# Note that the terms scientific notation, exponential notation, powers, exponents all mean the same thing.

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1 Rationale: why use scientific notation or powers? Writing very large numbers in scientific notation Writing very small numbers in scientific notation Practice converting between normal numbers and scientific notation Add and subtract in scientific notation Multiply and divide in scientific notation b A note about fractional powers Prefixes Practice with prefixes Supplementary material SI Units Converting between units for volume...11 Summary of Learning Outcomes Rationale: why use scientific notation or powers? In biology there are many instances where you might need to calculate and manipulate very large numbers or very small numbers. For example the number of nerve cells in an average brain might be On the other hand, the length of a cell under the microscope might be m. The number of cell surface receptors for hormones might be 0000 per cell whilst the concentration of peptide hormone in the extracellular space might be M. These very large or very small numbers are difficult to read and that is why we use scientific notation or powers. 1.2 Writing very large numbers in scientific notation Very large numbers can be rewritten as other numbers multiplied together. For example 0 is equal to times and we can write this as 2. The table shows how other larger numbers can be written. = = just one ten = 1 0 = x = 2 tens multiplied together = = x x = 3 tens multiplied together = = x x x = 4 tens multiplied together = = x x x x = 5 tens multiplied together = = tens multiplied together = Definition of terms: Note that the terms scientific notation, exponential notation, powers, exponents all mean the same thing. The numbers that you re multiplying together are called the base. The number of times you multiply them together is called the power or exponent. So in the last example, 000 is written as ten to the four or 4, is the base and 4 is the power or exponent. Creative Commons: Attribution Non-commercial Share Alike page 1 of 12

2 Some examples: Example 1.1 Write 6000 in scientific notation This is just 6 x 00 which is 6 x 3 Example 1.2 I have done an experiment to determine the concentration of drug in solution and the answer was molecules/l. Write this in scientific notation. Write and then count how many places you need to move the decimal point to the right In practice you would never be able to measure the concentration of drug to that degree of accuracy. Usually you would work out how many significant figures are appropriate in this instance. You may decide to write it in 4 significant figures instead, x 6. Creative Commons: Attribution Non-commercial Share Alike page 2 of 12

3 1.3 Writing very small numbers in scientific notation We can use the same ideas when writing very small numbers. 1/ = 0.1 = 1/ 1 = -1 1/0 = 0.01 = 1/ 2 = -2 1/00 = = 1/ 3 = -3 1/000 = = 1/ 4 = -4 there is a handy general rule to remember, 1 / a = -a Some examples: Example 1.3 Write in scientific notation Answer: 5.4 x -4 This time you had to count how many places to move the decimal place to the left. Example 1.4 Write in scientific notation Answer: 1.34 x -2 It is just a convention to put the decimal place after the first digit. You could, if you wanted to, write this number in many different ways including: x x x -3 All you are doing is moving the decimal place and changing the power to compensate. Creative Commons: Attribution Non-commercial Share Alike page 3 of 12

4 1.4 Practice converting between normal numbers and scientific notation It is important that you are familiar and confident with how to convert between normal numbers and scientific notation and vice versa. To write 6478 in scientific notation, write x 3. What you are doing is working out how many places to move the decimal point. The expression x 3 is just saying, write and move the decimal point three places to the right giving Or you can think of it as saying 6478 is the same as x 00 which is the same as x 3 To write in scientific notation, write 4.5 x -4 The expression 4.5 x -4 is saying, write 4.5 and move the decimal place four places to the left giving Or you can think of it as saying 4.5 / 4 or 4.5 / 000. Some Examples: Example 1.6 Write in scientific notation. Answer: 3.4 x 5 Example 1.7 Write in scientific notation. Answer: 8 x -6 Example 1.8 Fill in the gaps: can be written as x -2 and x -3 and x -4 Answer: x -2 and 4.75 x -3 and 47.5 x -4 Example 1.9 Write in scientific notation to two significant figures Answer: 9.9 x 6 (note that if the third digit is 5 or more, then the second digit is rounded up so in this case the third digit is 5 which means the second digit, 8, gets rounded up to 9. Creative Commons: Attribution Non-commercial Share Alike page 4 of 12

