What is the Method for Calculating Decimal Approximations of Fractions?

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Discussion Overview

The discussion revolves around the method for calculating decimal approximations of fractions, specifically focusing on the rounding process and the notation used to express errors in approximation. Participants explore the implications of the notation and the accuracy of decimal representations in mathematical contexts.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion regarding the rounding explanation in a math book, particularly about the error calculation for the fraction 23/24 and its decimal approximation.
  • Another participant questions the notation "+", suggesting that it indicates a value that is less than 0.00007 but greater than 0.00006, and provides a calculation to support this view.
  • A participant reiterates the confusion about the difference between 0.9584 and 0.95833..., emphasizing the significance of the repeating decimal in determining the error.
  • One participant concludes that the explanation makes sense, affirming that the "+" indicates additional figures in the approximation, thus supporting the earlier claims about the error values.

Areas of Agreement / Disagreement

Participants generally agree on the interpretation of the "+" notation and its implications for understanding the error in decimal approximations. However, there remains some uncertainty regarding the exact values and their relationships, particularly in the context of rounding and repeating decimals.

Contextual Notes

The discussion highlights potential limitations in understanding the notation and the implications of repeating decimals on error calculations. There is an acknowledgment of the complexity involved in rounding and approximating fractions.

OrbitalPower
I was reading about decimal approximations in one of my math books and I like his explanation of why 5 and over we round up and so on (as it's closer to rounding down). I even understood the explanation of how to calculate what the error could be given a series of decimal numbers that have been approximated.

However, this paragraph confused me:

"...Thus, by the method of article 34, 23/24 = .95833+. Expressed to four decimal places the real value of this fraction lies between .9583 and .9584; .9583 is .00003+ less than the true value, and .9584 is .00006+ greater. Therefore, .9583 is nearer the correct value and is said to be correct to four decimal places. Similarly, .958 is correct to three places and .96 to two."

I understand .95833 - .00003 = .9583. But shouldn't .9584 be .00007+ greater than .95833?
 
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I am not sure I understand his use of the + in this notation. Let me guess.

.9584 - .958333... = .0000666...

I think .00006+ discribes this number better then .00007+ clearly it is LESS then .00007 and greater then .00006.
 
OrbitalPower said:
I was reading about decimal approximations in one of my math books and I like his explanation of why 5 and over we round up and so on (as it's closer to rounding down). I even understood the explanation of how to calculate what the error could be given a series of decimal numbers that have been approximated.

However, this paragraph confused me:

"...Thus, by the method of article 34, 23/24 = .95833+. Expressed to four decimal places the real value of this fraction lies between .9583 and .9584; .9583 is .00003+ less than the true value, and .9584 is .00006+ greater. Therefore, .9583 is nearer the correct value and is said to be correct to four decimal places. Similarly, .958 is correct to three places and .96 to two."

I understand .95833 - .00003 = .9583. But shouldn't .9584 be .00007+ greater than .95833?
If you had 0.95833 exactly then 0.9584 would be exactly 0.00007 greater. But 23/24= 0.95833333... where the "3" keeps repeating. The "+" in 0.00006+" means "0.00006 plus more terms after that (in this case the "3333..."). The difference is a little less than 0.00007, again because of that continuing "3333...".
 
Thanks guys. There are usually few errors in his books so I didn't think it would be an error. The explanation above makes sense, the plus sign indicates that more figures are to be added, so .9583 is .00003+ less than the true value, which is the closest approximation at that level, and likewise for the other number, the .00006+, which is closer than .00007 because of the addition of the plus sign.
 

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