There are more sophisticated ways to handle uncertainties in quantities than significant figures. [For an extensive discussion, see the NIST website on Uncertainty of Measurement Results.]  However, using this relatively simple formalism should create a sensitivity and appreciation for the connections between measuring devices, their capabilities, and their limitations. Many exercises involve the determination and reporting of one or more quantitative properties. Both the value of a property and its uncertainty will be important. It is essential that each measurement that leads to the value has been made and recorded with the full capability of the device or apparatus used in its determination.

SUPL-001 contains a brief section on the concept of significant figures. As noted in that supplement:

  • close attention will be paid to the use of significant figures in laboratory reports
  • grading standards for exercises and quizzes invariably include a substantial number of points that deal with the proper implementation of this concept.
  • you are urged to master this concept early in the laboratory course.
We encourage the use of electronic calculators for facilitating the computations associated with experimental numbers, but, even inexpensive calculators are designed to deal with, and display, numbers having 10 or more digits. This places the burden of accounting for the appropriate representation of numerical data on the student investigator.
  • everyday laboratory measurements limit the number of digits that represent reality
  • the concept of significant figures seeks to limit the number of digits in a quantity to those that are justified based on its origin

Two features play a role in the determination of significant figures - measurement, and computation using measured values.



  • for a quantity that is the result of a direct measurement, the number of decimal digits (the number of digits in the representation of the number, both to the right and the left of the decimal point) in that quantity is generally fixed by the measuring device. The number of significant digits can depend on the value of the measurement.
  • Our rule of thumb is that, for a device whose output is linear, the eye can estimate values lying between two marks on a scale to one-fifth of the distance between the two marks.
For experimentally determined quantities, the number of significant figures is not simply related to the intrinsic precision of a device. Consider the buret and the analytical balance.

The buret permits a volume to be read to ± 0.02 mL (one-fifth of the smallest division on its scale). How many significant figures does such a quantity have?

  • If the volume is less than 1.00 mL, the number of significant figures in the volume is two (e.g., 0.57 mL). If the recorded volume is between 1.00 and 9.99 mL, the number of significant digits is three. If the volume is between 10.00 and 50.00 (the maximum capacity of the buret), the number of significant figures in the volume reading is four - the maximum number of significant digits which a 50 mL buret is capable of producing.
Buret readings must always be reported to two decimal places, regardless of the volume reading. The number of significant figures in the reading clearly depends on the magnitude of the reading.

Next, consider the analytical balance.

The analytical balance is an interesing case. It is capable of displaying weights to ± 0.0001 g and its digital display reports weights with four decimal places, no matter how large or small they may be. (But, see intrinsic precision below.)

The number of significant figures represented by weights varies with the total weight of an object. Nevertheless, weights determined on an analytical balance must always be reported to four decimal places, regardless of the weight. Again, the number of signficant figures in the weight depends on the weight. E.g.

For weights:

  • less than 1 mg (0.0010 g), the balance produces only 1 significant figure (e.g., 0.0006 g = 0.6 mg)
  • between 1.0 and 9.9 mg (0.0099 g), the balance produces 2 significant figures (e.g., 0.0066 g = 6.6 mg)
  • between 10.0 and 99.9 mg (0.0999 g), the balance produces 3 significant figures (e.g., 0.0666 g = 66.6 mg)
  • between 100.0 and 999.9 mg (0.9999 g), the balance produces 4 significant figures (e.g., 0.6666 g = 666.6 mg)
  • between 1.0000 g and 9.9999 g, the balance produces 5 significant figures, (e.g. 6.6666 g)
  • between 10.0000 g and 99.9999 g, the balance produces 6 significant figures (e.g., 66.6666 g)


  • for a computed quantity, the number of significant digits reflects the number of digits in the numbers from which it is computed.

The number of significant figures in a computed quantity may be the same or different from those of the quantities from which it is computed. Consider a container which together with its contents weighs 35.2749 g. A sample of the contents is transferred out, producing a final weight of 35.2235 g. Both the initial and final weights are known with a precision of 6 significant figures. The weight of the sample is:  

35.2749 g
-35.2235 g
0.0514 g

The 6 significant figures in the initial and final weights produce a number (their difference) with only 3 significant figures (but still with four decimal digits.  The zero after the decimal point, while important, is not a significant figure).

Suppose the sample weight is now used in the calculation of a concentration by dividing it by a volume of 20.32 mL. Despite the four significant figures in the volume, the resulting concentration should be reported to at most 3 significant figures, namely .00253 g/mL. In general, the number of significant figures in a multiplication or division of quantities will be that of the quantity with the least number of significant figures.


