Define Significant FiguresThe
Significant Figures (also called significant digits) of a number are those digits that carry
meaning contributing to its precision. This includes all digits except:
1. leading and trailing zeros where they serve merely as placeholders to indicate the scale of the
number.
2. 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 digits is often used in connection with rounding. Rounding to n
significant digits 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 digits (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.
Computer representations of floating point numbers typically use a form of rounding to significant
digits, but with binary numbers.
The term "significant digits" can also refer to a crude form of error representation based around
significant-digit rounding; for this use, see significance arithmetic.
Identifying significant digitsThe rules for identifying significant digits when writing or interpreting numbers are as follows:
* All non-zero digits are considered significant. For example, 91 has two significant digits (9 and
1), while 123.45 has five significant digits (1, 2, 3, 4 and 5).
* Zeros appearing anywhere between two non-zero digits are significant. Example: 101.12 has
five significant digits: 1, 0, 1, 1 and 2.
* Leading zeros are not significant. For example, 0.00052 has two significant digits: 5 and 2.
* Trailing zeros in a number containing a decimal point are significant. For example, 12.2300 has
six significant digits: 1, 2, 2, 3, 0 and 0. The number 0.000122300 still has only six significant
digits (the zeros before the 1 are not significant). In addition, 120.00 has five significant digits.
This convention clarifies the accuracy 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, 13 \bar{0} 0 has three significant digits (and hence indicates that the
number is accurate to the nearest ten).
* The last significant digit of a number may be underlined; for example, "2000" has one
significant digit.
* A decimal point may be placed after the number; for example "100." indicates specifically that
three significant digits 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.
Scientific notationGenerally, 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 digits) becomes 1.2x10-4, and 0.00122300
(six significant digits) becomes 1.22300x10-3.
In particular, the potential ambiguity about the significance of trailing zeros is eliminated. For
example, 1300 to four significant digits is written as 1.300x103, while 1300 to two significant digits
is written as 1.3x103.
RoundingTo round to n significant digits:
* If the first non-significant digit is a 5 followed by other non-zero digits, round up the last
significant digit (away from zero). For example, 1.2459 as the result of a calculation or
measurement that only allows for 3 significant digits should be written 1.25.
* If the first non-significant digit 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 digits, Round
half up rounds up to 1.3, while Round half to even rounds to the nearest even number 1.2.
* Replace any non-significant digits by zeros.
ArithmeticFor multiplication and division, the result should have as many significant digits as the measured
number with the smallest number of significant digits.
For addition and subtraction, the result should have as many decimal places as the measured
number with the smallest number of decimal places.
In a logarithm, the numbers to the right of the decimal point is called the mantissa and the number
of significant figures must be the same as the number of digits in the mantissa.
When taking antilogarithms, the resulting number should have as many significant figures as the
mantissa in the logarithm.
W
hen performing a calculation, do not follow
these guidelines for intermediate results; keep as
many digits as is practical to avoid rounding errors.
Thank You
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