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0.2 Mathematical basics  (Page 3/3)

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Number Exponential Form Common Logarithm
1000 10 3 3
10 10 1 1
1 10 0 0
0.1 10 −1 −1
0.001 10 −3 −3

To find the common logarithm of most numbers, you will need to use the LOG button on a calculator.

Rounding and significant digits

In reporting numerical data obtained via measurements, we use only as many significant figures as the accuracy of the measurement warrants. For example, suppose a microbiologist using an automated cell counter determines that there are 525,341 bacterial cells in a one-liter sample of river water. However, she records the concentration as 525,000 cells per liter and uses this rounded number to estimate the number of cells that would likely be found in 10 liters of river water. In this instance, the last three digits of the measured quantity are not considered significant. They are rounded to account for variations in the number of cells that would likely occur if more samples were measured.

The importance of significant figures lies in their application to fundamental computation. In addition and subtraction, the sum or difference should contain as many digits to the right of the decimal as that in the least certain (indicated by underscoring in the following example) of the numbers used in the computation.

Suppose a microbiologist wishes to calculate the total mass of two samples of agar.

4.38 3 _ g 3.002 1 _ ______ g 7.38 5 _ g

The least certain of the two masses has three decimal places, so the sum must have three decimal places.

In multiplication and division, the product or quotient should contain no more digits than than in the factor containing the least number of significant figures. Suppose the microbiologist would like to calculate how much of a reagent would be present in 6.6 mL if the concentration is 0.638 g/mL.

0.63 8 _ g mL × 6. 6 _ mL = 4.1 g

Again, the answer has only one decimal place because this is the accuracy of the least accurate number in the calculation.

When rounding numbers, increase the retained digit by 1 if it is followed by a number larger than 5 (“round up”). Do not change the retained digit if the digits that follow are less than 5 (“round down”). If the retained digit is followed by 5, round up if the retained digit is odd, or round down if it is even (after rounding, the retained digit will thus always be even).

Generation time

It is possible to write an equation to calculate the cell numbers at any time if the number of starting cells and doubling time are known, as long as the cells are dividing at a constant rate. We define N 0 as the starting number of bacteria, the number at time t = 0. N i is the number of bacteria at time t = i , an arbitrary time in the future. Finally we will set j equal to the number of generations, or the number of times the cell population doubles during the time interval. Then we have,

N i = N 0 × 2 j

This equation is an expression of growth by binary fission.

In our example, N 0 = 4, the number of generations, j , is equal to 3 after 90 minutes because the generation time is 30 minutes. The number of cells can be estimated from the following equation:

N i = N 0 × 2 j N 90 = 4 × 2 3 N 90 = 4 × 8 = 32

The number of cells after 90 minutes is 32.

Most probable number

The table in [link] contains values used to calculate the most probable number example given in How Microbes Grow .

