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SIGNIFICANT FIGURES
The term significant figures (sig figs or sf) and the more-or-less equivalent term significant digits (sig digs) are rather loosely defined; they mean different things to different people.
- In some cases, these terms refer simply to notions of place value.
In all cases, keep in mind that anything that can be done using significant figures can be done better using other means. In particular, if you want to express uncertainty, express it separately and explicitly; for example, in the expression 1.234±.0056, the uncertainty is ±.0056, which is absolutely clear (and could not have been clearly expressed using significant figures).
The concept of significant figures presumably 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.
Determining significant figures
If you are going to use any notion of significant digits, you should beware that there are multiple inconsistent conventions for assigning "significance" to digits.
- 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"). 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.
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:
This appears to have four significant digits, but the bar indicates that in fact the second zero is the last significant digit. This notation is extremely rarely used, and you should not expect readers to understand it without explanation.
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.
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.
See also
External links
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