Maths and Chemistry for Biologists

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Maths and Chemistry for Biologists
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Maths 1

• This section of the course covers
• why biologists need to know about maths and
• chemistry
• powers and units
• an introduction to logarithms
• the rules of logarithms
• the usefulness of logs to the base 10

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Why do biologists need to know about maths and
chemistry?

• The next slide describes a typical experiment in
  biology.
• It is written in four languages
• common English, biology, chemistry and maths
• You need to speak all four to understand it
• This part of the course aims to cover those bits
  of chemistry and maths that the biologist must
  know

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Powers and Units

• Chemical and biological systems often involve
  very large and very small numbers
• There are 602,000,000,000,000,000,000,000 atoms
  in 12 grams of carbon
• Each atom has a radius of
• 0.00000000000000275 m
• These numbers are very inconvenient easy to get
  wrong number of zeros
• This is where powers come in

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Powers

• Number multiplied by itself several times e.g.
• 2 x 2 x 2 x 2 written as 24
• (spoken as two to the power four)
• Special cases 22 is two squared and 23 is two
  cubed
• Powers can be negative e.g. 2-3 (two to the minus
  three)
• This means
• Special case is 20 1
• Any number raised to the power zero is equal to 1

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• How does this help with large and small numbers?
• 602,000,000,000,000,000,000,000
• is the same as 6.02 x 1023
• that is, 6.02 multiplied by 10 23 times
• (move the decimal point left 23 places)
• 0.00000000000000275
• is the same as 2.75 x 10-15
• that is, 2.75 divided by 10 15 times
• (move the decimal point right 15 places)

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Rules for powers

• When terms are multiplied powers are added
• so 32 x 33 35
• When terms are divided powers are subtracted
• so 33
• and
• 3(7-3-4) 30 1

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Units

• All chemical and physical quantities have units
• We could give a length as 0.005 m or 5 x 10-3 m
• Or we could give it as 5 mm (5 millimetres)
• So we can avoid using powers of ten by changing
  the size of the unit
• For example, you might buy 1000 g of sugar or
  alternatively 1 kg (1 kilogram)
• We add a prefix to the unit to change its size

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Prefixes to units

• The common ones are

So 10-6 m 1 ?m 3 x 10-9 g 3 ng 5 x 109 V
5 GV
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A word of warning

• Do not add, subtract, multiply or divide numbers
  with units with different prefixes
• e.g. to work out the area of a rectangle 1 m long
  by 5 mm wide cannot say the area is 5 because the
  units are not defined
• Change one of the lengths to have same prefix
• e.g. 1 m 103 mm so area is 5 x 103 mm2
• or 5 mm 5 x 10-3 m so area is 5 x 10-3 m2

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One for you to do

• The universe contains 1011 galaxies and each
  galaxy contains 1011 stars
• Suppose that 1 in 1000 of those stars has a
  planet with conditions suitable for life to
  develop
• Suppose that the probability of life developing
  on such a planet is 1 in 1,000,000,000,000
• How many planets might have developed life?

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Answer
101111-3-12 107
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Logarithms

• DEFINITION if a bc then c logba
• (spoken as log to the base b of a)
• Two important cases base 10 and base e
• (e is an irrational number equal to 2.71828.)
• Base 10 log10 2 0.3010
• What this means is that 100.3010 2

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Why are they useful?

• Change numbers with powers of 10 into simpler
  forms
• e.g. log10 5x106 6.699
• log10 2x10-4 -3.699
• As number goes up by a power of 10 the log goes
  up by unity
• e.g. log10 5 0.699
• log10 50 1.699
• (Note numbers less than 1 have negative logs
  negative numbers do not have logs)

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An example a dose/response curve
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In the experiment in the previous slide we
plotted the response against the log of the Dose.
This is because the dose covered a Very wide
range of values. We used the property of logs
that as the number goes up by 10 fold the log
goes up by unity. Try plotting the response
directly against The dose and you will see that
you get a Rather silly looking graph.
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Rules and results

• If no base is specified then it is assumed to be
  10
• log (a x b) log a log b
• log (a/b) log a log b
• log an n x log a
• It follows from the definition that log 10 1
• so log 10n n x log 10 n
• e.g. log 106 6
• log 10-3 -3

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Some for you to do

• Without using a calculator, work out
• log 1023
• log 1.2 given that log 120 2.0792
• (remember that these are all logs to the base 10)

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Answers

• log 1023 23 x log 10 23
• log 1.2 log 120 x 10-2 log 120 log 10-2
• 2.0792 (-2) 0.0792