The Road to Reality Study Notes: Difference between revisions

Penrose introduced the complex numbers, extending addition, subtraction, multiplication, and division of the reals to the system obtained by adjoining $i$, the square root of -1. Polynomial equations can be solved by complex numbers, this property is called ''algebraic closure'' and follows from the Fundamental Theorem of Algebra. Complex numbers can be visualized graphically as a plane, where the horizontal coordinate gives the real coordinate of the number and the vertical coordinate gives the imaginary part. This helps understand the behavior of power series; for example, the power series $1-x^2+x^4+\cdots$ converges to the function $1/(1+x²)$ only when $|x|<1$, despite the fact that the function doesn't seem to have "singular" behavior anywhere on the real line. This is explained by singularities at $x=i,-i$.