How to manage Zeros:

Perhaps the most confusing aspect of significant figures is how to deal with the digit zero (0).  That number is often used as a place holder to indicate the location of a decimal point.  When used in this manner, zero is not a significant figure.  The only times that 0 is significant is when one or more of them occur between non-zero digits, or when one or more of them in a row is at the end of a string of non-zero digits.

Consider the following examples:
0.654         The 0 is used only to highlight the occurence of a decimal point following it.  It is not a significant figure

0.0654        In this case, neither 0 is significant.  They are not preceded by any non-zero digit

0.605        The 0 is again not significant, but the 0, since it occurs between two non-zero digits, is significant

0.6005      The 0 is again not significant, but both the 0's are significant, since they occur between two non-zero digits.

10.654       This time, the 0 to the left of the decimal point is significant.  It occurs between two non-zero digits.

10.0654     In this number, both the zeros are significant digits since there are non-zero digits both before the first and after the second.

Now consider a number which is the result of an experimental measurement.  Suppose the invesigator reports the number as     3.16
       We can assume that she was certain that the result was less than 3.17 and more than 3.15.

Indeed. with the usual rules for rounding, we can assume that the result was less than 3.165 and more than 3.155

Suppose instead, she reports the number as            3.160  
What is the significance of the terminal 0

The fact that the number is written as it is, must indicate that the investigator was certain that the result was less than 3.161 and more than 3.159.  (If we include rounding - less than 3.1605 and more than 3.1595.) - a substantial decrease in the uncertainty of the result and an increase in the precision of the measurement over the previous representation.

If the number is reported as      3.1600       This must be an indication that the investigator was certain that the number was less than 3.1601 and more than 3.1599.      (You should be able to provide the relevant limits assuming the usual rounding rules.)

An additional decrease in the uncertainty.   These zeros are certainly significant!

If you understand the above, you should be able to determine which zeros are significant in the following number:    0.0010600

Integers, Rational and Irrational Numbers - A Special Case

For quantitites that are restricted to occur in discrete, indivisible units (e.g., people (excluding half-wits), atoms, etc.), The number of significant figures is infinite.  We always assume that integers always have as many significant figures as are necessary when used in a calculation.  E.g., in the calculation of the perimeter of a square (4 times the length of a side), we can assume that the number 4 used in the calculation has as many significant figures as the length measurement. One possible exception to this rule is Avogadro's number which, while one would like to think is an integer, has neither been defined, nor is known, to the requisite 23 significant figures.

Quantities that are calculated from integers (rational numbers) similarly have infinite precision. E.g., the numbers represented by the quotient and product of the integers 2 and 7, 2/7 and 2*7,can be written to as many places as are required.

The astute student will recognize that numbers such as pi, e (2.71828.....), log 2, etc., while not rational, can be represented to as many significant figures as may be required in any given computation (E.g., the calculation of the volume of a sphere from its radius, using V = (4/3) x pi x r3 ). [Indeed, in this formula, the 4, the two 3's and the pi are all known to an infinite number of significant figures.  All uncertainty in the volume, V is due only to the uncertainty in the radius r. The number of significant figures in V is fixed by those in r3 (which in turn is fixed by the precision in r)]

Significant Figures and Scientific Notation

Scientific notation (i.e, writing numbers in the form y.yyy X 10n) simplifies the use of significant figures considerably.  The issue of using zeros as place holders for decimal points can be made to disappear.  The multiplier (y.yyy) should always have exactly the number of significant figures as are justified.  This may involve a number of trailing zeros.  E.g.:

6.02214 x 1023 has 6 significant figures.  While scientific notation goes a long way toward solving the significant figure problem, it is not always easy to follow, particularly when additions and subtractions are involved. 

Consider for example the following example of two buret readings whose difference is the net volume. In scientific notation

  Final Volume   3.105 x 101 mL
  Initial Volume  3.2 x 10-1 mL
  Net Volume 3.073 X 101 mL
In normal numeric representation, that subtraction is represented as follows:
  Final Volume   31.05 mL
  Initial Volume 0.32 mL
  Net Volume    30.73 mL
This example demonstrates why, in the laboratory course, scientific notation will not always be the best way to handle simple computations - particularly in the addition and subtraction of numbers.
Significant Figures in Logarithms and Exponentials

A discussion of significant figures and uncertainties in logarithmic and exponential quantities is given in the web page on calculations for SUSB-010 which includes a link to a more exact treatment.

Intrinsic Precision

The intrinsic precision of a device refers to the precision determined by its design and construction - including its interface with the investigator. E.g., the four decimal places in the digital display of the balance notwithstanding, the manufacturer's specification of the intrinsic precision of the analytical balance is +/- 0.0002 g. This means that independent weighings of the same object will produce results that do not differ by more than 0.0002 g. Nevertheless, weights are always recorded with all four decimals.


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Robert F. Schneider (rschneider at notes.cc.sunysb.edu)
Last Update: 2012-07-12