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A number line is a picture of a ... more kinds of numbers can be placed on the line, including fractions, decimal ... for example, the length of a line segment ...
A repeating decimal or recurring decimal is a decimal representation of a number whose digits are eventually periodic (that is, after some place, the same sequence of digits is repeated forever); if this sequence consists only of zeros (that is if there is only a finite number of nonzero digits), the decimal is said to be terminating, and is not considered as repeating.
A vinculum can indicate a line segment where A and B are the endpoints: ¯. A vinculum can indicate the repetend of a repeating decimal value: 1 ⁄ 7 = 0. 142857 = 0.1428571428571428571... A vinculum can indicate the complex conjugate of a complex number: + ¯ =
That is, one has a repeating decimal. For example, 1 / 81 = 0. 012345679 012... (with the group 012345679 indefinitely repeating). The converse is also true: if, at some point in the decimal representation of a number, the same string of digits starts repeating indefinitely, the number is rational.
The proof given below is a direct formalization of the intuitive fact that, if one draws 0.9, 0.99, 0.999, etc. on the number line, there is no room left for placing a number between them and 1. The meaning of the notation 0.999... is the least point on the number line lying to the right of all of the numbers 0.9, 0.99, 0.999, etc.
Here the nth term in the sequence is the nth decimal approximation for pi. Though this is a Cauchy sequence of rational numbers, it does not converge to any rational number. (In this real number line, this sequence converges to pi.) Cauchy completeness is related to the construction of the real numbers using Cauchy sequences.
Also the converse is true: The decimal expansion of a rational number is either finite, or endlessly repeating. Finite decimal representations can also be seen as a special case of infinite repeating decimal representations. For example, 36 ⁄ 25 = 1.44 = 1.4400000...; the endlessly repeated sequence is the one-digit sequence "0".
The long real line pastes together ℵ 1 * + ℵ 1 copies of the real line plus a single point (here ℵ 1 * denotes the reversed ordering of ℵ 1) to create an ordered set that is "locally" identical to the real numbers, but somehow longer; for instance, there is an order-preserving embedding of ℵ 1 in the long real line but not in the real ...