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Let's start by considering the simplest possible geometry<ref>Well, the simplest possible ''interesting'' geometry, anyway.</ref>: the [[Euclidean plane]].

Let's start by considering the simplest possible geometry<ref>Well, the simplest possible ''interesting'' geometry, anyway.</ref>: the [[Euclidean plane]].

The Euclidean plane is an infinite, flat, two-dimensional surface. A sheet of paper is a good approximation of the Euclidean plane. Onto this plane, we can project a set of [[Cartesian coordinates]]. By "Cartesian," we mean that the coordinate axes are straight lines, that they are perpendicular, and that the unit lengths of the axes are equal. A fancier term for a Cartesian coordinate system is an ''orthonormal basis.''

The Euclidean plane is an infinite, flat, two-dimensional surface. A sheet of paper is a good approximation of the Euclidean plane. Onto this plane, we can project a set of [[Cartesian coordinate system|Cartesian coordinates]]. By "Cartesian," we mean that the coordinate axes are straight lines, that they are perpendicular, and that the unit lengths of the axes are equal. A fancier term for a Cartesian coordinate system is an ''orthonormal basis.''

Note carefully the distinction between the Euclidean plane and Cartesian coordinates. The plane exists as a thing in and of itself, just as a blank piece of paper does. It has certain properties, which we'll get into below. Those properties are ''intrinsic'' to the plane. That is, the properties don't have anything to do with the coordinates we project onto the plane. The plane is a geometric object, and the coordinates are the method by which we ''measure'' the plane. (The emphasis on the word ''measure'' there is not accidental; please keep this idea in the foreground of your mind as we continue.)

Note carefully the distinction between the Euclidean plane and Cartesian coordinates. The plane exists as a thing in and of itself, just as a blank piece of paper does. It has certain properties, which we'll get into below. Those properties are ''intrinsic'' to the plane. That is, the properties don't have anything to do with the coordinates we project onto the plane. The plane is a geometric object, and the coordinates are the method by which we ''measure'' the plane. (The emphasis on the word ''measure'' there is not accidental; please keep this idea in the foreground of your mind as we continue.)

Cartesian coordinates are not the only coordinates we can use in the Euclidean plane. For example, instead of having axes that are perpendicular to each other, we could choose axes that are straight lines, but that meet at some non-perpendicular angle. These types of coordinates are called ''oblique.''

Cartesian coordinates are not the only coordinates we can use in the Euclidean plane. For example, instead of having axes that are perpendicular to each other, we could choose axes that are straight lines, but that meet at some non-perpendicular angle. These types of coordinates are called ''oblique.''

For that matter, we're not bound to use straight-line coordinates at all. We could instead choose [[polar coordinates]], wherein every point on the plane is described by a distance from a fixed but arbitrary point and an angle from a fixed but arbitrary direction. Polar coordinates are often more convenient than Cartesian coordinates. For example, when navigating a ship on the ocean, the location of a fixed point is usually described in terms of a ''bearing'' and a ''distance,'' where the distance is the straight-line distance from the ship to the point, and the bearing is the clockwise angle relative to the direction in which the ship is sailing. Polar coordinates in two and three dimensions are often used in physics for similar reasons.

For that matter, we're not bound to use straight-line coordinates at all. We could instead choose [[polar coordinates]], wherein every point on the plane is described by a distance from a fixed but arbitrary point and an angle from a fixed but arbitrary direction. Polar coordinates are often more convenient than [[Cartesian coordinate system|Cartesian coordinates]]. For example, when navigating a ship on the ocean, the location of a fixed point is usually described in terms of a ''bearing'' and a ''distance,'' where the distance is the straight-line distance from the ship to the point, and the bearing is the clockwise angle relative to the direction in which the ship is sailing. Polar coordinates in two and three dimensions are often used in physics for similar reasons.

But there's a fundamental problem with polar coordinates that is not present with Cartesian coordinates. In Cartesian coordinates, every point on the Euclidean plane is identified by ''exactly one'' set of real numbers: there is precisely one set of <math>x</math> and <math>y</math> coordinates for every point, and every point corresponds to precisely one set of coordinates.

But there's a fundamental problem with polar coordinates that is not present with Cartesian coordinates. In Cartesian coordinates, every point on the Euclidean plane is identified by ''exactly one'' set of real numbers: there is precisely one set of <math>x</math> and <math>y</math> coordinates for every point, and every point corresponds to precisely one set of coordinates.