Exact figure . A digit d of a number x is said to be an exact digit or an exact figure if the absolute error of x is less than or equal to half the place value of d . That is, if
E (x ) ≤½ p (d)
Otherwise, the figure d is said to be inaccurate.
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Several approximate x numbers are given below . Determine in each case which figures of x are exact.
- x= 31.1416. It is known that x * = π = 3.141592653 ..
- x= 3,99999. It is known that x * = 4
- x= 4.20457. It is known that x * = 4.20451
- x= 0.00046384. It is known that E (x) = 0.000002
- x= 23.01241. It is known that E m (x) = 0.04
First of all we must find E (x) = | x * – x | = | 3,141592653-3,1415 | = 0.00000734 … To determine if a figure d is exact, we must check if this error satisfies that:
E (x) ≤½ p (d) In this case, proceeding from left to right, we have:
E ( x) ≤½ p (3 ) = 0.5 ………… 3 is an exact figure
E (x) ≤½p (1) = 0.05 …….. … 1 is an exact number
E (x) ≤½p (4) = 0.005 ……… 4 is an exact number
E (x) ≤½p (1)) = 0.0005 … … 1 is an exact number
E (x) ≤½p (6) = 0.00005 ….. 6 is an exact number
For didactic reasons, we have proceeded to analyze all the digits from left to right, but it is obvious that it would have been enough to prove that the number 6, which is the one with the lowest place value, was exact to affirm that she and all those to her left are exact. In summary in this case the 5 digits of the number x are exact.
In this case it is E (x) = | x * -x | = | 4-3,99999 | = 0.00001
If d represents the fifth 9 of x , ½ p (d) = 0.000005 and E (x) ≤½ p (d) is not satisfied. If d represents the fourth 9 of x , ½ p (d) = 0.00005 and we satisfy E (x) ≤½p (d)
Since the other digits of x have a higher place value, they will also be exact. In summary, in this case the exact figures for x are underlined below: 3.9999 9. The last 9 is not an exact figure
Since x * is known , the absolute error can be calculated: E (x) = | x * -x | = | 4,20451 -4,2O457 | = 0.00006 The digit 4 found in thousandths has place value 0.001. It is true that: E (x) ≤½ p (4) ) = 0.0005 thousandths and all the figures that appear to its left are exact. As for the digit 5, which appears in the fourth decimal digit: E (x) ≤½ p (5) = 0.00005 The digit 5 and with more reason, the 7 that appears to its right are not exact figures. As a summary, the exact figures for the number x are underlined below : 4,204 57
This case illustrates the fact that it is not necessary to know the exact value x * to determine the exact digits of x, it is enough to know the absolute error. As the error contains a 2 in the sixth decimal digit it is clear that the sixth digit of x cannot be exact. Consider the 6 that is in the fifth decimal place, its place value is p (6) = 0.00001 and it is true that: E (x) = 0.000002 ≤½ p (6) = 0.000005 It is concluded that the figures are exact 6 and those on the left, those underlined below 0.00046 384
In this case the absolute error of x is unknown , we only have the maximum absolute error, which is a higher bound of the absolute error. However, this information is useful: it indicates that the absolute error could be 0.04. In that case, it is evident that the second decimal number (a 1) may not be exact. Regarding the digit that is in the first decimal place, it is true that:
E (x) 0.04≤½ p (0) = 0.05 So this digit 0 is an exact number and, with more reason, those to the left. These exact figures are underlined below: 23.01 241 As the precise value of the absolute error is not in this example, the digits that are not underlined are unknown and may or may not be exact; these figures are usually called dubious .