What is the Method for Calculating Decimal Approximations of Fractions?

  • Thread starter OrbitalPower
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In summary, the conversation discussed the topic of decimal approximations and how to calculate the error given a series of approximated decimal numbers. A paragraph from a math book was shared, which contained confusing notation involving the use of a plus sign to indicate additional digits. It was explained that the plus sign represents the continuation of a repeating number in the decimal, resulting in a slightly smaller difference than if the number was rounded up. This clarified the confusion and provided a better understanding of the concept.
  • #1
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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  • #2
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.
 
  • #3
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...".
 
  • #4
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.
 

1. What are decimal approximations?

Decimal approximations are numbers that are rounded to a certain number of decimal places. They are an estimation of the actual value of a number and are often used when dealing with numbers that have a large amount of digits after the decimal point.

2. How do you calculate decimal approximations?

To calculate decimal approximations, you must first determine the desired number of decimal places. Then, you round the number to that number of decimal places. If the next digit after the desired decimal place is 5 or greater, you round up. If it is less than 5, you round down. For example, if rounding to 2 decimal places, 1.267 would become 1.27 and 1.263 would become 1.26.

3. What is the purpose of using decimal approximations?

Decimal approximations are often used in order to make numbers more manageable and easier to work with. They can also be used to simplify calculations, as working with numbers that have a large number of decimal places can be difficult.

4. Are decimal approximations accurate?

No, decimal approximations are not completely accurate. They are an estimation of the actual value of a number and may have some degree of error. The more decimal places that are included in the approximation, the more accurate it will be.

5. When should decimal approximations be used?

Decimal approximations should be used when dealing with numbers that have a large number of decimal places, as they can make calculations easier and more manageable. They are also commonly used in situations where an exact value is not necessary, such as in everyday financial transactions.

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