5 1.5 Add and subtract in scientific notation To add or subtract two numbers in scientific notation, you first need to convert them to the same power. For example, 5 x x 5 = 5 x x 3 = 405 x 3 = 4.05 x 5 This is just the same as what you would normally do, i.e. you would line them up The same idea is used when subtracting, 2 x -3 8 x -4 = 20 x -4 8 x -4 = 12 x -4 = 1.2 x x x 3 = x 3 This might be easier to visualise as x x -4 = x -4 Creative Commons: Attribution Non-commercial Share Alike page 5 of 12

6 1.6 Multiply and divide in scientific notation To multiply numbers with the same base, add the exponents a b x a c = a b + c Some examples: Example 1. 3 x 5 = 3+5 = 8 Example x 3 = 2 x 3 = 2+3 = 5 Here you have to convert 0 to 2 so you have the same base first before adding the powers. Example x 2 x 5 x Here you just multiply the 6 and 5 as you would normally do, then add the powers. =30 x 12 What is the power of a power? (a b ) c (b x c) = a Example 1.13 ( 3 ) 3 = 3 x 3 x 3 = 3 x 3 = 9 Example 1.14 ( -5 ) 2 = - Creative Commons: Attribution Non-commercial Share Alike page 6 of 12

7 To divide numbers with the same base, subtract the exponents b a b c = a c a Example 1.15 Example = = = = Example = = = 15 9 ( 15) Example 1.18 = = = 3 3 Here you have to convert 0.1 to -1 so you have the same base first before adding the powers Example 1.19 = 2.5 = Here you just divide 5 by 2 as you would normally do, then subtract the powers. But what if b c gives zero? If b - c is zero, then the exponents were the same and this is the same as dividing a number by itself which of course gives one. a 0 = 1 Example = = = b A note about fractional powers. Once we understand that a x b = a+b then it becomes clear that the values for a and b do not need to be integers. For example, consider the following, 0.5 x 0.5 = = 1 = This is the same as writing: 1/2 x 1/2 = 1/2+1/2 = which is the same as: = Similarly 1/3 x 1/3 x 1/3 = is the same as writing: = Creative Commons: Attribution Non-commercial Share Alike page 7 of 12

8 The symbols and 3 are typically only used for square roots and cubed roots i.e. 1/2 and 1/3 respectively. Otherwise we just use the decimal notation eg Fractional powers follow all the same rules as integer powers. multiplying a x b = a+b example: 0.3 x 0.8 = 1.1 dividing a = b a b example: = = 0. 9 powers of powers ( a ) b = axb example: ( 0.3 ) 0.5 = 0.15 For addition and subtraction we must convert to the same power, so: = x x -7 = x Prefixes Prefixes are a useful way of abbreviating even further for example -3 g = 1 mg (one milligram) Here is a summary of all of the standard prefixes. The main prefixes in use in biomedical science are shown in bold: learn them. Factor Prefix Symbol Factor Prefix Symbol 24 yotta Y -1 deci d 21 zetta Z -2 centi c 18 exa E -3 milli m 15 peta P -6 micro µ 12 tera T -9 nano n 9 giga G -12 pico p 6 mega M -15 femto f 3 kilo k -18 atto a 2 hecto h -21 zepto z 1 deca da -24 yocto y Creative Commons: Attribution Non-commercial Share Alike page 8 of 12

9 1.8 Practice with prefixes Some examples: Example 1.21 Convert scientific notation to a prefix: Convert 5 x -5 g to mg and then to µg. 5 x -5 g = 5 x -2 x -3 g = 0.05 mg or = 50 x -6 g = 50 µg Example 1.22 Example 1.23 Convert a prefix to scientific notation: 12pmol = 12 x -12 mol = 1.2 x -11 mol Write nm in scientific notation: 0.033nM = x -9 M = 3.3 x -11 M Example 1.24 Under the microscope, an epithelial cell looks quite rectangular and you can use the formula for the area of a rectangle to estimate the area of the cell. The dimensions you measure are width = 1 µm and length µm. Express the area in scientific notation with m 2 as the units. Area = width x length = 1 x -6 m x x -6 m = x -12 m 2 The height has been estimated from other studies to be approximately 5µm, what is the volume of the cell (in m 3 in scientific notation of the form a x b )? Volume = area x height = x -12 m 2 x 5 x -6 m = 50 x -18 m 3 = 5 x -17 m 3 Creative Commons: Attribution Non-commercial Share Alike page 9 of 12

