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Significant figures

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Significant figures (also called significant digits and abbreviated sig figs or sig digs, respectively) is a method of expressing errors in measurements. The term is also sometimes used to describe some rules-of-thumb, known as Significance arithmetic, which attempt to indicate the propagation of errors in a scientific experiment or in statistics when perfect accuracy is not attainable or not required. Scientific notation is often used when expressing the significant figures in a number.

The concept of significant figures is derived from the method of measuring a value so that the smallest accurately known decimal place is next to last and only one further is estimated; for example, if an object is measured with a ruler that is marked in millimeters and is known to be between six and seven millimeters and appears to the measurer to be approximately two-thirds of the way between them, an acceptable measurement for it could be 6.6 mm or 6.7 mm, but not 6.666666... mm. This rule based upon the principle of not implying more precision than can be justified when measurements are taken in this manner.

Counting significant figures

The number of significant figures in a number is typically interpreted from its written form using the following rule:

Ignoring any decimal point in a number, start at the left end of it and move right; when a non-zero digit is found, that digit and all digits to its right are significant.

For example,

1.2345
0012345
0.00012345

all have five significant digits.

Leading zeros are therefore not considered significant, except that conventionally a number with value zero is considered to have one significant digit.

In general, all digits in a number after the first non-zero digit are considered significant. There is one exception to this, the case where a 'conventional' trailing zero digit has been added to indicate that a value is floating-point rather than an integer, as in some computer programming languages. (For example 2.0 rather than 2). Multiple trailing zeros after the decimal point are (almost) always significant.

In order to correctly show which digits are significant, values such as two thousand should be expressed in scientific notation, if necessary, using the correct number of significant figures. If only two digits — the '2' and the first '0' — are significant (i.e., the true value could be anywhere from 1950 to 2049), the appropriate representation is 2.0 × 10³; if three are significant (the value is in the range 1995 to 2004) then it is 2.00 × 10³; if four are significant (from 1999.5 to 2000.4), then it could be either 2000 (two, zero, zero, zero) or 2.000 × 10³. (For clarity, the former form could be written 2000., with a decimal point; otherwise, some may read the number as having just one significant digit and three zeros for placement.) If five, it could be either 2000.0 or 2.0000 × 10³.

The same can be achieved by using another unit for the quantity expressed. A distance of 2000 m is supposed to have 4 significant digits, but 2 km has only one. More informally it can be done by using words to express numbers. The value 12 million has two significant digits, while officially 12,000,000 has 8. In practical situations it is wise to consider several trailing zeroes as not significant.

Sometimes a bar over a trailing zero is used to indicate that it is significant. For example, <math>20 \bar{0} 0<math> appears to have four significant digits; the bar indicates that in fact the second zero is the last significant digit.

Measuring with significant figures

As illustrated in the above example involving the length measurement in millimeters, the significant figures method is that, when measuring using a non-electronic instrument, the observer should estimate within the nearest tenth of a division marked on the instrument. For example, if a graduated cylinder were marked off at every millilitre (ml), the observer should measure the amount of volume contained in the cylinder to the nearest tenth of a millilitre.

In order to express the degree of precision to which a value was measured, decimal numerals are used. When using significant figures rules, it should be assumed that the last significant digit of every measurement was estimated. Using the previous example, if the observer read the amount of liquid in the cylinder to be exactly at the 12 ml mark, the observer would write the value as 12.0 ml, which would indicate that the tenths place was the precision obtained, and the 0 was estimated. If the cylinder were marked off to every tenth of a ml, the observer would write the value as 12.00 ml.

Note that exact numbers obtained by counting discrete objects are not subject to the rules of significant figures and should be expressed as exact integers. Similarly, mathematical constants (such as π) do not have significant figures—they should be treated as having an infinite number of significant figures. Empirically-determined 'constants', however, do have error bounds; sometimes these bounds can be ignored because the value has been determined to much higher precision than other numbers in the expression.

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