Math in the Biotechnology Laboratory: Basic Math Techniques

1
Chapter 13

• Math in the Biotechnology Laboratory Basic Math
  Techniques

2
Exponents

• Exponential notation or exponential form
• Commonly found in algebraic terms, expressions,
  and equations.
• Exponents are used to shorten or condense
  repeated multiplication.
• For example, a term containing an exponent is
  shown below74
• In this term, 7 is the base and 4 is the
  exponent.
• The exponent or power, indicates the number of
  times that the factor, or base, is multiplied.
• 74 7 7 7 7

3
Exponents

• When working with expressions containing only
  numbers
• Simply perform the indicated multiplication.
  Below are three examples
• 152 15 15 225 
• 31 3 
• 27 2 2 2 2 2 2 2 128
• Notice in the second example that a number raised
  to the first power is just that number (31 3).
• When you see a number that does not have an
  exponent, it is because the 1 is assumed.
• When variables are included in the terms, we work
  according to the same principles.

4
Exponents

• A Variable Raised to a Power
• If the term is simply one variable
• Exponent is handled just as it was in the
  previous section.
• If we want to raise one number or variable to a
  power we put the exponent as a superscript to
  that one number or variable.
• An algebraic term containing an exponent is shown
  next.

5
Exponents

• In this term, x is the base, and 2 is the
  exponent.
• The exponent, or power, indicates the number of
  times that the factor, or base, occurs.
• x2 x x
• Next, the term contains the variable to the
  fourth power.
• The numerical coefficient is not raised to the
  power.
• y is the base, and 4 is the exponent.
• 3y4 3 y y y y
• As with numbers, a variable that does not appear
  to have a power is raised the first power.
• z1 z

6
Exponents

• A Term Raised to a Power
• Remember, a term is either a number or a product
  of a number and one or more variables.
• Any powers to which the variables are raised.
• In a term, either the entire term, or just one
  variable within the term, may be raised to a
  power.
• To indicate that the entire term is raised to a
  power, we must enclose the term in parentheses
  and place the exponent outside the parentheses.
• The general rule for applying an exponent to a
  whole term contained within parentheses is
• (xa)b xab

7
Exponents

• The result is much different than when the
  parentheses are omitted.
• In the example, the entire term is raised to the
  second power this is commonly referred to as
  squared.
• Notice the parentheses around the entire term.
• (5x)2 51 2 x12 52x2 5 5 x x 25x2
• In this example there are no parentheses grouping
  5 and x.
• This means only the variable x is squared.
• 5x2 5 x x

8
Exponents

• General Rules for Using Exponents with Variables
  and Terms
• Any number or variable that appears to have no
  exponent is raised to the first power.
• x1 x
• If you raise any number or variable to the power
  of zero, the result is one.
• 50 1, and z0 1
• To apply an exponent to a term in exponential
  form, multiply the exponents.
• (ydxa)b ydbxab
• An exponent outside the parentheses applies to
  all parts of a product or quotient inside the
  parentheses
• (xy)a (x y)a xa ya
• When multiplying together two terms with the same
  variables, add the exponents.
• xc xd xc d
• A number or variable to a negative power means to
  move it to the denominator of a fraction and put
  it to the positive power.
• 53 1/53, and z2 1/z2

9
Exponents

• Exponents where the base is 10
• For numbers greater than 1
• The exponent represents the number of places
  after the number (and before the decimal point)
• The exponent is positive
• The larger the positive exponent, the larger the
  number
• 103 1000
• 106 1,000,000

10
Exponents

• Exponents where the base is 10
• For numbers less than 1
• The exponent represents the number of places to
  the right of the decimal point including the
  first nonzero digit
• The exponent is negative
• The larger the negative exponent, the smaller the
  number
• 10-2 0.01
• 10-5 0.00001
• 10-9 0.000000001

11
Scientific Notation

• Scientific notation is simply a method for
  expressing, and working with, very large or very
  small numbers. 
• Short hand method for writing numbers. 
• Numbers in scientific notation are made up of
  three parts
• The coefficient
• The base
• The exponent.
• Observe the example below
• 5.67 x 105
• This is the scientific notation for the standard
  number, 567 000. 
• Now look at the number again, with the three
  parts labeled.5.67   coefficient       10
  base       5 exponent
• In order for a number to be in correct scientific
  notation, the following conditions must be true
• 1. The coefficient must be greater than or equal
  to 1 and less than 10.
• 2. The base must be 10.
• 3. The exponent must show the number of decimal
  places that the decimal needs to be moved to
  change the number to standard notation. 
• A negative exponent means that the decimal is
  moved to the left when changing to standard
  notation.