10 1.9 Supplementary material SI Units The standard units of measurement are the SI Units (Systeme Internationale) are given in the table below. A good website for reference is the National Physical Laboratory ( measurement SI Unit standard abbreviation time second s length metre m mass kilogram kg electric current ampere A thermodynamic temperature kelvin K amount of substance mole mol luminous intensity candela cd There are quite a few units which are in everyday use which are not in the above table. The important ones for biologists are the units for time, temperature and volume: name symbol value in SI units minute min 1 min = 60 s hour h 1 h = 60 min = 3600 s day d 1 d = 24 h = s degrees Celsius C temp in C = (temp in K) litre l, L 1 l = 1 dm 3 = -3 m 3 Note that the litre can be abbreviated as l or L but L is often used because of the potential for confusion of l ( ell ) and 1 ( one ). Sometimes dm 3 is used instead of L and cm 3 is used instead of ml although L and ml are more common. There are a few conventions and it is a good idea to follow them to avoid confusion. The key ones are: Unit symbols are unaltered in the plural (i.e. write 8 m not 8 ms to mean 8 metres.) Abbreviations such as sec (for either s or second) or mps (for either m/s or meter per second) are not allowed. A space is left between the numerical value and unit symbol (25 kg but not: 25- kg or 25kg). Mathematical operations should only be applied to unit symbols (kg/m 2 ) and not unit names (kilogram/cubic metre). kg/m 2 can also be written as kg.m -2 Creative Commons: Attribution Non-commercial Share Alike page of 12

11 1. Converting between units for volume Volume is sometimes expressed in dm 3, or cm 3 or litres or ml it is essential to be able to convert between them. Converting volumes between m 3 and L. A spherical bacterial cell has a diameter of approximately 4 µm, what would its volume be (in m 3 and in L)? There are (at least) two possible approaches to this. Method 1: The easiest way to do this is to change the units of the dimensions to dm first. The volume of a sphere is 4/3 πr 3. The radius is 2 µm = 2 x -6 m = 2 x -5 dm Volume = 4/3 x 3.14 x (2 x -5 dm) 3 = 33.5 x -15 dm 3 = 33.5 fl Method 2: An alternative method is to calculate the volume in µm 3 and convert the answer from µm 3 directly to L. Volume = 4/3 x 3.14 x (2 µm) 3 = 33.5 (µm) 3 I have included the bracket around the µm here to point out that it is a micrometer cubed that is, ( -6 m) 3 which is -18 m 3. This is not the same thing as µ(m) 3 which would be -6 m 3. Commonly when you see µm 3 written down it means (µm) 3. Volume = 33.5 x ( -6 ) 3 = 33.5 x -18 m 3 So now we need to convert 33.5 x -18 m 3 to litres. One way to do this is to say, 33.5 x -18 m 3 x ( dm.m -1 ) 3 x 1L.dm -3 translating this into words we take 33.5 x -18 m 3 and multiply by decimeters per m three times and by one litre per decimetre cubed. You can see that the units cancel out to leave us with x -18 x () 3 L So we are left with 33.5 x -15 L = 33.5 fl Creative Commons: Attribution Non-commercial Share Alike page 11 of 12

12 Summary of Learning Outcomes At the end of this section, you should be able to do the following confidently and reliably. 1. Convert between normal numbers and scientific notation and vice versa 2. Use the basic rules of manipulating powers and scientific notation, i.e. add, subtract, multiply and divide. 3. To know the common prefixes. 4. To be able to use these prefixes in calculations. 5. Recognise SI Units, 6. Convert between units for volume dm 3 L Creative Commons: Attribution Non-commercial Share Alike page 12 of 12

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