Source: Wikipedia. Pages: 105. Chapters: Numeral system, SI prefix, Roman numerals, Binary-coded decimal, Maya numerals, Scientific notation, Binary prefix, Computer numbering formats, Gray code, Counter, Elias gamma coding, Elias delta coding, Babylonian numerals, Greek numerals, List of numbers, Table of bases, 0.999..., Long and short scales, Positional notation, Names of large numbers, Repeating decimal, HindüArabic numeral system, History of the HindüArabic numeral system, Counting, Number prefix, Chronogram, Egyptian numerals, Genealogical numbering systems, Numerical digit, Bijective numeration, Ones' complement mathematics, Nicolas Chuquet, Jacques Pelletier du Mans, Slashed zero, History of writing ancient numbers, Suzhou numerals, Non-standard positional numeral systems, Midy's theorem, Offset binary, Algorism, Units place, Radix, Elias omega coding, List of numeral systems, Prehistoric numerals, Bi-quinary coded decimal, Names of small numbers, Cyrillic numerals, Eastern Arabic numerals, Leading zero, Pace count beads, Exponential-Golomb coding, Engineering notation, Tally marks, Tally counter, Excess-3, Tallyman, Levenstein coding, Radix point, Sign-value notation, Subtractive notation, List of numeral system topics, Hundred, Chuvash numerals, Aegean numerals, Thermometer code. Excerpt: This article is about "bases" as that term is used in discussion of certain numeral systems. This table of bases gives the values of 1-100 in bases 2-20. In mathematics, the repeating decimal 0.999... which may also be written as 0., or 0.(9), denotes a real number that can be shown to be the number one. In other words, the symbols 0.999... and 1 represent the same number. Proofs of this equality have been formulated with varying degrees of mathematical rigour, taking into account preferred development of the real numbers, background assumptions, historical context, and target audience. That certain real numbers can be represented by more than one digit string is not limited to the decimal system. The same phenomenon occurs in all integer bases, and mathematicians have also quantified the ways of writing 1 in non-integer bases. Nor is this phenomenon unique to 1: every nonzero, terminating decimal has a twin with trailing 9s, such as 8.32 and 8.31999... The terminating decimal is simpler and is almost always the preferred representation, contributing to a misconception that it is the only representation. The non-terminating form is more convenient for understanding the decimal expansions of certain fractions and, in base three, for the structure of the ternary Cantor set, a simple fractal. The non-unique form must be taken into account in a classic proof of the uncountability of the entire set of real numbers. Even more generally, any positional numeral system for the real numbers contains infinitely many numbers with multiple representations. The equality 0.999... = 1 has long been accepted by mathematicians and taught in textbooks to students. In the last few decades, researchers of mathematics education have studied the reception of this equality among students, many of whom initially question or reject it. Many are persuaded by an appeal to authority from textbooks and teachers, or by arithmetic reasoning as below to accept that the two are equal. However,
