Iterative and recursive control structures are the most fundamental mechanisms of computing that make programming more effective and expressive. However, these constructs are perhaps the most diverse and confusable instructions in programming languages at both syntactic and semantic levels. This article introduces the big-R notation that provides a unifying mathematical treatment of iterations and recursions in computing. Mathematical models of iterations and recursions are developed using

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... al inductions. Based on the mathematical model of the big-R notation, fundamental properties of iterative and recursive behaviors of software are comparatively analyzed. The big-R notation has been adopted and implemented in Real-Time Process Algebra (RTPA) and its supporting tools. Case studies demonstrate that a convenient notation ma�� dramaticall�� reduce the difficult�� and complexit�� in expressing a frequentl�� used and highly recurring concept and notion in computing and software engineering. Keywords: basic control structures�� cognitive informatics�� computing�� formal methods�� iteration�� loop�� mathematical notations�� RTPA�� recursion�� semantics�� software engineering�� syntax�� the big-R notation