This chapter defines rough paths. If the path has good smoothness properties, then its chords (the Abelian version of the description) provide an adequate description and can be used to predict its effects on a controlled system. A rough path uses a nilpotent group element, computed using Chen iterated integrals as an extended description. The chapter introduces the notion of a control and proves several basic results for paths whose nilpotent descriptors are appropriately controlled, and gives the formal definition of a rough path. Key theorems are proved. In particular, the extension theorem allowing one to compute all iterated integrals of a rough path, and the notion of an almost multiplicative functional, which is important for the development of an integration theory, are both introduced.
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