Post by Arlon10 on May 9, 2020 23:18:18 GMT
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.
One does not need to use the approximately equal symbol here (≈).
First of all no, many decimal representations are not exact, including that of 1/7.
Secondly. even if they were exact, no one has that many digits. Most Calculators do not store more than 15 digits, less expensive ones only 8 digits. Microsoft Excel only uses as many as 15 digits. I wrote a program that can do 100 digit calculations, but I knew full well at the time it is only good for philosophical investigations into the properties of numbers. There is no real world measurement that precise. You are talking about even larger numbers of digits. No matter how small you write you can't get that many digits on a ream of paper. Even Watson the supercomputer can't work with that many digits. It then becomes irrelevant what that many digits might "equal," does it not? In any actual use of mathematics for real world computations it is not a matter of exactitude, it is a matter of necessary precision, for example ±10-8 is beyond anything you will likely need.
You may of course use any notation you wish as long as you and your associates understand each other.