This shows m_e to 7 sig. figs.  we know that the last digit is uncertain but we don't know by how much.  To properly indicate the uncertainty in the numbers, we need to be able to incorporate the uncertainty into the number itself.  One way gives the following notation.

 9.109534 × 10⁻³¹ kg ± 0.000047×10⁻³¹kg.  The uncertainty is herein expressed via the second number with the plus/or/minus symbol.  This indicates the centre of the range (the first number) and the size of the range (the second number).  The true value is expected to lie somewhere within that range.

This method is awkward and the better way to express an uncertainty when using scientific notation is to incorporate the error as parenthetical digits at the end of the main number.

9.109534(47) × 10⁻³¹ kg. 

This now expresses the same number (range-centre and width) as before in a compact way.

Significant Figures.

When we are reporting measured amounts, we need to specify the uncertainty limits of our number, i.e., we need to specify how accurately we know the value.  Our calculator might be able to report to us 9 or 10 digits but perhaps only two of them have any meaning.  When writing the number into a report, we should indicate this limit to the reader.  The simplest method is to only write the "significant" figures, i.e., those digits in the number that are actually meaningful. 

Take as an example, an individual walking the street with some money in his/her pocket.  This individual has not kept exact track of his/her expenditures during the day but knows that there is about 15 dollars give or take  maybe 50¢ (even this is only a guess in this example but is often calculated more accurately in scientific experiments).  We write this amount in proper notation as $15.0±.5 or $15.0(5). In other words, we have defined a range of possible amounts that this person is carrying as being as low as $14.50 and as high as $15.50.

 Now suppose this person notices a penny on the ground and picks it up.  He/she now still has about $15. the penny is relatively insignificant since the known amount of money was uncertain to a larger amount than just 1¢.  Mathematically, we could write this as $15.01(50)  The range of possible amounts this number could represent is now $14.51 to 15.51 and is almost identical to the initial amount.  The problem here is the the limit of accuracy is only about 50¢ not exactly 50¢ so we cannot even justify including the penny in the second range since we are so uncertain as to the actual size of the range.  Hence, before and after picking up the penny, this person has about $15 give or take about 50¢.  The penny was insignificant.  While this example may seem a bit complex, most of the time we do calculations, we don't actually calculate or propagate our errors (see your lab manual for instructions on propagation of errors).  We just keep track of significant figures.  This method is easiest but also the least accurate way to keep track of our limits of uncertainty.

Let's first try a few examples of simply counting significant figures in numbers to ensure we can properly determine them.

How many?

Note that when we write the number in scientific notation that we must always and only write significant digits.

When combining numbers, we must also combine the uncertainties.  Using the method of propagation of errors, we could accurately keep track of the uncertainty in our final calculated number.  However, an easier (but less precise) way is to simply keep track of significant figures.  To do this, we follow two simple rules. 

1. when numbers are added or subtracted, we keep track of decimal places, not sig. figs. and the answer has the same number of decimal places as the number with the lowest amount of decimal places.  Of course, we must first re-write all the numbers to the same order of magnitude

2. In multiplication and division, we keep track of sig. figs and the answer must have only as many sig figs as the least accurately known number (least number of sig. figs)

Dimensional Analysis

Dimensional analysis affords us a way to analyze measured values and combine them to produce new values by merely watching that dimensions are properly considered.  For example, the dimension of Area is length squared so we can quickly see that if we have two lengths, we can multiply them to get a value with dimensionality of area.

We can thus, create our own formulae as needed without memorizing.

[L]² = [L²] Þ area  e.g. m², in²

[L]³ = [L³] Þ volume  e.g. cm³ (sometimes called cc = cubic centimetre º mL)

[L t⁻¹] Þ speed  e.g. m s⁻¹ mph

[L t⁻²] Þ acceleration  e.g. m s⁻²

Consider force:

F = ma Þ [M] [L t⁻²] = [M L t⁻²]   e.g. kg·m·s⁻² º Newton, N

Conversion Factors

We see that a value may have several units but still have the same dimension.

In any equation, dimensions on both sides Must be the same.  However, the units (and hence the number itself) may differ.
    e.g. 1 m = 100 cm

    e.g.  1 L-atm = 101.325 Joules

Whenever we see these simple equalities, we can rewrite them into conversion factors.  These conversion factors are extremely useful throughout your career in Chemistry, etc. when the units we are given in a problem and the units we need for the answer are different.  Conversion factors always have an absolute magnitude of 1 (dimensionless but not unitless)  and hence, they are used to convert from one unit to another without changing the absolute magnitude of the value we are representing.  For example, we can measure the duration of a day in hours, minutes or seconds and get a different numerical value for each unit (24h, 1440 min, 86400s) but a day is still a day no matter what units we use to measure it.

