Significant figures

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The idea of significant figures (sig figs or sf), also called significant digits (sig digs) is a method of expressing error in measurement.

The most significant digit is the "first" digit of a number (the left-most digit). Similarly, the least significant digit is the "last" digit of a number (the right-most digit). A number is called more significant because it carries more weight. In the decimal number system (base-10), the weight of each digit to the left increases by a multiple of 10, and conversely the weight of each digit to the right decreases by a multiple of 10. A similar thing happens in the binary (base-2) number system - see most significant bit.

Sometimes the term "significant figures" is 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 originated from measuring a value and then estimating one degree below the limit of the reading; for example, if an object, measured with a ruler marked in millimeters, is known to be between six and seven millimeters and can be seen to be approximately 2/3 of the way between them, an acceptable measurement for it could be 6.6 mm or 6.7 mm, but not 6.666666 mm as a recurring decimal. This rule is based upon the principle of not implying more precision than can be justified when measurements are taken in this manner. Teachers of engineering courses have been known to deduct points when scoring papers if excessive significant figures are given in a final answer.

1 Determining significant figures
2 Measuring with significant figures
3 See also
4 External links

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Determining significant figures

Significant figures conventionally follow certain sets of rules. Such that:

All non-zero digits are significant: for example, 87.636 has five significant figures. In addition, any zeros that are between non-zeros are also considered significant; for example, 40.02 has four significant figures. Any zeros that follow immediately to the right of the decimal place in numbers smaller than one are not considered significant, e.g., 0.00057 has two sf. The situation regarding trailing zero digits that fall to the left of the decimal place in a number with no digits provided that fall to the right of the decimal place is less clear, but these are typically not considered significant unless the decimal point is placed at the end of the number to indicate otherwise (e.g., "2000." versus "2000"). However, any zeros that follow the last non-zero digit to the right of the decimal point are significant, e.g.: 0.002400 has four significant figures.

Conventionally, a number with value 0 is considered to have one significant figure.

In order to indicate exactly which digits are significant, values such as two thousand can 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 2049.999...), the conventional representation is 2.0 × 10³; if three are significant (the value is in the range 1995 to 2004.999...) then it is 2.00 × 10³; if four are significant (from 1999.5 to 2000.4999...), 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 four 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 multiple trailing zeroes as insignificant.

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.

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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 are therefore expressed as integers. Similarly, mathematical constants (such as π) do not have significant figures—they are treated as having an infinite number of significant figures. The same is true of a value defined as an integral multiple of some other value, such as the atomic mass of carbon 12. Empirically-determined 'constants' (such as conductivities of materials), however, do have error bounds; sometimes these bounds can be ignored because the value has been determined to much higher precision than is needed for a given calculation. The speed of light in vacuum expressed in SI units is a special case, since the meter is defined in terms of that speed.

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