Maths and Chemistry for Biologists

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

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

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 m These numbers are very inconvenient – easy to get wrong number of zero’s This is where powers come in

Powers Number multiplied by itself several times e.g. 2 x 2 x 2 x 2 written as 2 4 (spoken as two to the power four) Special cases 2 2 is two squared and 2 3 is two cubed Powers can be negative e.g (two to the minus three) This means Special case is 2 0 = 1 Any number raised to the power zero is equal to 1

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

Rules for powers When terms are multiplied powers are added so 3 2 x 3 3 = 3 5 When terms are divided powers are subtracted so = 3 3 and = 3 (7-3-4) = 3 0 = 1

Units All chemical and physical quantities have units We could give a length as m or 5 x 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

Prefixes to units The common ones are FracPrefixSymbolMultPrefixSymbol millim10 3 kilok micro  10 6 megaM nanon10 9 gigaG picop10 12 teraT femtof attoa So m = 1  m; 3 x g = 3 ng; 5 x 10 9 V = 5 GV

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 = 10 3 mm so area is 5 x 10 3 mm 2 or 5 mm = 5 x m so area is 5 x m 2

One for you to do The universe contains galaxies and each galaxy contains 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?

Answer = = 10 7

Logarithms DEFINITION: if a = b c then c = log b a (spoken as log to the base b of a) Two important cases base 10 and base e (e is an irrational number equal to ….) Base 10: log 10 2 = What this means is that = 2

Why are they useful? Change numbers with powers of 10 into simpler forms e.g. log 10 5x10 6 = log 10 2x10 -4 = As number goes up by a power of 10 the log goes up by unity e.g. log 10 5 = log = (Note – numbers less than 1 have negative logs; negative numbers do not have logs)

An example – a dose/response curve Dose (ng)Log (dose) Resp , , , ,000,

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.

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 a n = n x log a It follows from the definition that log 10 =1 so log 10 n = n x log 10 = n e.g. log 10 6 = 6 log = -3

Some for you to do Without using a calculator, work out log log 1.2 given that log 120 = (remember that these are all logs to the base 10)

Answers log = 23 x log 10 = 23 log 1.2 = log 120 x = log log = (-2) =