I found this, related to rounding rules. Maybe as a math major you could explain where these rounding rules came from and how they were accepted to be a valid mathmatical rule? Obvious, they were adopted becasue of delemia's like you discribe 1=.9 example. These rules must come into play converting from fractions to decimals, a division process, but multiplying to proove your answer does not work? It produces a delemia like the 1=.9, until you follow the accepted rules of rounding, than it comes out right. Somewhere mathmatics discovered these things happen, converting whole numbers or fractions into decimals. In robotics we controled machine accuracy with encoders that counted inches of travel with very high counting numbers. the more numbers you divided up a 1 inch increment, the closer you could return to the same position. This was called resolution per inch. Decimals allow us to count in very high resolution in parts of whole numbers. Fractions at best are very low resolution and reduced to very crude approximations. You know when you are about about half way walking home from school. But if you knew school is 4834 steps from home half way is 2417 steps. Using fractions your resolution is a few larger segments, rather than many many small segments. 1=.9 if you are using 1/3 as the fraction. 1=1 if you use 1/4 as the fraction becasue it will end in a zero.
When rounding whole numbers there are two rules to remember:
I will use the term rounding digit - which means: When asked to round to the closest tens - your rounding digit is the second number to the left (ten's place) when working with whole numbers. When asked to round to the nearest hundred, the third place from the left is the rounding digit (hundreds place).
Rule One. Determine what your rounding digit is and look to the right side of it. If the digit is 0, 1, 2, 3, or 4 do not change the rounding digit. All digits that are on the right hand side of the requested rounding digit will become 0.
Rule Two. Determine what your rounding digit is and look to the right of it. If the digit is 5, 6, 7, 8, or 9, your rounding digit rounds up by one number. All digits that are on the right hand side of the requested rounding digit will become 0.
Rounding with decimals: When rounding numbers involving decimals, there are 2 rules to remember:
Rule One Determine what your rounding digit is and look to the right side of it. If that digit is 4, 3, 2, or 1, simply drop all digits to the right of it.
Rule Two Determine what your rounding digit is and look to the right side of it. If that digit is 5, 6, 7, 8, or 9 add one to the rounding digit and drop all digits to the right of it.
Rule Three: Some teachers prefer this method:
This rule provides more accuracy and is sometimes referred to as the 'Banker's Rule'. When the first digit dropped is 5 and there are no digits following or the digits following are zeros, make the preceding digit even (i.e. round off to the nearest even digit). E.g., 2.315 and 2.325 are both 2.32 when rounded off to the nearest hundredth. Note: The rationale for the third rule is that approximately half of the time the number will be rounded up and the other half of the time it will be rounded down.
1000 when asked to round to the nearest thousand (1000)
800 when asked to round to the nearest hundred (100)
770 when asked to round to the nearest ten (10)
765 when asked to round to the nearest one (1)
765.4 when asked to round to the nearest tenth (10th)
765.37 when asked to round to the nearest hundredth (100th.)
765.368 when asked to round to the nearest thousandth (1000th)