12
Scientific Notation

• Changing numbers from scientific notation to
  standard notation.
• Change 6.03 x 107 to standard notation.
• Remember,  107 10 x 10 x 10 x 10 x 10 x 10 x 10
  10,000,000
• So, 6.03 x 107 6.03 x 10,000,000 60,300,000
• Answer 60,300,000
• Instead of finding the value of the base, we can
  simply move the decimal seven places to the right
  because the exponent is 7.
• So, 6.03 x 107 60,300,000

13
Scientific Notation

• Now let us try one with a negative exponent.
• Ex.2 Change 5.3 x 10-4 to standard notation.
• The exponent tells us to move the decimal four
  places to the left.
• So, 5.3 x 10-4 0.00053

14
Scientific Notation

• Changing numbers from standard notation to
  scientific notation
• Ex.1  Change 56,760,000,000 to scientific
  notation
• Remember, the decimal is at the end of the final
  zero.
• The decimal must be moved behind the five to
  ensure that the coefficient is less than 10, but
  greater than or equal to one.
• The coefficient will then read 5.676
• The decimal will move 10 places to the left,
  making the exponent equal to 10.
• Answer equals 5.676 x 1010

15
Scientific Notation

• Now we try a number that is very small.
• Ex.2  Change 0.000000902 to standard notation
• The decimal must be moved behind the 9 to ensure
  a proper coefficient.
• The coefficient will be 9.02
• The decimal moves seven spaces to the right,
  making the exponent -7
• Answer equals 9.02 x 10-7 

16
Scientific Notation

• Not only does scientific notation give us a way
  of writing very large and very small numbers, it
  allows us to easily do calculations as well. 
• Calculators are very helpful tools, but unless
  you can do these calculations without them, you
  can never check to see if your answers make
  sense.  
• Any calculation should be checked using your
  logic, so don't just assume an answer is
  correct. 
• Rule for Multiplication - When you multiply
  numbers with scientific notation, multiply the
  coefficients together and add the exponents. 
• The base will remain 10.

17
Scientific Notation

• Ex 1.  Multiply  (3.45 x 107) x (6.25 x 105)
• first rewrite the problem as    (3.45 x 6.25) x
  (107 x 105)
• Then multiply the coefficients and add the
  exponents    21.5625 x 1012
• Then change to correct scientific notation and
  round to correct significant digits  2.16 x 1013
• NOTE - we add one to the exponent because we
  moved the decimal one place to the left.
• Remember that correct scientific notation has a
  coefficient that is less than 10, but greater
  than or equal to one

18
Scientific Notation

• Ex 1.  Multiply  (3.45 x 107) x (6.25 x 105)
• First rewrite the problem as    (3.45 x 6.25) x
  (107 x 105)
• Then multiply the coefficients and add the
  exponents    21.5625 x 1012
• Then change to correct scientific notation and
  round to correct significant digits  2.16 x 1013
• NOTE - we add one to the exponent because we
  moved the decimal one place to the left.
• Remember that correct scientific notation has a
  coefficient that is less than 10, but greater
  than or equal to one.

19
Scientific Notation

• Ex. 2.  Multiply (2.33 x 10-6) x (8.19 x 103)
• Rewrite the problem as (2.33 x 8.19) x (10-6 x
  103)
• Then multiply the coefficients and add the
  exponents  19.0827 x 10-3
• Then change to correct scientific notation and
  round to correct significant digits 1.91 x 10-2
• Remember that -3 1 -2

20
Scientific Notation

• Rule for Division - When dividing with scientific
  notation, divide the coefficients and subtract
  the exponents. 
• The base will remain 10.
• Ex. 1 Divide 3.5 x 108 by 6.6 x 104
• Rewrite the problem as                 3.5 x 108
                                                  
  -----------                             
                               6.6 x 104        
• Divide the coefficients and subtract the
  exponents to get      0.530303 x 104
• Change to correct scientific notation and round
  to correct significant digits to get 5.3 x 103
• Note - We subtract one from the exponent because
  we moved the decimal one place to the right.