A table is titled Most Probable Number Table. For each row, it states the number of tubes giving a positive reaction for a 5-tube set for 10 mL, 1 mL and 0.1 mL tubes, followed by the MPN per 100 mL, and the 95% confidence limits for low and high. For row 1, the reactions are 10 mL = 0, 1 mL = 0, 0.1 mL = 0; the MPN is <2, and the low and high confidence limits are <1 and 7. For row 2, the reactions are 10 mL = 0, 1 mL = 1, 0.1 mL = 0; the MPN is 2, and the low and high confidence limits are <1 and 7. For row 3, the reactions are 10 mL = 0, 1 mL = 2, 0.1 mL = 0; the MPN is 4, and the low and high confidence limits are <1 and 11. For row 4, the reactions are 10 mL = 1, 1 mL = 0, 0.1 mL = 0; the MPN is 2, and the low and high confidence limits are <1 and 7. For row 5, the reactions are 10 mL = 1, 1 mL = 0, 0.1 mL = 1; the MPN is 4, and the low and high confidence limits are <1 and 11. For row 6, the reactions are 10 mL = 1, 1 mL = 1, 0.1 mL = 0; the MPN is 4, and the low and high confidence limits are <1 and 11. For row 7, the reactions are 10 mL = 1, 1 mL = 1, 0.1 mL = 1; the MPN is 6, and the low and high confidence limits are <1 and 15. For row 8, the reactions are 10 mL = 2, 1 mL = 0, 0.1 mL = 0; the MPN is 5, and the low and high confidence limits are <1 and 13. For row 9, the reactions are 10 mL = 2, 1 mL = 0, 0.1 mL = 1; the MPN is 7, and the low and high confidence limits are 1 and 17. For row 10, the reactions are 10 mL = 2, 1 mL = 1, 0.1 mL = 0; the MPN is 7, and the low and high confidence limits are 1 and 17. For row 11, the reactions are 10 mL = 2, 1 mL = 1, 0.1 mL = 1; the MPN is 9, and the low and high confidence limits are 2 and 21. For row 12, the reactions are 10 mL = 2, 1 mL = 2, 0.1 mL = 0; the MPN is 9, and the low and high confidence limits are 2 and 21. For row 13, the reactions are 10 mL = 2, 1 mL = 3, 0.1 mL = 0; the MPN is 12, and the low and high confidence limits are 3 and 28. For row 14, the reactions are 10 mL = 3, 1 mL = 0, 0.1 mL = 0; the MPN is 8, and the low and high confidence limits are 1 and 19. For row 15, the reactions are 10 mL = 3, 1 mL = 0, 0.1 mL = 1; the MPN is 11, and the low and high confidence limits are 2 and 25. For row 16, the reactions are 10 mL = 3, 1 mL = 1, 0.1 mL = 0; the MPN is 11, and the low and high confidence limits are 2 and 25. For row 17, the reactions are 10 mL = 3, 1 mL = 1, 0.1 mL = 1; the MPN is 14, and the low and high confidence limits are 4 and 34. For row 18, the reactions are 10 mL = 3, 1 mL = 2, 0.1 mL = 0; the MPN is 14, and the low and high confidence limits are 4 and 34. For row 19, the reactions are 10 mL = 3, 1 mL = 2, 0.1 mL = 1; the MPN is 17, and the low and high confidence limits are 5 and 46. For row 20, the reactions are 10 mL = 3, 1 mL = 3, 0.1 mL = 0; the MPN is 17, and the low and high confidence limits are 5 and 46. For row 21, the reactions are 10 mL = 4, 1 mL = 0, 0.1 mL = 0; the MPN is 13, and the low and high confidence limits are 3 and 31. For row 22, the reactions are 10 mL = 4, 1 mL = 0, 0.1 mL = 1; the MPN is 17, and the low and high confidence limits are 5 and 46. For row 23, the reactions are 10 mL = 4, 1 mL = 1, 0.1 mL = 0; the MPN is 17, and the low and high confidence limits are 5 and 46. For row 24, the reactions are 10 mL = 4, 1 mL = 1, 0.1 mL = 1; the MPN is 21, and the low and high confidence limits are 7 and 63. For row 25, the reactions are 10 mL = 4, 1 mL = 1, 0.1 mL = 2; the MPN is 26, and the low and high confidence limits are 9 and 78. For row 26, the reactions are 10 mL = 4, 1 mL = 2, 0.1 mL = 0; the MPN is 22, and the low and high confidence limits are 7 and 67. For row 27, the reactions are 10 mL = 4, 1 mL = 2, 0.1 mL = 1; the MPN is 26, and the low and high confidence limits are 9 and 80. For row 28, the reactions are 10 mL = 4, 1 mL = 3, 0.1 mL = 0; the MPN is 27, and the low and high confidence limits are 9 and 80. For row 29, the reactions are 10 mL = 4, 1 mL = 3, 0.1 mL = 1; the MPN is 33, and the low and high confidence limits are 11 and 93. For row 30, the reactions are 10 mL = 4, 1 mL = 4, 0.1 mL = 0; the MPN is 34, and the low and high confidence limits are 12 and 93. For row 31, the reactions are 10 mL = 5, 1 mL = 0, 0.1 mL = 0; the MPN is 23, and the low and high confidence limits are 7 and 70. For row 32, the reactions are 10 mL = 5, 1 mL = 0, 0.1 mL = 1; the MPN is 31, and the low and high confidence limits are 11 and 89. For row 33, the reactions are 10 mL = 5, 1 mL = 0, 0.1 mL = 2; the MPN is 43, and the low and high confidence limits are 15 and 110. For row 34, the reactions are 10 mL = 5, 1 mL = 1, 0.1 mL = 0; the MPN is 33, and the low and high confidence limits are 11 and 93. For row 35, the reactions are 10 mL = 5, 1 mL = 1, 0.1 mL = 1; the MPN is 46, and the low and high confidence limits are 16 and 120. For row 36, the reactions are 10 mL = 5, 1 mL = 1, 0.1 mL = 2; the MPN is 63, and the low and high confidence limits are 21 and 150. For row 37, the reactions are 10 mL = 5, 1 mL = 2, 0.1 mL = 0; the MPN is 49, and the low and high confidence limits are 17 and 130. For row 38, the reactions are 10 mL = 5, 1 mL = 2, 0.1 mL = 1; the MPN is 70, and the low and high confidence limits are 23 and 170. For row 39, the reactions are 10 mL = 5, 1 mL = 2, 0.1 mL = 2; the MPN is 94, and the low and high confidence limits are 28 and 220. For row 40, the reactions are 10 mL = 5, 1 mL = 3, 0.1 mL = 0; the MPN is 79, and the low and high confidence limits are 25 and 190. For row 41, the reactions are 10 mL = 5, 1 mL = 3, 0.1 mL = 1; the MPN is 110, and the low and high confidence limits are 31 and 250. For row 42, the reactions are 10 mL = 5, 1 mL = 3, 0.1 mL = 2; the MPN is 140, and the low and high confidence limits are 37 and 340. For row 43, the reactions are 10 mL = 5, 1 mL = 3, 0.1 mL = 3; the MPN is 180, and the low and high confidence limits are 44 and 500.

