Powers
Applications of Powers - including SI units
POWERS AND EXPONENTS
Powers are simply numbers multiplied by themselves.
10 x 10 = 100 or 102
the small 2 is the exponent or power of 10
10 x 10 x 10 = 1000 = 103
10 x 10 x 10 x 10 = 10, 000 = 104
NOTE : the exponent is the number of times the number is multiplied by itself and in the case of 10, it is also equal to the number of zeros after the one in the final number.
Note also that
10 x 1 = 10 = 101
100 = 1
Why is this notation used? It is used as a shorthand way of expressing very large or very small numbers.
1 million = 1,000,000 = 106
5 million = 5,000,000 = 5 x 106
5,300,000 = 53 x 100,000 = 53 x 105 = 5.3 x 106
NOTE : the last example that as I moved the decimal point one place to the left for the 53, I increased the power of 10 by 1 (5 to 6). That is because essentially I divided 53 by 10 and multiplied 105 by 10 to balance. That brings us to the rules for manipulating powers.
ADDITION
You can only add numbers expressed as powers of 10 if the exponent is the same; you add the digits and the power remains the same e.g.
(3 x 105) + (5 x 105) = (3+5) x 105 = 8 x 105
How then do you add numbers such as
(3 x 104) + (6 x 105) ?
You must change them into the same power and then add the digits, with the power of 10 remaining the same; there are two ways of doing this:
3 x 104 = 0.3 x 105
0.3 x 105 + 6 x 105 = 6+0.3 x 105 = 6.3 x 105
or
6 x 105 = 60 x 104
60 x 104 + 3 x 104 = 60+3 x 104 = 63 x 104 = 6.3 x 105
note that in both cases I have returned the final number to a form in which there is a single whole number followed by decimal points. Although there is no single correct way of writing numbers as powers, the above form is the most widely used.
SUBTRACTION
As with addition, the numbers must be in the same power of 10
(6 x 104)- (2 x 104) = 6-2 x 104 = 4 x 104
6 x 104 - 4 x 103 = 6 x 104 - 0.4 x 104 = 6-0.4 x 104 = 5.6 x 104
MULTIPLICATION
When you want to multiply numbers expressed as powers of 10, you must multiply the digits and add the exponents. The numbers do not have to be in the same power i.e.
(5 x 102) x (5 x 103) = 500 x 5000 = 2500000.0 = 2.5 x 106
Use brackets to separate each component of the equation
(3 x 104) x (5 x 103) = (5 x 3) x 10 (4+3) = 15 x 107 = 1.5 x 108
DIVISION
As in multiplication, the digits must be divided as normal and the exponents are subtracted. Again the powers need not be the same i.e.
(5 x 104) / (5 x 102) = 50,000 / 500 = 500/5 = 100 = 5/5 x 104-2 = 1 x 102
What happens if you have a denominator larger than a numerator? You are left with a number which is less than 1 e.g.
(6 x 102) / (2 x 103) = 6/2 x 102-3 = 3 x 10-1
This brings us to negative exponents.
Negative exponents are used to express numbers between 0 and 1 i.e. the numbers to the right of a decimal point.
= 1/10 = 10-1 (one place to the right of the decimal point)
= 1/100 = 10-2 (two places to the right of the decimal point)
0.001 = 1/1000 = 10-3 (three places to the right of the decimal point)
hence in the example above, 3 x 10-1 = 0.3
What happens if you divide with a negative exponent?
1x 104 / 1 x 10-2 = 1 x 104-(-2) = 1 x 104+2 = 1 x 106
if you expand this out to see what is happening
10,000
1/100
invert the 1/100 and multiply = 10,000 x 100/1 = 1 x 106
(count the zeros)
alternatively multiply the top and bottom of the equation by 100
10,000 x 100
1/100 x 100 = 1,000,000 = 106
APPLICATIONS OF EXPONENTS
Working with very large numbers e.g. bacteria in a culture or blood cells etc.
For example, the numbers of red cells in blood is in the range 3.8-6.5 x 1012 per litre and it is easier to determine if the level is in this normal range if the numbers are expressed as powers of 10 and not fully expanded e.g.
5 x 1012 = 5,000,000,000,000 (normal) does not look significantly different to
2 x 1012 = 2,000,000,000,000 (low)
It is easier to compare 5 and 2 than the whole numbers.
Some bacteria can grow very quickly, doubling their numbers every 20 minutes. This means that a culture containing only a few bacteria can easily reach numbers of 108 or 109 bacteria per ml of culture when incubated overnight. It is hard to manipulate these sorts of numbers in ways other than by using powers of 10.
MANIPULATIONS
Say you have a culture of bacteria with a concentration of 5 x 108 bacteria per ml and a total of 10 mls.
Total number of bacteria in the culture = 5 x 108 x 10 = 5 x 109
If you remove 1 ml you have 5 x 109 – 5 x 108 remaining = 4.5 x 109 bacteria in a total of 9 mls.
If you dilute 1 ml of the culture by 1/100 you divide the concentration by 100
5 x 108/100 = 5 x 108-2 = 5 x 106 per ml
Working with very small numbers:
As well as being useful for manipulating very large numbers, powers are very effective for working with very small numbers i.e. numbers less than 1.
