Scientific notation is method abbreviating numbers that are too large or too small.
Sometimes numbers can be too big or too small to handle. For instance one of the fundamental numbers in chemistry is: 602,300,000,000,000,000,000,000. This number is the atoms in a mole of any substance. Chemists use it all the time, but it would not be possible to do easily if it wasn't for Scientific Notation. What is Scientific Notation, and how it works is what this article is about. Starting first with the example given of the number of atoms in a mole of a substance; in Scientific Notation it would be written as:
6.023 X 10 (stated as 6.023 times 10 raised to the 23rd power). The exponent 23 is the order of magnitude or power of ten. It is also the number of places that the decimal is moved to the left in order to achieve the 6.023. Scientific Notation states that any number can be written as a number between 1.0 and 10.0 multiplied by 10 raised to some power. The number between 1.0 and ten is called the base number. The power of ten are exponents.
To understand this let's first start with a one
digit number 3. This could be written as 3 x 10
or 3 x 1. 30 would be written as 3.0 x 10 ;
300 would be 3.00 x 10 ; 3000 would be
3.000 x 10 . Notice that the power or exponent is equal to the number of places the decimal must be moved to the left to get the number to one digit to the left of the decimal point.
The reason for all this manipulation is to make it easier to work with extremely large or small numbers. When a number is changed to Scientific Notation form it becomes much easier to multiply, divide, or find roots. The rules are relatively simple for multiplication you multiply the base
numbers and and the exponents. For example if I were to multiply 50,000 by 20,000 first change the numbers to Scientific Notation:
(5 x 10 ) x (2 x 10 )
Multiplying the base numbers and adding the
powers of ten you get (10 x 10 ) moving the decimal that is to the right eight places (one for each power of ten) you get 1,000,000,000!
Division works much the same way, you would write the numbers in Scientific Notation divide the base numbers and subtract the exponents.
For example using the same numbers and get:
(5 x 10 )divided by (2 x 10 ) using the rules and dividing base numbers and subtracting
exponents the result is (2.5 x 10 ) or 2.5 (the 0 exponent means the decimal doesn't move). The rules for multiplication and division are simply to treat the base numbers as you always would and add the exponents if multiplying and subtract the exponents if dividing. The same is true for raising numbers to powers (raise base number to the power and multiply the exponents) finding roots, (by finding the root of a one or two digit base number and dividing by the exponent).
The square root of 160,000 is found by moving the decimal so you get:
16 x 10 finding the square root of 16 and dividing the power by 2 the result is:
4 x 10 or 400.
These rules are simple but what about extremely small numbers. Well the difference is that if you have to move the decimal to the right to get the proper Scientific Notation form. For instance O.00025 is:
2.5 x 10 the rules are the same, treat the base number regularly, and treat the exponents as the same.
Using this method you can work with extremely large or small numbers with a sure, straight forward method to simplify and work problems out.