Specifically, the rules for identifying significant figures when writing or interpreting numbers are as follows:[1] 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: 101.1203 has seven significant figures: 1, 0, 1, 1, 2, 0 and 3. 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 since it has three trailing zeros. This convention clarifies the precision of such numbers; for example, if a measurement precise to four decimal places (0.0001) is given as 12.23 then it might be understood that only two decimal places of precision are available. Stating the result as 12.2300 makes clear that it is precise to four decimal places (in this case, six significant figures). 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 precise 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 figure; any trailing zeros following this are insignificant. For example, 1300 has three significant figures (and hence indicates that the number is precise to the nearest ten). The last significant figure of a number may be underlined; for example, 2000 has two significant figures. A decimal point may be placed after the number; for example 100. indicates specifically that three significant figures are meant.[2] In the combination of a number and a unit of measurement, the ambiguity can be avoided by choosing a suitable unit prefix. For example, the number of significant figures in a mass specified as 1300 g is ambiguous, while in a mass of 13 hg or 1.3 kg it is not. 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 with a plus-minus sign, as in 20 000 ± 1%, so that significant-figures rules do not apply. This also allows specifying a precision in-between powers of ten (or whatever the base power of the numbering system is). Scientific notation[edit] In most cases, the same rules apply to numbers expressed in scientific notation. However, in the normalized form of that notation, placeholder leading and trailing digits do not occur, so all digits are significant. For example, 0.00012 (two significant figures) becomes 1.2×10−4, and 0.00122300 (six significant figures) becomes 1.22300×10−3. In particular, the potential ambiguity about the significance of trailing zeros is eliminated. For example, 1300 to four significant figures is written as 1.300×103, while 1300 to two significant figures is written as 1.3×103. The part of the representation that contains the significant figures (as opposed to the base or the exponent) is known as the significand or mantissa. Alternatively: 1. All non-zero digits are significant 2. In a number without a decimal point, only zeros between non-zero digits are significant. 3. In a number with a decimal point, all zeros to the right of the first non-zero digits are significant. Rounding and decimal places[edit] Question book-new.svg This section does not cite any references or sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (October 2013) Main article: Rounding The basic concept of significant figures is often used in connection with rounding. Rounding to 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. For example, the population of a city might only be known to the nearest thousand and be stated as 52,000, while the population of a country might only be known to the nearest million and be stated as 52,000,000. The former might be in error by hundreds, and the latter might be in error by hundreds of thousands, but both have two significant figures (5 and 2). This reflects the fact that the significance of the error (its likely size relative to the size of the quantity being measured) is the same in both cases. To round to n significant figures: If the first non-significant figure is a 5 followed by other non-zero digits, round up the last significant figure (away from zero). For example, 1.2459 as the result of a calculation or measurement that only allows for 3 significant figures should be written 1.25. If the first non-significant figure is a 5 not followed by any other digits or followed only by zeros, rounding requires a tie-breaking rule. For example, to round 1.25 to 2 significant figures: Round half up (also known as 5/4) rounds up to 1.3. This is the default rounding method implied in many disciplines if not specified. Round half to even, which rounds to the nearest even number, rounds down to 1.2 in this case. The same strategy applied to 1.35 would instead round up to 1.4. Replace non-significant figures in front of the decimal by zeros. In financial calculations, a number is often rounded to a given number of places (for example, to two places after the decimal separator for many world currencies). Rounding to a fixed number of decimal places in this way is an orthographic convention that does not maintain significance, and may either lose information or create false precision. As a
Posted on: Sun, 06 Jul 2014 10:27:04 +0000
Trending Topics
Recently Viewed Topics
© 2015