0.999…
In mathematics, 0. 999… denotes the repeating decimal consisting of infinitely many 9s after the decimal point. This repeating decimal represents the smallest number no less than every decimal number in the sequence. This number is equal to 1. There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs.
About 0.999… in brief
In mathematics, 0. 999… denotes the repeating decimal consisting of infinitely many 9s after the decimal point. This repeating decimal represents the smallest number no less than every decimal number in the sequence. This number is equal to 1. There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs. The technique used depends on the target audience, background assumptions, historical context, and preferred development of the real numbers, the system within which 0.999.. is commonly defined. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation. For this and other reasons, rigorous proofs relying on non-elementary techniques, properties, or disciplines may find the equality sufficiently counterintuitive that they question or reject it. This has been the subject of several studies in mathematics education. In the standard number systems, there is no positive number that is less than 110n for all n. This is the Archimedean property, which can be proven to hold in the system of rational numbers.
Therefore, 1 is the smallest numbers that is greater than all 0. 9, 0. 99, 999, etc., and so 1 = 0.9999. 1. For this, it suffices to prove that if a number is not larger than 1, then no number is smaller than 1. One has to show that 1 has two equal representations, which is a property of all base representations. The previous explanation is not a proof, as one cannot define properly the relationship between a number and its representation as a point on the number line. For accuracy, the number 0.. 9, with nines after the decimal point, is denoted 0, 0,. 9, 0, 9, and so on on 3. As 110n = 0. 01, the addition of n digits after thedecimal point implies that n is the number after the point, with 0. 01, n, n. 1, 0. 9,. 0, 2, n,. 1, and n. 2, so on 3 = 0, 3, 0, 3, and 0. 3, 1, 1,. 1,. 2, 2,. 2,. 0. 1,. 0,. 3, 2. 2,. 1. 2. 1… 2, 1. 0, 1. 1,… 1. 1, 1. The distance to 1 from the nth point is 110n. 1 = 110, the distance from 0.99 to 1 is 0.01 = 1102, and 1. 01 = 1101, and 2. 0 = 1103, and 3. 0. 2 = 1104, and 4. 0. 3. 2, 0. 4. 2. 1, etc., is the point at the right of all of the numbers 0. 8, 8, 7, 7. 7, 6. 6, 5. 5, 4. 4, 3. 3.
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