If we take the equality we wrote (above) between values in meters and centimeters, we can convert it to a conversion factor and use it.

Note that the conversion factor has units of cm m^-1 but is in effect dimensionless since both cm and m are units of the dimension [L] length.

Example

A certain synthetic process yields 7.83 * 10^-2 g of product per second. After 5.00 days of continuous reacting, how many kilograms will be produced?

Here, we used a set of conversion factors derived from commonly known equalities like 60 seconds in a minute (60s=1min), 60 minutes in an hour (60 min = 1h) and 24 hours in a day (24h=1day).

Example

In this next example, we probably need to look up one equality between kilometres and miles before we can create the appropriate conversion factor.  The others are probably created from commonly known equalities.

The speed of a car is 50.0 mph.  What is its speed equal to in meters per second?

Several other combinations of conversion factors are possible.

Conversion factors and sig. figs.

When using conversion factors, we need to be aware that there are different kinds of these factors.  Some are defined quantities, such as 1 foot = 12 inches, or 1 kg = 1000g.  These conversion factors are whole-number ratios and  have an infinite number of sig figs.  Other conversion factors are measured conversions and therefore have a limited number of sig figs.  for example, 1 inch = 2.54 cm.  There is no defined relationship between these two different measurement systems.  It merely turns out that to 3 sig figs, the measured length of something that is 1 inch long using centimeters is 2.54.  If we measured that same 1" long object using more accurate equipment, we might find a conversion factor that has more sig. figs.

In Summary:

•  defined conversion factors have infinite number of sig. figs.
•  measured conversion factors have a limited number of sig. figs.

SI units

The International System of units (SI) is generally used for scientific measurements.  We are familiar with some of them, such as kilograms, kilometers, etc.  There are generally two classes of SI units, SI Base units and SI derived units.

SI Base Units.

These units are the basis upon which the entire SI system is built.  Their definitions are often build on some measurable physical property.  These definitions have changed over time in some cases.  Take for example, the metre.  Originally, it was defined as some unit fraction of the distance from the north pole to the equator along the meridian that passes through Paris.   Later, it was defined as length of 9 192 631 770 wavelengths of the microwave radiation of a certain transition of ¹³³Cs. Now, it's defined to be the distance traveled by light in a vacuum in 1/299 792 458 of a second. (see http://en.wikipedia.org/wiki/SI_base_unit or http://physics.nist.gov/cuu/Units/current.html )

There are many other SI units that are used regularly which are derived from combinations of these base units.  Of the extremely large possible number of such combinations, there are 22 which are named. The table below shows a few of these used in this course.

Because of the huge range of values, we often simplify the representation of a measured amount by using a prefix to represent a power of 10.  For example, we may measure the length of a field in meters but the length of a finger in centimeters and of a city street in kilometers.  Here is a more complete list of accepted SI prefixes.

These prefixes should be memorized, especially the ones between 10¹² and 10^-12.

(These tables are taken from http://physics.nist.gov/cuu/Units/units.html )

Logarithms/Exponentials:

Logarithms are commonly used to compress the scale of numbers to an easily manageable range.  For example, we are familiar with the Richter scale for earthquakes where every unit represents an order of magnitude in earthquake intensity.  Thus, a quake measuring 5 on the Richter scale is 10 times stronger than one measuring 4 on that scale.  Using simple numbers, we can express the energy of an earthquake that can span 8 or 9 orders of magnitude. 

To manipulate logarithms and exponential numbers, there are a few simple rules.

When multiplying exponential numbers, add the exponents if same base.

When dividing exponential numbers, subtract the exponents if same base.

Logarithms are like exponentials:

When you multiply two logarithmic numbers, add their logs if same base.

When you divide two logarithmic numbers, subtracts their logs if same base.

What is a logarithm?

It depends on the Base used.

e.g. 

If not even power of the base:

To go from a log to a real number:

find the anti-log of the

In general,

Natural logs        base e         e = 2.718281828...

log to base e is often abbreviated as ln.

ln e = 1                 anti ln  (also written as ln^-1)º e^x

ln e² = 2

etc...

To convert between log and ln: 

Let's say that we need to calculate the natural log of a number as in ln y = x but our calculator can only calculate base 10 logarithms (log y).  In that case, we can convert from log y to ln y quite easily knowing the equality

ln y  = 2.303 × log y

thus, we can write

      x = 2.303×log y

Michael J. Mombourquette.
Copyright © 1997
Revised: February 23, 2011.

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