Addition, Subtraction, Multiplication and Division of Scientific Notation

Very large and small numbers are difficult to deal with in calculation.  Let say for example you want to multiply a small number like 0.000 000 000 000 000 233 with a large number like 6,000,000,000,000,000,000.  Do you think you can multiply such numbers easily without converting those numbers in scientific notation?  For me, I don't think so.

To do such calculations easily we can change the number first to scientific notation and follow the rules in multiplying scientific notations.  Let' us have the rules in doing arithmetic calculations of scientific notation.

Adding and subtracting scientific notation

In adding and subtracting scientific notation, we then change the exponent to a common exponent.
Meaning the exponent of the two given  must have the same exponent.  Then, once the exponent is already the same you can also do the required operation.

Let us have an example:

1.  Add 24,000,000,000,000,000 to 5,000,000,000,000,000

We first change the numbers to scientific notation.

       2.4 x 10¹⁶
 +    5.0 x 10¹⁵

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Since we cannot add immediately the two values because of their different exponents, we can change either of them  and still you will get the same result.  Let us try following the higher exponent 16 meaning we need to change the second scientific notation 15 to 16.  So, how do we do it?
We just move the decimal point once to the left the exponent 15 will become 16. Moving the decimal point to the left will increase the positive exponent and will cause the decrease of the negative exponent.

  2.4 x 10¹⁶
  0.5 x 10¹⁶

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 2.9 x 10¹⁶

Just add the value of N and just copy x 10 and the common exponent.

How about if we will use the lower exponent, 15 as the common exponent?  To decrease the exponent from 16 to 15 we need to move the decimal point to the right once.  Just see the example below:

24.0 x 10¹⁵

+ 5.0 x 10¹⁵

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 29.0 x 10¹⁵    or  2.9 x 10¹⁶

Since the value of N does not follow the standard form which should be not less than 1 and not more than 10 we need to move the decimal point again to follow the standard form of scientific notation.  We move the decimal point to the left and we increase the exponent to 16.  As you can see we still have the same answer.

Let us have an example of subtraction with a negative exponent.

2.    Subtract 0.000 000 000 000 000 345 from 0.000 000 000 000 002 4

Change the numbers first to scientific notation

  2.40 x 10^-15

- 3.45 x 10^-16

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Before we calculate the problem above we need to change the exponent to a common exponent.  Let us use the higher exponent so that we do not move again the decimal point in the final answer.

   2.400  x 10^-15

-  0.345  x 10^-15

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   2.055  x 10^-15

We did not move the decimal point since the answer is already expressed in correct form of scientific notation.

Multiplication and Division of Scientific Notation

In multiplying scientific notation, the value of N is multiplied and the exponent is being added.  You have to follow addition of signed number since the exponent, n, is either positive or negative integer.

Example:

                     (8.0 x 10⁴)(5.0 x 10²) = (8.0 x 5.0)(10⁴⁺²)
                                                         = 40 x 10⁶
                                                         = 4.0 x 10⁷

   
                  (4.0 x 10^-5)( 7.0 x 10³) = (4.0 x 7.0)(10^-5+3)
                                                        = 28 x 10^-2
                                                        = 2.8 x 10^-1


In dividing scientific notation the value of N is divided but the exponents are subtracted.  Also follow the rules in subtracting signed numbers.

Example:

                (6.9 x 10⁷) ÷  ( 3.0 x 10^-5)  =  (6.9 ÷ 3.0)(10^7-(-5)
                                                            =  2.3 x 10¹²


                (8.5 x 10⁴) ÷  (5.0 x 10⁹)  = (8.5 ÷ 5.0)(10^4-9)
                                                          = 1.7 x 10^-5