Agroecosystems Home Page | SI Units & Conversion Factors

Numbers, Units and Conversions

The Tasks:
(i) to express numbers in scientific notation;
(ii) to express numbers with a specified number of significant digits;
(iii) to express measurements in SI units.

These tasks involve use of exponents, rounding off, reference to the SI system, and conversion of non-SI units to SI units.

This page introduces these topics and provides links to some helpful web resources.

CONTENTS
  1. Scientific Notation
  2. Significant Digits
  3. SI Units
  4. Conversions
 Example

A U.S. publication reports that the yield of a new variety of wheat on a test field was 2.347 tons/acre. Express the yield in SI units with 3 significant digits.

Ans: 525 Mg km-2 or 525 g m-2


1. Scientific Notation

 In scientific notation, numbers are expressed as a number between 1.000.. and 9.999.. multiplied by a power of 10. This makes working with the numbers easier, especially when there are many digits before or after a decimal place. It also removes possible ambiguity about the number of significant digits ( Section 2 ). Some examples are shown below.

Number Expressed in Scientific Notation
456 4.56 x 102
0.0504 5.04 x 10-2
1 1, or 1 x 100, or 100
22 2.2 x 101


With scientific notation, we can quickly check our calculations to see if they are "in the right ball park". For example, if we write out a multiple of all of the numbers in the left column above as
146 x 0.0504 x 1 x 22
we get 505.6 with a calculator.

It is a little tricky for most of us to do that in our head, even approximately. However it is much easier when the calculation is written out as
4.56 x 102 x 5.04 x 10-2 x 1 x 2.2 x 101
First, excluding for the moment the tens and their exponents, round off the numbers to one significant figure and multiply them out: 5 x 5 x 1 x 2 = 50.

Then add the exponents of 10 to get the number of zeros to add before or after the decimal place: the sum of the exponents is 2 +(-2) + 0 + 1 = 1

So the approximate answer is 50 x 101 = 500. which is close to what we got with the calculator.

Web Resources:
  • SCIENTIFIC NOTATION
    (http://www.nyu.edu/pages/mathmol/textbook/scinot.html)
    New York University, Spons. (Viewed 15 Sep. 2003). A brief explanation of how to express numbers in scientific notation, with self quizzes.

  • EXPONENTS / POWERS
    (http://www.purplemath.com/modules/exponent.htm) E. Stapel (Western International University) Auth., Maint. (Viewed 15 Sep. 2003). This page describes use of exponents and scientific notation.

    • 2. Significant digits and rounding off

     In the biological sciences, calculations are typically made to 3 significant digits. This means that we "round off" the 3rd digit, according to the digit that follows. Calculations are commonly made using 4 or more significant digits, and only the formally reported, final result is rounded off to 3 significant digits.
    See Examples


    In principle, the number of significant digits reported for the results of a calculation should be the same as the minimum number of significant digits in any individual number involved in the calculation. For example,
    3 * 0.0053 * 3.6557 is equal to 0.0581256, when multiplied out on a calculator.

    Rounded off to 3 significant digits it is 0. 0581 or 5.81 x 10-2

    However since "3" has only one significant digit, the true accuracy of the result is conveyed by expressing the result to only one significant digit in which case it would be reported as 0.06 or 6 x 10-2
    Often this convention is overlooked.

    Similarly, numbers are often expressed to a certain number of decimal places, even though this involves different numbers of significant digits.


    For example in the table at right, the distance is reported with 1 significant digit for Bird A, 2 significant digits for Bird B and 3 for Bird C. Strictly speaking this implies that the distances were measured more accurately as they got longer, which is unlikely to have been the case. Nevertheless this format is frequently used because it is easy to read when there is a wide range in the magnitude of individual numbers.    		  
    Distance traveled by individual birds in one day.
    Bird
    Distance
    (km)
    Bird A 0.1
    Bird B 1.1
    Bird C10.1
    Web Resources:
  • SIGNIFICANT FIGURES TUTORIAL
    (http://ist-socrates.berkeley.edu/~chem1a/sigfigs/sigfig2.htm) University of California, Berkeley, College of Chemistry, Spons. (Viewed 15 Sep. 2003). A guided tutorial with self-quizzes.

    • 3. SI Units

     "Weights and measures" or systems for measuring things are as old as commerce and the origins of human societies. Traditional measures relied on familiar units such as the "foot", and varied between cultures and place. The system employed by modern science is a modified metric system called SI, standing for Système Internationale d'Unités. As well as in science, it has been adopted in commerce by most countries, a notable exception being the United States. Canada adopted it in 1970, however old units or mixed Imperial and metric are still used in certain commercial sectors, for example, fertilizer application rates are often given in units of kg per acre.

