Significant figures

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Significant figures are a way of expressing precision in measurement. They may also be referred to as sig figs, significant digits, or sig digs.

In some cases, these terms refer simply to notions of place value.

In other cases, these terms refer to some highly problematic rules-of-thumb, known as significance arithmetic, which attempt to express the significance or accuracy or precision or tolerance in a scientific experiment or in statistics when perfect accuracy is not attainable or not required.

In almost all cases, significant figures and the uncertainty they represent can be better represented. For example, in the expression 1.234±.056, the uncertainty is ±.056, which is absolutely clear, and could not have been clearly expressed using significant figures.

The concept of significant figures originated from some confusion between roundoff and uncertainty; sometimes roundoff error is the dominant contribution to the uncertainty, and sometimes not. This is related to the ill-founded rule that when measuring a value you should estimate 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 involves some confusion between readability (or resolution) and uncertainty; sometimes readability is the dominant contribution to uncertainty, and sometimes not.

One should always uphold the principle of not implying more precision than can be justified. Counting digits is a terribly clumsy way of implying anything about precision or accuracy or tolerance or uncertainty, and is more commonly encountered in the classroom than in the real world. Examiners in mathematics, physics and engineering courses in some cases deduct points when scoring papers if "too many" figures are given in a final answer. In contrast, in the research laboratory and in real-world situations, notions of significant figures are rarely applied at all; it is expected that guard digits will be used, and that the uncertainty will be expressed separately and explicitly. You can verify this by looking through the pages of Physical Review. You can also verify this by looking at the NIST compendium of physical constants -- none of the given values adhere to any notion of significant figures.

1 Determining significant figures
- 1.1 Methods for clearly showing significant figures
2 Measuring with significant figures
3 Handling integers, counts and constants
4 Example
5 See also
6 External links

[edit] Determining significant figures

If one is going to use any notion of significant digits, one should be aware that there are multiple inconsistent conventions for assigning "significance" to digits. Significant figures can also refer to volumetric devices and instruments for example a graduated cylinder or perhaps a volumetric flask.

According to one convention, commonly used for writing numerals by hand, the most significant digit is the "first" digit of a number (the left-most non-zero digit). Similarly, the least significant digit is the "last" digit of a number (sometimes, but not always, 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. This notion of relative significance involves nothing more than the notion of place value.

When using the fixed-point representation of numbers in a computer register or memory word, the significance of each bit is fixed by its position in the word, independent of whether it is zero or not, and independent of whether other bits are zero or not - see most significant bit. Again, this is based on simple, reliable place value ideas.

We now venture into the regime of counting the so-called significant digits. 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 whose absolute value is 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"). To make things more clear, trailing zeros are only recognized as significant figures if the number they are a part of has a decimal point. For example, 450 only has two sig figs, but 450. has three. 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.

This notion of digit-counting involves far more than mere place value. It is typically intended to express something about significance, uncertainty, tolerance, or something like that.

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

The digit-counting discussion dealt mainly with decimal numerals, but corresponding conventions apply when writing binary (base 2) numerals by hand.

[edit] Methods for clearly showing significant figures

Scientific notation is normally used when it is desired to attribute some notion of significant figures to a numeral. Otherwise, all sorts of problems arise, mainly concerning the interpretation of trailing zeros, as we now discuss.

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 through 2050), the conventional representation is 2.0 × 10³; if three are significant (the value is in the range 1995 through 2005) then it is 2.00 × 10³; if four are significant (from 1999.5 through 2000.5), 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 an exact integer or 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. According to most conventions, trailing zeros to the left of the decimal point are considered insignificant. The sensible way to solve this problem is to use scientific notation: 1.2e7 m or 1.2000000e7 m are each unambiguous.

It has been suggested that a bar over a trailing zero can be used to indicate that it is significant. Take for example the number 2,000 with a bar over the second zero:

<math>2,0\overline{0}0</math>

This number appears to have four significant digits, but in fact, the bar indicates that the second zero is the last significant digit. This notation is rarely used, and generally needs explanation for the average reader to understand its meaning, especially since a bar also can be used to represent repeating decimals (as in <math>\frac{1}{3}=0.\overline{3}</math>). An unambiguous and widely-understood alternative is to express the uncertainty explicitly, e.g. writing 2000±5.

[edit] 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.

[edit] Handling integers, counts and constants

Note that integers obtained by exactly counting discrete objects have zero uncertainty. Similarly, rational numbers have zero uncertainty. This is a problem, because certain rationals (such as 1/3rd) cannot be represented as a decimal numeral with any finite number of digits. Similarly, irrational numbers such as π have an exact value but cannot be represented exactly by any finte number of digits. More importantly, note that a simple decimal number such as 2.54 is formally a rational number (namely 254/100) and as such should be considered exact, in contravention of all notions of significant figures. Such numbers commonly arise in practice; for example, there are exactly 2.54 centimeters per inch, by definition -- a fact that cannot be expressed using significant figures. Similarly, the speed of light in a vacuum, which, when expressed in SI units, has no uncertainty because the metre is defined in terms of the speed of light.

In contrast, there are plenty of empirically-determined physical constants that do have nonzero uncertainty.

In all cases, the best procedure is to express the uncertainty separately and explicitly, for example 2.54±0 cm per inch, not relying on significant figures to express anything.

[edit] Example

How would you convert 125 yards to millimeters? There are two most simplified paths. One is to convert to feet, inches, centimeters and then millimeters. The other is to go directly from yards to meters to millimeters. Both possibilities will be addressed here.

1) At first glance, the first option seems longer and more tedious. However, this option uses all exact figures, so there would be no need to worry about significant figures.

2) Here you will need to apply the principles of significant figures. You can use the measurement of 1.094 yards equals roughly 1 meter. Therefore, you would divide your yards by 1.094. Then, merely multiply by 1000 to get your answer. You should get an answer of about 114259.5978 mm. However, the amount of significant figures you can use is limited by the conversion of 1.094. You can only use 4 significant figures at this point. Therefore, you will need to round to 114300 mm, or, in correct notation, 1.143 x 10^5 mm.