Post by Arlon10 on May 9, 2020 19:40:46 GMT
When you speak of the condition "in infinity" it is important to remember there is no such place. Infinity never becomes a quantity in any place. It is best to consider which are and which are not quantities. Infinity is better understood as an "ongoing process" than a quantity.
While the assumption of a quantity is not particularly hazardous in this specific instance, it is generally wise to avoid algebraic manipulations involving expressions that are not quantities.
Since the use of digital calculators became commonplace it is not unusual to find endeavors that "equate" any digital representative with more accurate notation arbitrarily. Casio has some interesting and inexpensive calculators that can use various modes of expressions.
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.