Aproximate calculation in decimals

Question

I am trying to refresh on precision of calculations. If we have the decimal fractions: $.234673$, $.322135$, $.114342$, $.563217$ each known to be correct to six figures why are each of the decimals may not be accurate by as much as $.0000005$? How do we come up with this number? If I recall if e.g. he have $32.71$ the $1$ represents 1 hundrenth instead of 1 unit but is still $1/3271$ of the total, right? But using $3/234673$ for the first decimal, $5/322135$ for the second $2/114342$ for the third and $7/563217$ for the last I don't get $0.0000005$. So how do we find the approximation?

Answer 1

The thing to note is that numbers in the range $0.2346725-0.2346734\dot{9}$ all round to $0.235673$. The maximum error here is $0.0000005$. It is a similar argument for all of the other numbers.

In general, a number $a\times10^p$, with $a$ an integer (no decimal part), has maximum error $5\times10^{p-1}$.

Edit in response to below comments

The absolute and relative error (defined here) measure the error in two different ways.

Absolute error measured the difference between the actual and approximate value, taking no regard to the size of the numbers. 30 to approximate 31 and 1470 to approximate 1469 both have an absolute error of 1.

Relative error measures the percentage that the error is of the actal value. Using the same numbers as the above example, the relative error using 30 to approximate 31 is $1/31\approx0.03225$ and using 1470 to approximate 1469, the relative error is $1/1469\approx0006807$. So even though both of these approximations have an absolute error of $1$, they have very different relative errors.

Answer 2

well, it depends on what you mean by correct "to six figures". Usually it is intended to six significant digits, so that the leading 0 is not counted among them; but since you are saying that this is not the case I'll change my answert.

The first number is greater than or equal to $0.23467$ and less than $0.23468$, so the maximum possible error is $0.000007$.

Moreover, this error is absolute, that is, the difference between the actual value and the given one. $3/234673$ or $7/234673$ is a relative error, that is, the ratio between the error and the given value. In this case the difference is small, but if you have the number $2346734$ with six correct digits the result changes.