How many decimal places accuracy has an approximation?

Question

(This is a basic but very fundamental question about terminology, so no stupid or disrespectful answers please.)

To how many decimal places accuracy is an approximation 'a' to a quantity 'x'?

Specifically tell me your answer to this example:
a = 3.16
x = pi = 3.14159...

Note that to 1 decimal place, a≈3.2, whereas x≈3.1
So: which of the following would be correct:
"a is x correct to 0 decimal places"
or:
"a is x correct to 1 decimal place"

What is confusing is that the error ε = a-x = +0.0184
which is < 5x10^-2, thus implying an accuracy of 2 dps.
Explain all that if you can.

Explain the reasoning behind your response clearly (and get your terms correct).

Ok, both of you failed to answer the question:
"a is x correct to n decimal places"
What is n: 0 or 1?
Try again.

"2 dps" means "2 decimal places", obviously.

Bonjour: my answer is very correct, and your answer is woeful.
I never claimed "a is x".
I said "a is an approximation to x (calculated by one of various methods which is not relevant to discuss here)."

"There is no error in this approximation."
Nonsense. All approximations have an error ε = a-x, and I explicitly showed how to calculate it. (If they didn't have an error, they wouldn't be an approximation, now would they, they'd be an exact value?)

"Errors occur in the realm of probability and statistics."
That's a wildly sloppy false claim.
(Where do you get this stuff from?)

[Hy messaged me to claim]
The statement "a is x correct to 1 decimal place" hasn't any formal recognition in Mathematics.
Total nonsense.

Encyclopedias and reference books are full of statements like "In the year X, Y computed pi correct to n decimal places."
So, my question simply asks, what does that precisely mean?
Specifically, say the computed value was 3.16, would it be correct to say a=3.16 was pi correct to 0 or 1 decimal place?

The approximation error ε = a-x =0.0184 is a distinct quantity to a. I pointed out the interesting fact that ε is beyond 2 dps approximation error whereas a is apparently 0 dp.
This apparent paradox is because ε straddles the 1-dp-accuracy boundary of 3.15.

Answer 1

by convention,
a is correct to 2 decimal place, or 3 significant digits,
x is correct to 5 decimal places or 6 significant digits,

a is x correct to 0 decimal places since a rounded to 1 place is 3.2 and x rounded to 1 place is 3.1
Only at 0 decimal places are the two numbers equal, since we use rounding, and not truncation.

a - x = 0.02 even though you can string out more decimal places.
3.155 < a < 3.165
3.141585 < x < 3.141595
arithmetically,
0.013405 < a - x < 0.02315
statistically,
0.0150375 < a - x < 0.0217825
or
0.015 < a - x < 0.022
or, rounding to the least accurate number of decimal places, as is customary,
a - x = 0.02

Answer 2

Yes, getting terms correct is fundamental.
"a is x correct to 1 decimal place" is questionable.
"The approximation of a to 1 decimal place is equal to the approximation of x to 1 decimal place" is clear, and clearly incorrect.

However "a-x = 0 correct to 1 decimal place" is a correct statement about the number a-x.
I see that the second answerer has agreed with my first paragraph, in different words.

The apparent paradox you raised is a reminder of the perils of operating on quantities which have been approximated. e.g. a+a = 6.3 correct to 1 decimal place (I do understand dps!!), but if you add the 1decimal place approximation of a to itself you get 6.2. So operations after rounding can lead to significant errors, and I guess you've just shown that comparisons after rounding are also unreliable -- unless, that is, you round to more decimal places than you need in the subsequent operation or comparison.

Answer 3

This depends on how good an approximation do you need. Your terms are not very useful and I have no idea of hat a "dps" is. Absolute error is probably more useful; when approximating pi, you are approximating all of it. So 3.16 is within a linear error of 0.02, but is 0.02/3.14 in terms of absolute error.

Answer 4

a is simply not x. a rounded to the nearest integer = x rounded to the nearest integer.

There is no error in this approximation. Errors occur in the realm of probability and statistics.

Your question is grammatically incorrect.