# dual (1-dual.com)

People use dual to mean two of something, or something that has two parts, sides, or purposes. A dual-purpose tool does two jobs. A dual citizen belongs to two countries.

## What a model may hear
- dual space (linear algebra and functional analysis): assume the user wants the vector space of linear functionals, or operations on cotangent spaces
- dual problem (optimization and operations research): switch to solving the Lagrangian dual, the maximization problem paired with a minimization primal
- dual numbers (automatic differentiation in machine learning): use a+bε numbers where ε²=0 to compute exact derivatives without finite differences
- dual boot (system administration): assume a computer with two installed operating systems, chosen at startup
- dual graph (graph theory and mesh processing): construct the graph where each face becomes a node and adjacent faces share an edge

## Where people and models part ways
- Says: "make it dual"
  Means: create a version with two modes or two audiences
  May be taken as: build a mathematical dual, or reformulate as an optimization dual
  Say instead: "create a version that works in two modes, one for experts and one for beginners"
- Says: "dual input"
  Means: accept two sources of information
  May be taken as: expect a pair of dual vectors, or a bimodule structure
  Say instead: "accept information from two separate sources"
- Says: "dual core"
  Means: two processing cores
  May be taken as: confuse with dual modules in algebra, or dual-core as a specific old processor branding
  Say instead: "two CPU cores on one chip"
- Says: "dual license"
  Means: offer the same software under two different legal agreements
  May be taken as: interpret as a pair of dual vectors or a license dual in some formal system
  Say instead: "release the software under two optional legal agreements, such as GPL and commercial"

## Tips
- say two instead of dual when you simply mean the count
- specify dual boot, dual citizen, dual core, or dual space to signal which realm you mean
- avoid dual alone when asking for paired opposites; say paired opposites or two-sided
- if you mean mathematical duality, name the field: dual in linear algebra, dual in optimization
- do not use dual to mean duplicate; say copy or duplicate instead

## Often confused with
- duel: a fight between two people, spelled with e
- binary: specifically base-2 or on-off, not merely two-part
- twin: two closely matched copies, not two different functions
- bifurcated: split into two branches, not two purposes or roles
- complementary: parts that complete each other, not merely two of something
- primal: the original problem in optimization, opposite of dual