21
Scientific Notation

• Rule for Addition and Subtraction - when adding
  or subtracting in scientific notation, you must
  express the numbers as the same power of 10. 
• This will often involve changing the decimal
  place of the coefficient.
• Ex. 1  Add 3.76 x 104 and 5.5 x 102
• Move the decimal to change 5.5 x 102 to 0.055 x
  104
• Add the coefficients and leave the base and
  exponent the same  3.76 0.055 3.815 x 104
• Following the rules for rounding, our final
  answer is 3.815 x 104        
• Rounding is a little bit different because each
  digit shown in the original problem must be
  considered significant, regardless of where it
  ends up in the answer.

22
Scientific Notation

• Ex. 2  Subtract (4.8 x 105) - (9.7 x 104)
• Move the decimal to change 9.7 x 104 to 0.97 x
  105
• Subtract the coefficients and leave the base and
  exponent the same  4.8 - 0.97 3.83 x 105
• Round to correct number of significant digits
  3.83 x 105  

23
Units of Measure

• The Metric system was developed in France during
  the Napoleonic reign of France in the 1790's.
• The metric system has several advantages over the
  English system which is still in place in the
  U.S.
• The scientific community has adopted the metric
  system almost from its inception.
• In fact, the metric system missed being
  nationalized in this country by one vote in the
  Continental Congress in the late 1700's or early
  1800's.
• The advantages of the Metric system are
• It was based on a decimal system (i.e. powers of
  ten).
• Simplifies calculations by using a set of
  prefixes
• It is used by most other nations of the world,
  and therefore, it has commercial and trade
  advantage.
• If an American manufacturer that has domestic and
  international customers is to compete, they have
  to absorb the added cost of dealing with two
  systems of measurement.

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Units of Measure

• Prefixes" used in the metric system sometimes
  referred to as the System Internationale (SI).
• One of the mathematical advantages of the metric
  system is its combination of metric terminology
  with its decimal organization.
• There are several prefixes that are associated
  with a decimal position and can be attached to
  the base metric unit in order to create a new
  metric unit.
• The knowledge of the decimal meaning of the
  prefix establishes the relationship between the
  newly created unit and the base unit.
• For example the prefix "kilo" means 103 or 1000
  so if I take a mythical base unit like the
  "bounce" and I attach the kilo prefix in front, I
  create a new unit called the "kilobounce".
• In addition, the relationship between the two
  units is now well established.
• Since I know that "kilo" means 1000 then one
  kilobounce unit is the same as (or equal to) 103
  bounce units.

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Prefix Symbol Multiply the base by
exa- E 1 000 000 000 000 000 000
peta- P 1 000 000 000 000 000
tera- T 1 000 000 000 000
giga G 1 000 000 000
mega M 1 000 000
kilo k 1000
hecto- h 100
deca- da 10
deci- d 0.1
centi c 0.01
milli- m 0.001
micro- u 0.000 001
nano- n 0.000 000 001
pico- p 0.000 000 000 001
femto- f 0.000 000 000 000 001
atto- a 0.000 000 000 000 000 001
 

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Mass Measurement

• The measure of mass in the metric system has
  several units that scientists use most often.
• The kilogram is the standard unit of mass in the
  metric or SI system.
• The Kilogram (Kg) is roughly analogous to the
  English pound.
• It takes approximately 2.12 pounds to equal one
  Kilogram.
• A smaller mass unit analogous to the English
  ounce is the gram.
• The gram represents approx. 30 dry ounces in
  mass. Other metric mass units include
• The centigram (cg) 0.01 g
• Milligram (mg) 0.001 g
• Microgram (µg) 0.000001 g
• Nanogram (ng) 0.000000001 g
• The basic instrument used to measure mass is the
  mass balance.
• There are some digital balances today that can
  display the mass of an object in several
  different mass units both in the English and
  Metric systems

27
Dimensional Measurement

• Dimensional measurement that is measure of
  length, width, and height.
• The basic metric unit of dimension is the meter
  (m).
• The meter is analogous to the English yard.
• A meter is equal to slightly more than a yard
  (about 10 larger).
• One meter is equal to 1.09 yards or 39.36 inches.
  