    If you are not familiar with SI units, consult one of the web resources below. Following are a few key points.

    • The SI defines seven base units and prescribes symbols for the units (e.g., the base unit for length is the meter, symbolized by "m").

    • Standard prefixes are used with the base units to identify larger and smaller units (e.g. the prefix micro, symbolized by µ multiplies a base unit by 10-6; hence 1 µm = 10-6 m).

    • There are a large number of "derived units" which are derived from the base units (e.g., a derived unit for area is the square meter, symbolized by m2).

    • Prefixes modify units by factors of 1000. In general, when reporting a result as XXX units, XXX should be a number between 1 and 999 (with digits following a decimal place as appropriate); if the number is less than 1 (or 1.000...) or greater than 999 (or 999.999...), modify the unit with the appropriate prefix. For example, instead of reporting a result as 0.506 g, you would report it as 506 mg; instead of reporting a number as 23,000 g it would be reported as 23 kg (or 23.0.. according to the number of significant digits).

      The SI to some extent obviates the need for scientific notation in reporting results. However, it may still be appropriate to use scientific notation when we want to describe quantities in familiar units. For example, we would normally report the distances to planets in kilometers, rather than gigameters or yottameters. When scientific notation is used with SI units, it is recommended that exponents of 10 be multiples of 3. For example, the minimum distance from Earth to Mars would be reported as 54.7 x 106 km, rather than 5.47 x 107 km. Thus when used in conjunction with SI units, numbers written in scientific notation can be between 1.000.. and 999.999.. rather than just between 1.000.. and 9.999..

    • Compound symbols for derived units formed by multiplication of base units should be separated by spaces or periods, e.g., the derived unit for speed or velocity is meters per second, symbolized as m s-1 or m.s-1. A single forward slash (/) may be used to indicate division, however, there should not be more than one slash in a multiple unit. For example, g/m2/d is incorrect; it should be written as g/(m2.d) or g/(m2 d). In general, use of negative exponents with symbols is encouraged to indicate division, e.g., g m-2 d-2.
    Web Resources:
  • SI UNITS & CONVERSION FACTORS
    (http://www.dal.ca/~dp/reports/mcpheeic.htm)
    Dalhousie University Agroecosystems Class, Spons., M. McPhee, Auth. (Viewed 15 Sep. 2003). The site provides a quick reference for looking up base units, prefixes, derived units and conversion factors of interest to agroecologists.

  • A DICTIONARY OF UNITS
    http://www.ex.ac.uk/cimt/dictunit/dictunit.htm) University of Exeter, Centre for Innovation in Mathematics Teaching, Spons. (Viewed 15 Sep. 2003) "This [site] provides a summary of most of the units of measurement to be found in use around the world today (and a few of historical interest), together with the appropriate conversion factors needed to change them into a 'standard' unit of the SI."

  • Anonymous. 1998. Use of SI (Metric) Units ASAE (American Society of Agricultural Engineers) Standards, 1998. Available as pdf document at www.beeh.unp.ac.za/courses/ASAE/102.pdf

  • THE INTERNATIONAL SYSTEM OF UNITS (SI)
    (http://www1.bipm.org/en/si/) Bureau International des Poids et Mesures (BIPM), Spons. (Viewed 15 Sep. 2003). "The task of the BIPM is to ensure world-wide uniformity of measurements and their traceability to the International System of Units (SI)."

    • 4. Conversions

     Frequently, older publications and non-scientific publications make use of non-SI units. Even many textbooks, especially those written in the U.S., employ non-SI units when the non-SI units are likely to be more familiar than the SI units. However, most scientific journals today require strict adherence to SI units.

    So it is often necessary to convert non-SI units to SI units. This involves simple arithmetic calculations, however they can be tricky when there are a lot of different units involved. It is usually advised that you write out the calculations with all of the units stated, and then cancel out the units to make sure you have them the right way around.

    See example. If you have difficulties following this example, or just generally doing these sorts of calculations, consult the web resources below, and develop an approach that works for you.

    Web Resources
    • UNITS (http://chem107.chem.tamu.edu/brown/units.html) texas A & M University, Spons, Drs. Brown and Mawk, Auth. (Chemistry 107, General Chemistry for Engineers). (Viewed 15 Sep. 2003). A set of pages to help engineers handle calculations involving units.

    • UNIT CONVERSIONS (http://www.towson.edu/~ladon/unit.html) Towson University, Spons. L. Ladon, Auth. (Viewed 15 Sep. 2003). Brief description of how to convert units.

    • Cottam, C.A. 2002. Comprehensive method for converting units. Physics Education 37: 259-260. Abstract and pdf version available at http://www.iop.org/EJ/abstract/0031-9120/37/3/404

    Page posted Oct 4, 2003 by