Numbers are features of the world around us. The usual aim is to invent new theorems, not new numbers. The square of any number is always positive. So negative numbers cannot have square roots.
Numbers are artificial inventions made by human beings, capturing many aspects of nature. Imaginary numbers were inevitable, useful and somehow on a par with the more familiar real ones.
Mathematical ideas almost always emerge from something far more complicated. Descartes and Newton interpreted these imaginary numbers as a sign that a problem has no solution. They could be used to find solutions, they could ocurr when solutions did exist.
When something works, and you feel you need it, you generally stop asking whether it makes sense. Cauchy theorme concerns definite integrals of complex functions, expressions where a and b are complex numbers. Ideas are useless if no one gets to hear them.
The entire structure of a complex function is determined by its singularities- the points at which it is badly behaved. And the most important singularities are its poles, the places where it becomes infinite.
Numbers are artificial inventions made by human beings, capturing many aspects of nature. Imaginary numbers were inevitable, useful and somehow on a par with the more familiar real ones.
Mathematical ideas almost always emerge from something far more complicated. Descartes and Newton interpreted these imaginary numbers as a sign that a problem has no solution. They could be used to find solutions, they could ocurr when solutions did exist.
When something works, and you feel you need it, you generally stop asking whether it makes sense. Cauchy theorme concerns definite integrals of complex functions, expressions where a and b are complex numbers. Ideas are useless if no one gets to hear them.
The entire structure of a complex function is determined by its singularities- the points at which it is badly behaved. And the most important singularities are its poles, the places where it becomes infinite.