presenting the same argument, also state that it does not explain the equality, indicating that such an explanation would likely involve concepts of infinity and completeness. , citing , also conclude that the treatment of the identity based on such arguments as these, without the formal concept of a limit, is premature. The same argument is also given by , who notes that skeptics may question whether is cancellable that is, whether it makes sense to subtract from both sides. Analytic proofs Since the question of 0.999... does not affect the formal development of mathematics, it can be postponed until one proves the standard theorems of real analysis. One requirement is to characterize real numbers that can be written in decimal notation, consisting of an optional sign, a finite sequence of one or more digits forming an integer part, a decimal separator, and a sequence of digits forming a fractional part. For the purpose of discussing 0.999..., the integer part can be summarized as b0 and one can neglect negatives, so a decimal expansion has the form The fraction part, unlike the integer part, is not limited to finitely many digits. This is a positional notation, so for example the digit 5 in 500 contributes ten times as much as the 5 in 50, and the 5 in 0.05 contributes one tenth as much as the 5 in 0.5. Infinite series and sequences A common development of decimal expansions is to define them as sums of infinite series. In general: For 0.999... one can apply the convergence theorem concerning geometric series: Since 0.999... is such a sum with a = 9 and common ratio r = , the theorem makes short work of the question: This proof appears as early as 1770 in Leonhard Euler's Elements of Algebra. The sum of a geometric series is itself a result even older than Euler. A typical 18th-century derivation used a term-by-term manipulation similar to the algebraic proof given above, and as late as 1811, Bonnycastle's textbook An Introduction to Algebra uses such an argument for geometric series to justify the same maneuver on 0.999... A 19th-century reaction against such liberal summation methods resulted in the definition that still dominates today: the sum of a series is defined to be the limit of the sequence of its partial sums. A corresponding proof of the theorem explicitly computes that sequence; it can be found