Scientific Notation

Scientific Notation

 

A scientific notation is also called standard form or otherwise known as exponential notation.  In the world of numbers, this is the manner of writing numbers that contain values ranging from largest values up to the smallest values, which are written in standard decimal notation.  This is comprised of useful number properties that are also used in calculators, as well as by the scientists, mathematicians, statisticians, doctors and engineers.   In scientific notation all numbers are written like this:  this is read as “times ten and raised to the power of b” or to the power of 10.  The   exponent b is an integer, while coefficient a is any real number.  In addition, if the number is negative, a minus sign must be placed before the number.   Below is a table that contains some examples of ordinary number/decimal notation converted into scientific notation in a normalized manner.

Ordinary decimal notation

Scientific notation (normalized)

300

3×102

4,000

4×103

5,720,000,000

5.72×109

0.0000000061

6.1×10−9

 

 

As shown in the table above, any given number can be written in the form of a×10b in various ways.  For instance, you can write 350 as 3.5×102 or 35×101 or 350×100. In normalized scientific notation, the exponent b is chosen such that the absolute value of a remains at least one but less than ten (1 ≤ |a| < 10). For example, 350 is written as 3.5×102. This form allows easy comparison of two numbers of the same sign in a, as the exponent b gives the number’s order of magnitude. In normalized notation the exponent b is negative for a number with absolute value between 0 and 1 (e.g., minus one half is −5×10−1). The 10 and exponent are usually omitted when the exponent is 0. Note that 0 itself cannot be written in normalized scientific notation since the mantissa would have to be zero and the exponent undefined.

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Moreover, the usage of scientific notation is to convert ordinary decimal notation into scientific notation.  And, in order to do this correctly, move the decimal separator to the desired number of places or digits to the left or right direction.  By this way, the significant will be put in the desired range; let’s say between 1 and 10 for the normalized form.  On the other hand, if you are going to move the decimal point  n places to the left, then , it means that you multiply it by 10n; .  And, if you will move the decimal point n places to the right, it means that you multiply it by 10−n.  A good example for this is 1230000, if you move the decimal point six places going to the left it will give you 1.23.  But, if you multiply it by 106, it will give you an answer of 1.23×106. In the same manner, with 0.000000456, if you move the decimal point seven places to the right, you will get 4.56, then, if you multiply it by 10−7, you will get a result of 4.56×10−7.  In this example, 1.234×100 is just written as 1.234. You will notice that the decimal separator does not move, therefore, the exponent multiplier is logically 100, and equivalent to 1. In order to convert scientific notation into ordinary decimal notation, only have to do is to take the significant and move the decimal separator by the number of places which is indicated by the exponent.   Remember, left, if the exponent is negative, or right if the exponent is positive. You can add zeroes as needed.  For example: given 9.5 × 1010, move the decimal point ten places to the right to yield 95000000000.  If you noticed, the conversion between different scientific notation representations of the same number is being gained through doing opposite operations of multiplication or division by a power of ten on the significant, as well as the exponents.  The decimal separator is shifted n places to the left (or right), corresponding to division (multiplication) by 10n, and n is added to (subtracted from) the exponent, corresponding to a canceling multiplication (division) by 10n. Just take a look at this example:

1.234×103 = 12.34×102 = 123.4×101 = 1234

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Subsequently,   it is normal in scientific notation to record all the significant digits from all measurements, at the same time to guess one additional digit if there is any hint available to the observer to guess the missing digit.  The resulting number is valuable because it contains some information which leads to greater precision in measurements and in aggregations of adding them or multiplying them together.

(http://en.wikipedia.org/wiki/Scientific_notation)

 

References:

 (http://whatis.techtarget.com/definition/0,,sid9_gci526200,00.html)

(http://en.wikipedia.org/wiki/Scientific_notation)

 

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