Twonumbers

In mathematics and computer science, twonumbers denotes an ordered pair of numbers, typically written as (a, b). As a basic data construct, a twonumber can represent a point in the two-dimensional plane, a two-component vector, or a pair of related numerical values used in algorithms.

Operations: Twonumbers support coordinate-wise addition and subtraction: (a, b) + (c, d) = (a+c, b+d). Scalar multiplication is defined by k*(a, b) = (ka, kb). Equality holds when a=c and b=d. The set of all twonumbers with these operations forms a two-dimensional real vector space, isomorphic to R^2. A common inner product is the dot product: (a, b) • (c, d) = ac + bd, with norm ||(a, b)|| = sqrt(a^2 + b^2). The distance between (a, b) and (c, d) is sqrt((a-c)^2 + (b-d)^2).

Geometry and coordinates: Every twonumber corresponds to a unique point in the Cartesian plane and to a two-dimensional vector from the origin. Many geometric operations, such as transformations by matrices, can be applied to twonumbers via linear algebra. The pair can also be converted to polar coordinates (r, θ) where r = sqrt(a^2 + b^2) and θ = atan2(b, a).

Relation to other concepts: If one equips twonumbers with a specific multiplication rule, (a, b) * (c, d) = (ac − bd, ad + bc), the resulting algebra is isomorphic to the complex numbers. This illustrates how twonumbers can represent numbers with two real components in multiple ways.

Applications: Twonumbers underpin algorithms in computer graphics, robotics, physics simulations, data encoding, and any domain involving 2D coordinates or vectors. They provide a minimal, language-agnostic abstraction for two-element numerical data.