Summary significant figures

The significant figures (also called significant digits and abbreviated sig figs, sign.figs or sig digs) of a number are those digits that carry meaning contributing to its precision (see entry for Accuracy and precision). This includes all digits except:

* leading and trailing zeros (unless a decimal point is present) where they serve merely as placeholders to indicate the scale of the number.
* spurious digits introduced, for example, by calculations carried out to greater accuracy than that of the original data, or measurements reported to a greater precision than the equipment supports.

The concept of significant figures is often used in connection with rounding. Rounding to n significant figures is a more general-purpose technique than rounding to n decimal places, since it handles numbers of different scales in a uniform way. A practical calculation that uses any irrational number necessitates rounding the number, and hence the answer, to a finite number of significant figures. Computer representations of floating point numbers typically use a form of rounding to significant figures, but with binary numbers.



The rules for identifying significant digits when writing or interpreting numbers are as follows:

* All non-zero digits are considered significant. Example: the number 1 has one significant figure. In 20 and 300, the first figure is significant while the others may or may not be (see below). 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, 0.00012 has two significant figures: 1 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.
* Alternatively, the above can be summarized by three rules.
o 1. All non-zero digits are significant.
o 2. In a number without a decimal point, only zeros BETWEEN non-zero digits are significant (unless a bar indicates the last significant digit--see below).
o 3. In a number with a decimal point, all zeros to the right of the first non-zero digit are significant.
* 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, 13 \bar{0} 0 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.[1]

However, these conventions are not universally used, and it is often necessary to determine from context whether such trailing zeros are intended to be significant. If all else fails, the level of rounding can be specified explicitly. 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.

A number with all zero digits (e.g. 0.000) has no significant digits, because the uncertainty is larger than the actual measurement.



example :
Postage Scale 3 ±1 g 1 significant figure
Two-pan balance 2.53 ±0.01 g 3 significant figures
Analytical balance 2.531 ±0.001 g 4 significant figures




Rules for counting significant figures are summarized below.

Zeros within a number are always significant. Both 4308 and 40.05 contain four significant figures.

Zeros that do nothing but set the decimal point are not significant. Thus, 470,000 has two significant figures.

Trailing zeros that aren't needed to hold the decimal point are significant. For example, 4.00 has three significant figures.

If you are not sure whether a digit is significant, assume that it isn't. For example, if the directions for an experiment read: "Add the sample to 400 mL of water," assume the volume of water is known to one significant figure.