined the term 'imaginary roots'.
Advance to the 18th century and two prime mathematicians in the form of Abraham de Moivre and Leonhard Euler solidify the concepts in this field. Since then, complex numbers has reached a stage in its development where it is vital in understanding other fields such as quantum mechanics.
Abraham de Moivre
Leonhard Euler
de Moivre's Theorem
nth roots of a complex number
Quantum Mechanics
Relativity
Complex numbers are built around the idea of the square root of a negative number. Of course, such a number does not exist on the real plane and so one must
The Mandelbrot using complex numbers
Complex numbers later showed its existence in the 16th century when two mathematicians, Tartaglia and Cardano, were caught in an endless dispute of who was the first to solve a cubic equation. Their search for a solution to a cubic equation brought them to complex plane where square roots of negative numbers started to show. More work was spent on investigating these mysterious roots and it was Rene Descartes who in 1637 first coined the term 'imaginary roots'.
Advance to the 18th century and two prime mathematicians in the form of Abraham de Moivre and Leonhard Euler solidify the concepts in this field. Since then, complex numbers has reached a stage in its development where it is vital in understanding other fields such as quantum mechanics.
CONCEPT CONTENTS
> ADDITION AND MULTIPLICATION
> CONJUGATE AND MAGNITUDE
> THE COMPLEX PLANE
> ADDITION AND MULTIPLICATION ON THE COMPLEX PLANE
> PROOF USING TRIGONOMETRY IDENTITIES
> FURTHER COMPLEX NUMBERS RESULTS
> DE MOIVRE'S THEOREM
> NTH ROOTS OF A COMPLEX NUMBER
> GEOMETRICAL INTERPRETATION OF NTH ROOTS
AMC QUESTIONS
> 1977 AHSME #16
> 1990 AHSME #22 solution 1
> 1990 AHSME #22 solution 2
Notable Mathematicians
Rene DescartesAbraham de Moivre
Leonhard Euler
Important Theorems
Euler's Formulade Moivre's Theorem
nth roots of a complex number
Applications
Control TheoryQuantum Mechanics
Relativity
