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Number & Operations for Teachers Copyright David & Cynthia Thomas, 2009 |
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Understanding the meaning
of 0.9999999 … Every decimal number consists of an infinite string of digits. While the meaning of some such strings is clear (e.g., 0.500 indicating exactly one-half), the meaning of others is less clear. For instance, consider the decimal number 0.999 … Upon seeing this notation, we are supposed to imagine an infinite number of 9’s arranged to the right of a decimal point. One way to build this picture is to begin with the decimal number 0.9, then add 0.09 to obtain 0.99, then add 0.009 to obtain 0.999, and so on forever. Clearly, each 9 added to this string of 9’s increases the growing value of the string. But what is it growing toward? Most students would say that this process produces decimal numbers that get closer and closer to the number 1.000 … with the caveat that the sequence of 9’s never actually attains the value 1.000 … The following algebraic argument should cause you to question this caveat. Let x = 0.999 … Multiplying both sides of this equation by 10, we obtain 10x = 9.999 … Subtracting the first equation from the second, we obtain 9x = 9.000 … Dividing both sides of this equation by nine, we obtain x = 1.000 … Since 0.999 … = x = 1.000 …, 0.999 … = 1.000 … In other words, the notations 0.999 … and 1.000 … refer to the same number! How are we to understand this argument? Is it a trick? Not really. If there are an infinite number of 9’s in the sequence 0.999 …, there is no last 9. Until there is a last 9, it is impossible to specify the separation between the numbers 0.999 … and 1.000 … Intuitively, this situation might be interpreted as saying, “If you cannot find a separation between where you are and where you want to be … you have arrived!” |
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