Origin (mathematics)

In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. It helps us describe where everything else is located in relation to this central point.

In physical problems, the choice of origin is often arbitrary, meaning any choice of origin will ultimately give the same answer. This allows us to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry. Choosing the right origin can make solving problems much easier and more straightforward.

Cartesian coordinates

In a Cartesian coordinate system, the origin is the point where the axes meet. This special point helps us find other points in space. We can describe any point’s location by its coordinates, which tell us how far along each axis the point is, either forward or backward from the origin. The origin itself always has coordinates of zero, like (0,0) on a flat grid or (0,0,0) in three-dimensional space.

Other coordinate systems

In a polar coordinate system, the origin is also called the pole. It doesn’t have clear polar coordinates because the angle isn’t defined at the origin itself.

In Euclidean geometry, you can pick any point as the origin that works best for you.

The origin of the complex plane is where the real axis and imaginary axis meet. This point matches the complex number zero.