• A larger metric unit used often is the kilometer
  (km) which is analogous to the English mile.
• One kilometer is equal to 0.62 miles.
• In countries where the metric system is the
  national standard, signposts and posted speed
  limits are in km or km per hour.
• For example, the most common speed limit in
  Mexico is 100, but that is 100 km/h or about 60
  miles per hour!!
• Other dimensional units include the
• Decimeter (dm) 0.1 m
• Centimeter (cm) which is analogous to the English
  inch. 0.01 m
• One inch is equal to 2.54 cm
• Millimeter (mm) 0.001 m
• Micrometer (µm) 0.000001 m 10-6 m
• Nanometer (nm) 0.000000001 m 10-9
• The nanometer is used when very small
  inter-atomic or intermolecular distances are
  called for

28
Dimensional Measurement

• The main instrument in the science lab that
  measures dimension is the metric ruler.
• The metric ruler comes in various sizes.
• There is the 150 mm ruler and a metric meter
  ruler which are used most.
• However, all metric rulers are calibrated the
  same.
• The numerically numbered positions (major
  calibrations) are equal to centimeter marks, and
  then there are ten equally spaced positions
  (minor calibrations) in between each of the
  numbered positions each of which are equal to 0.1
  cm(1 mm).
• According to this calibration, one can record
  measurements with one position of estimation to
  the nearest 0.01 cm.
• Another instrument most often used in Physics
  labs is called a micrometer. As the name implies
  it can measure to the nearest micrometer and is
  used for very precise measurements of diameters.

29
Volume Measurement

• The third type of measure is measure of volume.
• We can break this down into the measure of
• Solid volume (regular and irregular)
• Fluid (liquid and gas) volume
• Volumes of Regular Solids
• Regular Solids are those that have well defined
  dimensions of length, width, height, and
  diameter.
• These can first be measured with a suitable
  dimensional instrument like a metric ruler.
• Then a suitable geometrical formula might be
  applied to get the volume.
• For example, if the solid was rectangular shaped,
  you would measure the dimensions of the rectangle
  and then use the formula
• V l X w X h in order to determine the volume of
  the rectangle.

30
Volume Measurement

• Volumes of Irregular Solids
• Irregularly shaped solids do not have well
  defined dimensions and therefore can't use the
  above method of determining its volume.
• However, one can use the principle of liquid
  displacement that says since two chunks of matter
  can't occupy the same space at the same time that
  when placed together one object will displace the
  other.
• If we measure a certain volume of water in a
  graduated cylinder to be 5.0 cm3, and we immerse
  some pieces of metal into the water, the reading
  on the graduated cylinder might read 14.0 cm3.
• By subtracting the two readings we now have how
  much displacement of the water there was when the
  metal fragments were immersed.
• That displacement would be equal to the volume of
  the metal fragments.
• 14.0 - 5.0 9.0 cm3 volume of metal fragments

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Measurement of Fluid Volumes

• The basic metric unit of measure for volume is
  the liter (l) unit.
• The liter is analogous to the English quart.
• One liter being the same as 1.06 quarts.
• It is basically a fluid volume unit as is the
  smaller metric unit called the milliliter (ml).
• The milliliter is analogous to the English fluid
  ounce.
• One fluid ounce is equal to about 30 ml.
• Other metric units of volume that are more often
  associated with volumes of solids is the cubic
  centimeter(cm3) which is equal to a milliliter.
• To a careless observer the cm3 may look like a
  dimensional unit since it has the symbol for
  "centimeter" in it.
• However, it also has the word "cubic" which
  always indicates a volume unit.