Euclidean Plane

Abstract

The word “geometry” is derived from the Greek words geos and metron, meaning earth and measure, whose definition is generally attributed to the fact that the ancient Egyptians regularly utilized geometry to resurvey the fertile farmlands of the Nile river floodplain in late summer. The concepts of “distance” and “area” need not be defined; they are already given by nature. A plane with this concept of distance is called the Euclidean plane, denoted by ${\mathbb E}^2$. It does not have special points or directions. When this plane is equipped with a coordinate system, it is given the origin 0 and x- and y-axes. The Euclidean plane with a coordinate system can be identified by $\mathbb {R}^2$, the set of all ordered pairs (x, y) of real numbers. We will not distinguish $\mathbb {R}^2$ and ${\mathbb E}^2$ in this book.

“The study of mathematics, like the Nile, begins in minuteness but ends in magnificence.”

Charles Caleb Colton (1780–1832)

“Equations are just the boring part of mathematics. I attempt to see things in terms of geometry.”

Stephen Hawking (1942–2018)