Up until the last few centuries, the real number line, which includes the integers, rational numbers, and irrational numbers, was believed to be enough to explain the world. However, as individuals continued to explore the subject, namely through finding the solutions to polynomials, the inadequacies of real numbers became evident. For instance, the square root of negative numbers cannot be found on the real number line: what significance, if any, do these quantities bear in the real world?
To describe these seemingly impossible quantities mathematically, mathematicians named them "complex" numbers and created a plane to demonstrate such numbers graphically.
My research investigates what complex numbers represent, the geometry of operating on them, and various preliminary concepts crucial to understanding field of complex analysis.
To describe these seemingly impossible quantities mathematically, mathematicians named them "complex" numbers and created a plane to demonstrate such numbers graphically.
My research investigates what complex numbers represent, the geometry of operating on them, and various preliminary concepts crucial to understanding field of complex analysis.
What are Complex Numbers?
A complex number, denoted as z, can be written mathematically with the expression z=a+ib, where a and b are real numbers and i=sqrt(-1), the unit imaginary number. Complex numbers exist on the complex plane, which has a real-valued horizontal axis and an imaginary-valued vertical axis. We can graph z by traveling a units across the real axis and b units across the imaginary axis:
Complex numbers can take on a variety of different, but equivalent, forms depending on the types of problems mathematicians are dealing with.