Scientific Notation

Scientific notation, also known as standard form or as exponential notation, is a way of writing numbers that accommodates values too large or small to be conveniently written in standard decimal notation. Scientific notation has a number of useful properties and is often favored by scientists, mathematicians and engineers, who work with such numbers.

In scientific notation all numbers are written like this:

a × 10^b

("a times ten to the power of b"), where the exponent b is an integer, and the coefficient a is any real number (but see normalized notation below), called the significand or mantissa (though the term "mantissa" may cause confusion as it can also refer to the fractional part of the common logarithm). If the number is negative then a minus sign precedes a (as in ordinary decimal notation).

Ordinary decimal notation
300
4,000
5,720,000,000
0.0000000061

Scientific notation (normalized)
3×10^2
4×10^3
5.72×10^9
6.1×10^−9

Scientific notation is the way that scientists easily handle very large numbers or very small numbers. For example, instead of writing 0.0000000056, we write 5.6 x 10^-9. So, how does this work?

Here are some examples of scientific notation.
10000 = 1 x 10^4
1000 = 1 x 10^3
100 = 1 x 10^2
10 = 1 x 10^1
1 = 10^0
1/10 = 0.1 = 1 x 10^-1
1/100 = 0.01 = 1 x 10^-2
1/1000 = 0.001 = 1 x 10^-3
1/10000 = 0.0001 = 1 x 10^-4

24327 = 2.4327 x 10^4
7354 = 7.354 x 10^3
482 = 4.82 x 10^2
89 = 8.9 x 10^1 (not usually done)
0.32 = 3.2 x 10^-1 (not usually done)
0.053 = 5.3 x 10^-2
0.0078 = 7.8 x 10^-3
0.00044 = 4.4 x 10^-4

As you can see, the exponent of 10 is the number of places the decimal point must be shifted to give the number in long form. A positive exponent shows that the decimal point is shifted that number of places to the right. A negative exponent shows that the decimal point is shifted that number of places to the left.

In scientific notation, the digit term indicates the number of significant figures in the number. The exponential term only places the decimal point. As an example,

46600000 = 4.66 x 10^7
This number only has 3 significant figures. The zeros are not significant; they are only holding a place. As another example,
0.00053 = 5.3 x 10^-4
This number has 2 significant figures. The zeros are only place holders.

Scientific notation is simply a method for expressing, and working with, very large or very small numbers. It is a short hand method for writing numbers, and an easy method for calculations. Numbers in scientific notation are made up of three parts: the coefficient, the base and the exponent. Observe the example below:

5.67 x 10^5

This is the scientific notation for the standard number, 567 000. Now look at the number again, with the three parts labeled.
5.67 x 10^5
coefficient,base,exponent

In order for a number to be in correct scientific notation, the following conditions must be true:

1. The coefficient must be greater than or equal to 1 and less than 10.
2. The base must be 10.
3. The exponent must show the number of decimal places that the decimal needs to be moved to change the number to standard notation. A negative exponent means that the decimal is moved to the left when changing to standard notation.

Changing numbers from scientific notation to standard notation.

Ex.1 Change 6.03 x 107 to standard notation. Remember, 10^7 = 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10 000 000 so,6.03 x 10^7 = 6.03 x 10 000 000 = 60 300 000
Answer = 60 300 000

Instead of finding the value of the base, we can simply move the decimal seven places to the right because the exponent is 7. So, 6.03 x 10^7 = 60 300 000

Now let us try one with a negative exponent.

Ex.2 Change 5.3 x 10^-4 to standard notation. The exponent tells us to move the decimal four places to the left. So, 5.3 x 10^-4 = 0.00053

Changing numbers from standard notation to scientific notation

Ex.1 Change 56 760 000 000 to scientific notation. Remember, the decimal is at the end of the final zero.The decimal must be moved behind the five to ensure that the coefficient is less than 10, but greater than or equal to one. The coefficient will then read 5.676. The decimal will move 10 places to the left, making the exponent equal to 10. Answer equals 5.676 x 10^10

Now we try a number that is very small.

Ex.2 Change 0.000000902 to scientific notation. The decimal must be moved behind the 9 to ensure a proper coefficient. The coefficient will be 9.02.The decimal moves seven spaces to the right, making the exponent -7. Answer equals 9.02 x 10^-7

Rule for Multiplication - When you multiply numbers with scientific notation, multiply the coefficients together and add the exponents. The base will remain 10.

Ex 1. Multiply (3.45 x 10^7) x (6.25 x 10^5).

First rewrite the problem as: (3.45 x 6.25) x (10^7 x 10^5).

Then multiply the coefficients and add the exponents:21.5625 x 10^12.

Then change to correct scientific notation and round to correct significant digits: 2.16 x 10^13.

NOTE - we add one to the exponent because we moved the decimal one place to the left.


Rule for Division - When dividing with scientific notation, divide the coefficients and subtract the exponents. The base will remain 10.

Ex. 1. Divide 3.5 x 10^8 by 6.6 x 10^4

Rewrite the problem as:
3.5 x 10^8
---------
6.6 x 104

Divide the coefficients and subtract the exponents to get:0.530303 x 10^4

Change to correct scientific notation and round to correct significant digits to get: 5.3 x 10^3

Note - We subtract one from the exponent because we moved the decimal one place to the right.

Rule for Addition and Subtraction - when adding or subtracting in scientific notation, you must express the numbers as the same power of 10. This will often involve changing the decimal place of the coefficient.

Ex. 1. Add 3.76 x 10^4 and 5.5 x 10^2

Move the decimal to change 5.5 x 10^2 to 0.055 x 10^4

Add the coefficients and leave the base and exponent the same: 3.76 + 0.055 = 3.815 x 10^4

Following the rules for rounding, our final answer is 3.815 x 10^4

Rounding is a little bit different because each digit shown in the original problem must be considered significant, regardless of where it ends up in the answer.

Ex. 2. Subtract (4.8 x 10^5) - (9.7 x 10^4)

Move the decimal to change 9.7 x 10^4 to 0.97 x 10^5

Subtract the coefficients and leave the base and exponent the same: 4.8 - 0.97 = 3.83 x 10^5

Round to correct number of significant digits: 3.83 x 10^5

Reference:
http://en.wikipedia.org/wiki/Scientific_notation
http://www.chem.tamu.edu/class/fyp/mathrev/mr-scnot.html
http://www.fordhamprep.org/gcurran/sho/sho/lessons/lesson25.htm