This is particularly evident when using SI units.
| Unit | Multiple of Base Unit | Abbreviation |
| Tera | 1012 | T |
| Giga | 109 | G |
| Mega | 106 | M |
| Kilo | 103 | K |
| Hecto | 102 | H |
| Basic Unit |
1 gram (mass) 1 litre (volume) 1 metre (length)
|
g
l m |
| deci | 10-1 | d |
| centi | 10-2 | c |
| milli | 10-3 | m |
| micro | 10-6 | μ |
| nano | 10-9 | n |
| pico | 10-12 | p |
| femto | 10-15 | f |
| atto | 10-18 | a |
Examples 1 millimetre = 1 mm = 1/1000 of a metre
1 microlitre = 1μl = 1/1,000,000 of a litre
1 picogram = 1 pg = 1/1,000,000,000,000 of a gram
If you think that the term atto is too small to ever be used, consider Avogadro’s number, the number of atoms/molecules etc in a mole of any substance. Avogadro’s number is 6.02 x 1023.
In an attomole (aM) of sodium chloride for instance there would be
6.02 x 1023/1018 molecules of NaCl = 6.02 x 105 molecules
Picogram, picomoles, femtomoles etc are routinely used quantities in molecular biology.
UNIT CONVERSIONS
Note that the most commonly used units are multiples of 10-3 of the base or primary unit.
Eg 1 milligram = 10-3 grams
1 microgram = 10-6 grams
1 microgram = 10-3 milligrams
Also remember which way you are converting i.e. there are more milligrams in a gram (1000) than grams in a milligram (0.001). This means that if you are converting milligrams to grams you will get a smaller answer than when converting grams to milligrams.
The most commonly used units in biological sciences are the primary units
Grams
Litres
Metres
Moles
And the 10-3 multiples of these i.e.
Milli (10-3 or 1/1000)
Micro (10-6 or 1/1,000,000)
Nano (10-9 or 1/1,000,000,000)
So, to convert
nano to micro you divide by 1000
micro to milli you divide by 1000
milli to primary unit you divide by 1000
primary unit to milli you multiply by 1000
milli to micro you multiply by 1000
micro to nano you multiply by 1000
If you want to convert by more than one step, say nano to milli, you would divide by 1,000,000 (106) or go through the individual steps from nano to micro and from micro to milli. The general rule is when converting from small to larger, divide by the factor (because the answer is going to be smaller) and when going from large to smaller, multiply by the factor (because the answer is bigger).
LOGARITHMS
An extension of the use of exponentials is logarithms. Logarithms are simply a shorthand way of expressing exponential functions and are useful in biological sciences in a number of ways, most importantly in the calculation of pH.
DEFINITIONS
Logarithms are related to exponentials in that the log of a number is equal to the exponent:
102 = 100
log10100 = 2
The exponent of 100 (102) is two and the log of 100 is two. The log is the number of times the base (10 – represented by the subscript 10 next to the word log) has to be multiplied by itself to give 100.
Note that:
log10 1 = 0 (100=1)
log10 10 = 1 (101=10)
Logs are expressed as a whole number followed by 4 or 5 decimal points e.g.
Log10100 = 2.00000
The whole number is called the characteristic and the decimals are the mantissa.
MANIPULATING LOGS
MULTIPLICATION
Log (a x b) = log a + log b
E.g. log (100 x 1000) = log 100 + log 1000
= log 102 + log 103
= 2 + 3
= 5
DIVISION
Log (a/b) = log a – log b
Eg log (10,000/1000) = log 10,000 – log 1000
= log 104 – log 103
= 4-3
= 1
LOGS OF NUMBERS BETWEEN 1 AND 10
Logs of numbers between 1 and 10 are always a number greater than 0 but less than 1. You cannot calculate these yourselves but can find them in tables in log books or from your calculator.
E.g. log 5.7 = 0.75587
Note that the characteristic for these numbers is always 0.
LOGS OF NUMBERS GREATER THAN 10
Express the number as a function of 10 and use the multiplication rule e.g.
= 5.7 x 103
log 5700 = log (5.7 x 103) = log 5.7 + log 103 = 0.75587 + 3 = 3.75587
LOGS OF NUMBERS LESS THAN 1
Express the number as a fraction or a decimal and use the division rule e.g.
= 7.5 x 10-1
log 0.75 = log (7.5 x 10-1) = log 7.5 – 1 = 0.87506 – 1
= - 0.12494
OR
0.75 = ¾
Log (3/4) = log 3 – log 4 = 0.47712-0.60206 = - 0.12494
When expressing negative logs the minus sign is placed over the characteristic and the number is called bar...
_
Log 0.01 = log 10-2 = -2 ( 2 or bar 2)
Having performed a calculation in logs, the antilog can be looked up in tables or on your calculator, to give the numerical value of the solution to the problem. The mantissa is looked up to give the numbers and the decimal point assigned according to the characteristic. If the log is negative, as in the example above, the mantissa must first be subtracted from one and the result looked up. Most of the applications of logs in biological science do not require you to look up antilogs.