Questions & Answers

how does Neisseria cause meningitis
Nyibol Reply
what is microbiologist
Muhammad Reply
what is errata
Muhammad
is the branch of biology that deals with the study of microorganisms.
Ntefuni Reply
What is microbiology
Mercy Reply
studies of microbes
Louisiaste
when we takee the specimen which lumbar,spin,
Ziyad Reply
How bacteria create energy to survive?
Muhamad Reply
Bacteria doesn't produce energy they are dependent upon their substrate in case of lack of nutrients they are able to make spores which helps them to sustain in harsh environments
_Adnan
But not all bacteria make spores, l mean Eukaryotic cells have Mitochondria which acts as powerhouse for them, since bacteria don't have it, what is the substitution for it?
Muhamad
they make spores
Louisiaste
what is sporadic nd endemic, epidemic
Aminu Reply
the significance of food webs for disease transmission
Abreham
food webs brings about an infection as an individual depends on number of diseased foods or carriers dully.
Mark
explain assimilatory nitrate reduction
Esinniobiwa Reply
Assimilatory nitrate reduction is a process that occurs in some microorganisms, such as bacteria and archaea, in which nitrate (NO3-) is reduced to nitrite (NO2-), and then further reduced to ammonia (NH3).
Elkana
This process is called assimilatory nitrate reduction because the nitrogen that is produced is incorporated in the cells of microorganisms where it can be used in the synthesis of amino acids and other nitrogen products
Elkana
Examples of thermophilic organisms
Shu Reply
Give Examples of thermophilic organisms
Shu
advantages of normal Flora to the host
Micheal Reply
Prevent foreign microbes to the host
Abubakar
they provide healthier benefits to their hosts
ayesha
They are friends to host only when Host immune system is strong and become enemies when the host immune system is weakened . very bad relationship!
Mark
what is cell
faisal Reply
cell is the smallest unit of life
Fauziya
cell is the smallest unit of life
Akanni
ok
Innocent
cell is the structural and functional unit of life
Hasan
is the fundamental units of Life
Musa
what are emergency diseases
Micheal Reply
There are nothing like emergency disease but there are some common medical emergency which can occur simultaneously like Bleeding,heart attack,Breathing difficulties,severe pain heart stock.Hope you will get my point .Have a nice day ❣️
_Adnan
define infection ,prevention and control
Innocent
I think infection prevention and control is the avoidance of all things we do that gives out break of infections and promotion of health practices that promote life
Lubega
Heyy Lubega hussein where are u from?
_Adnan
en français
Adama
which site have a normal flora
ESTHER Reply
Many sites of the body have it Skin Nasal cavity Oral cavity Gastro intestinal tract
Safaa
skin
Asiina
skin,Oral,Nasal,GIt
Sadik
How can Commensal can Bacteria change into pathogen?
Sadik
How can Commensal Bacteria change into pathogen?
Sadik
all
Tesfaye
by fussion
Asiina
what are the advantages of normal Flora to the host
Micheal
what are the ways of control and prevention of nosocomial infection in the hospital
Micheal
what is inflammation
Shelly Reply
part of a tissue or an organ being wounded or bruised.
Wilfred
what term is used to name and classify microorganisms?
Micheal Reply
Binomial nomenclature
adeolu
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Source:  OpenStax, Microbiology. OpenStax CNX. Nov 01, 2016 Download for free at http://cnx.org/content/col12087/1.4
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