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a result that is accurate to within 10^-n

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PG1995

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Hi

Could you please help me with **broken link removed** query? Thank you.

Regards
PG
 
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Accurate to N decimal places does not necessarily mean N digits have to be same; it actually means that the result must be within half of the (N+1)th digit... i.e. you have to round the Nth digit using the (N+1)th digit.

e.g. PI (3.14159265...) to 3 decimal places is 3.142 and not 3.141. The first has an error of ~-4E-4, and the second ~5.9E-4. The second value is less correct than the first, and has an error of more than half the (N+1)th digit i.e. ~5E-4.

The accuracy within X means that the value you present must be within X of the actual result. As you can see in the above examples, the values are out by ~4E-4 and ~5.9E-4 respectively. If asked for PI within 0.0001, we could write 3.1416 which has an error ~7.3E-6, which is obviously less than (i.e. within) 0.0001.
 
Thank you, Dougy.

I have been trying to get conceptual understanding of it but no improvement. If the accuracy should be to N decimal places, then why should the error be within half of (N+1)th digit?

In **broken link removed** document, you can see it's easy to understand the notion of absolute error. **broken link removed** page could also be helpful. Could you please come up with such an explanation? Could someone please help me with it? Thank you.

Regards
PG
 
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Can you give two examples showing: 1) what you understand; and 2) what is confusing you?

If you take this statement as absolute fact: "The absolute error is always equal to half of the value of the smallest unit used," it should be clear that if one does not round up the value of pi, then the error is more than half of the smallest unit.

Of course rounding "5" is always a problem when you don't know the next digit or when there have been successive roundings ( e.g., 2.2349 rounds to 2.345; now if you wanted to round to two decimals and didn't know the history, which is more accurate, 2.23 or 2.24?).

John
 
I have been trying to get conceptual understanding of it but no improvement. If the accuracy should be to N decimal places, then why should the error be within half of (N+1)th digit?
If you want the accuracy to N decimal places, then the number you end up with will have N decimal places and be as close to the actual result as possible given those N decimal places.

To do this, you just round the number based on the (N+1)th digit. You already know that to round to a whole number, if the fractional part is <0.5, you round down; if it's >=0.5, then you round up. It's basically the same for when you're rounding to the Nth decimal.
 
I'm sorry to ask this again. The question statement says something like, "Determine an approximation to the root of the given equation that is accurate to at least within 10^-4". Here, what do I make of "10^-4" accuracy? I see that 10^-4 = 0.0001. Please help me with this. Thank you.

Regards
PG
 
Thanks, KISS.

But I'm sorry that I couldn't understand exactly what you were saying. By the way, I have solved **broken link removed** and it took me many iterations to reach the required accuracy. I hope I have it right. Thank you.

Regards
PG

PS: @KISS, I think I kind of understand what you said in your post. For instance, if the answer should be accurate to within 10^-5, then that would mean: -10^-5 <= answer <= 10^5.
 
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