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Pengalaman Belajar fisika di SMAN BI 1 Banjar
(Sekolah Bertaraf Internasional)
8.2 ´ 103 has two significant digits
(This phenomenon is known as "round-off error.")
- Writing more digits in an answer (intermediate or final) than justified by the number of digits in the data.
- Rounding-off, say, to two digits in an intermediate answer, and then writing three digits in the final answer.
Identifying significant digits
- All non-zero digits are considered significant. For example, 91 has two significant figures (9 and 1), while 123.45 has five significant figures (1, 2, 3, 4 and 5).
- Zeros appearing anywhere between two non-zero digits are significant. Example: 101.12 has five significant figures: 1, 0, 1, 1 and 2.
- Leading zeros are not significant. For example, 0.00052 has two significant figures: 5 and 2.
- Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has six significant figures: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only six significant figures (the zeros before the 1 are not significant). In addition, 120.00 has five significant figures. This convention clarifies the precision of such numbers; for example, if a result accurate to four decimal places is given as 12.23 then it might be understood that only two decimal places of accuracy are available. Stating the result as 12.2300 makes clear that it is accurate to four decimal places.
- The significance of trailing zeros in a number not containing a decimal point can be ambiguous. For example, it may not always be clear if a number like 1300 is accurate to the nearest unit (and just happens coincidentally to be an exact multiple of a hundred) or if it is only shown to the nearest hundred due to rounding or uncertainty. Various conventions exist to address this issue:
- A bar may be placed over the last significant digit; any trailing zeros following this are insignificant. For example, has three significant figures (and hence indicates that the number is accurate to the nearest ten).
- The last significant figure of a number may be underlined; for example, "20000" has two significant figures.
- A decimal point may be placed after the number; for example "100." indicates specifically that three significant figures are meant.
The abbreviation s.f. is sometimes used, for example "20 000 to 2 s.f." or "20 000 (2 sf)". Alternatively, the uncertainty can be stated separately and explicitly, as in 20 000 ± 1%, so that significant-figures rules do not apply.
- Start with the leftmost non-zero digit (e.g. the "1" in 1200, or the "2" in 0.0256).
- Keep n digits. Replace the rest with zeros.
- Round up by one if appropriate. For example, if rounding 0.039 to 1 significant figure, the result would be 0.04. There are several different rules for handling borderline cases — seerounding for more details.
- Officially if the first not significant figure is a 5 not followed by any other digits or followed only by zeros, the last significant figure should be rounded down or up to the nearest even number. So to round 1.25 to 2 significant figures should be 1.2 and 1.35 should be 1.4. If the first non-significant digit is a 5 followed by other non-zero digits, the last significant digit should be rounded up. For example, 1.2459 as the result of a calculation or measurement that only allows for 3 significant figures should be written 1.25.
This is particularly appropriate when the uncertainty itself is important and precisely known (here, 100 m is presumed to be precise, and the time is 11.71±0.005 s, or an uncertainty of nearly 430 ppm).
In this case, it is safe and indeed advantageous to provide more digits than would be called for by the significant-figures rules.
Reporting the result using three significant figures (8.54 m/s) might be interpreted as implying that the speed is somewhere between 8.535 and 8.545 m/s.
This is actually very close to the true precision, the actual speed being somewhere between 8.5360 and 8.5434 m/s.
Reporting the result using two significant figures (8.5 m/s) would introduce considerable roundoff error and degrade the precision of the result.
For example, a pair of 0.05-metre-wide (2-inch) lines at the start and end would introduce a separate uncertainty of ±0.05 m (2 in) or 500 ppm to the above calculation.
Now the total uncertainty has risen to 500 + 430 = 930 ppm, since both sources must be added together.
Applied to the speed, that now becomes 8.5397±0.0080 or 8.5397(80) m/s, the actual speed being somewhere between 8.5317 and 8.5477 m/s.)