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The work of scientists often involves the use of very large and very small numbers. Microbiologists, for example, work with very large quantities of very tiny organisms. Expressed numerically, microbiologists often work with billions of cells (>1,000,000,000) that can be as small as one millionth of a meter (0.000001) in diameter.
In order to facilitate effective communication and simplify the experimental process, the scientific community has established a universal notation to express numbers of all sizes.
The following is the formula for scientific notation:
A x 10B
Where A is any number reduced to its significant digits, and B is any integer.
Note: This notation is also sometimes expressed as AE(±)B, as often is the case with calculators (e.g., 1E+9).
To exemplify the use of scientific notation, take the number and size of the cells described above. Instead of 1,000,000,000 cells, microbiologists write 1 x 109 cells. In this instance, the number 9 is a positive integer, which means that in order to translate to normal numbers, nine zeros are placed afterthe 1. Conversely, if we want to describe the size of a cell that had a diameter of 0.000001 meters, then the exponent would be negative and the scientific notation would be 1 x 10-6 meters. This time, the zeros are placed in front of the 1.
One benefit of scientific notation is it enables researchers to reduce the number of zeros occupying space on computer screens and paper. In our example above, we were able to reduce the amount of zeros in a billion from 9 to 1, and in one millionth from 5 to 1.
Here are some additional real life examples of scientific notation being used to represent numbers:
- The number 100 becomes 1 x 102
- The number 1,000,000 becomes 1 x 106
- The population of Earth, ~7,100,000,000, becomes 7.1 x 109.
- The number of Stars in the Milky Way, ~200,000,000,000 becomes 2 x 1011.
- The weight of an electron, 0.00000000000000000000000000000091093822 kg, can be expressed as 9.1093822 x 10-31 kg.
- The hours in a life span of 75 years, 657,000, simplifies to 6.57 x 105 hours.
- The number of DNA base pairs in a human, ~3,000,000,000 per cell multiplied by ~10,000,000,000,000 cells, can be expressed more simply as (3 x109) x (1 x 1013) = 3 x 1022total base pairs.
It is difficult to have a discussion of scientific notation without mentioning signifcant digits (also called significant figures). In all of these examples, large and small numbers are reduced to a manageable form to make it easier for us to understand, but some of the multipliers have more digits than others. In the cell example, for instance, we have a muliplier 1 for both cell number and size. This reflects the relatively low level of confidence we have in our measurements of cell size and number.
The more numbers in "A" of the scientific notation formula, the more precise the number is, and the more confidence we have in the data generated to determine that number. Notice that we have more confidence in how many people there are on Earth (2 significant digits) than how many stars are in our galaxy (1 significant digit). This reflects the fact that right now it is easier to measure human population than the number of stars in our galaxy. Following this logic, we can deduce that the equipment/devices used to measure the weight of an electron are highly precise since the accepted value contains 8 significant figures.
An example of a number with many significant digits is the speed of light, 299,792,458 meters/second. The speed of light has been defined with 9 significant digits, which means experimenters are very confident in the measurements that led to this number. Notice that scientific notation cannot be used to reduce this value since there are no zeros to consolidate.
Microbiologists often use scientific notation to gauge how many cells in a solution can form visible colonies when grown on agar plates. These colonies are referred to as colony forming units, or CFU. The precision of counting CFU is inherently limited for technical reasons. A confidence of 2 significant digits is usually achievable with current plating and counting techniques (e.g., 1.2 x 108).
In summary, scientists often use scientific notation to make very large and very small numbers easier to express and understand. Microbiologists are no exception. For microbiologists, scientific notation is valuable to express great numbers of cells determined from dilution and plating to agar, as well as to express the small size of the organisms we study.