Stewart essential calculus early transcendentals 7th edition pdf

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The repeating decimal continues with infinitely many nines. This number can be shown to equal 1. 999″ and “1” represent the same number. 999 can have the same meaning, a different definition, or be undefined. The utilitarian preference for the terminating decimal representation contributes to the misconception that it is the only representation. 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. 1, and that they get closer and closer to 1. More precisely, the distance from 0. Therefore, if 1 were not the smallest number greater than 0.

999, etc, then there would be a point on the number line that lies between 1 and all these points. Therefore, 1 is the smallest number that is greater than all 0. This least upper bound is one way to define infinite decimal expansions: the real number represented by an infinite decimal is the least upper bound of its finite truncations. The argument here does not need to assume completeness to be valid, because it shows that this particular sequence of rational numbers in fact has a least upper bound, and that this least upper bound is equal to one. 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. When using such systems, notation 0. The matter of overly simplified illustrations of the equality is a subject of pedagogical discussion and critique.