Decimal approximations

1. Jul 21, 2008

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?

2. Jul 22, 2008

Integral

Staff Emeritus
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. Jul 22, 2008

HallsofIvy

Staff Emeritus
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. Jul 22, 2008

OrbitalPower

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.