S.O.S. #4 The Rubbish Heap

S.O.S. #4 The Rubbish Heap May 9, 2020 20:18:48 GMT

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Post by general313 on May 9, 2020 20:18:48 GMT

May 9, 2020 19:40:46 GMT Arlon10 said:

May 9, 2020 18:23:20 GMT general313 said:

If using the idea of "limit" makes you feel more comfortable with the definition, then fine. You can say that the series never reaches that limit if you stop after a finite number of addition steps. Today mathematicians are perfectly comfortable in stating that the "sum" of the infinite series is equal to 1.

Interestingly, history is full of examples of discomfort with the idea of certain numbers. In the middle ages and perhaps later, negative numbers were often avoided in mathematical calculations, until it was realized that they were quite handy for intermediate results.

The Ancient Greeks were famously uncomfortable with irrational numbers like sqrt(2) because they believed that all numbers should be expressible as a fraction (a ratio of two whole numbers). Even our use of "rational" and "irrational" when applied to thinking reflects this history. This inconvenient truth was apparently so disturbing to the Ancient Greeks that it hampered their development of algebra (they focused on geometry instead).

   That is unless they studied calculus which uses the correct terminology of "limit" I noted earlier.

   The problem remains which are quantities and which are not. The expression 1/7 is an exact quantity. It's decimal representation is an ongoing division without end. Since it is a repeating chain of digits "142857" the temptation is to assume there is an "arrival" at some quantity once the repeating pattern is known. It can be important to remember that the division never ends though. The expression 1/8 is likewise an exact quantity, however its decimal representation is not ongoing. The division yields an exact quantity of 0.125 because that division is completed at that point. Since calculators often use 8 to 15 digits and truncate the rest, the error caused by decimal representations that are not quantities are not very far off the mark. Remember the radius of a proton. The errors can accumulate in extended calculations though.

   Notice that even though 1/7 can be expressed as a fraction of integers and its decimal representation is periodic, it will still accumulate errors in extended calculations that depend on decimal representations.

It doesn't matter whether they studied calculus, and I very much doubt that anyone can find a mathematician who hasn't. Since you mentioned the decimal representation of 1/7, it should be pointed out that that representation is exact. Hence one can use the equation 1/7 = 0.142857142857...
One does not need to use the approximately equal symbol here (≈).