Scientific Notation
Copyright, 1993 Dr. F.W. Boyle, Jr.

Converting numbers into scientific notation at first appears complex but is in reality simply being able to count spaces and learning the sign based on the direction the decimal point is moved.

   Scientific numbers all have several things in common.  

   1) They represent relatively large or small real numbers.

   2) The value in front of the decimal point is required to be 
      between 1 and 9.  

   3) The sign of the exponent (power) of the 10 (ten) is for the 
      purpose of converting the number from scientific notation back 
      to the original number when necessary, i.e. adding or 
      subtracting two numbers.  The direction to move the decimal 
      is to the right, adding zeros (0) as needed, when the sign 
      is positive or absent and left, placing zeros (0) in front as 
      needed when the sign is negative.

   4) These numbers are always expressed as: 

                           a.bc x 10Y  

       where a, b, c are numbers and Y is a number representing 
       the number of zeros (decimal places) that needed to be 
       added to the left (-Y) or to the right (+Y or Y) to 
       return the number to its more common form. 

       An easy way to remember how to expand scientific notation 
       is to look at the sign of the power.  If the sign is 
       negative (-Y), then write the number portion to the right 
       and add zeros IN FRONT of the numbers.  Then count from 
       old decimal position toward the left to locate the new 
       decimal point position.  If the sign is positive (+Y or 
       Y), write the number to the left and add zeros AFTER the 
       number.  Then count from the old decimal position toward 
       the right to find the new decimal position.

   5) a number which fits the format as shown in No. 4 is 
      already in scientific notation and does not need 
      transformation.  

   6) the exponent sign is negative if the decimal point position 
      is moved to the right when going into scientific notation.  
      The exponent sign is positive is the decimal point position 
      is moved to the left going into scientific notation.  These 
      are the reverse of the moves discussed in No. 4 so be 
      careful not to confuse which way the number is being 
      transformed.
To figure the transformation from a real number into scientific notation, you must look carefully at the number being transformed.

For example:

1234567

Even though there is no decimal present, standard mathematical rules state the decimal is assumed to be at the right end of the number.

1234567.

Since scientific notation requires the number in front of the decimal to be between 1 and 9, the decimal must be moved to the left. Let's see how many places the decimal must move.


  One (1) place  123456.7
 
  Two (2) places 12345.67
 
Three (3) places 1234.567

 Four (4) places 123.4567

 Five (5) places 12.34567

  Six (6) places 1.234567
Now the number in the position to the left of the decimal has a value between 1 and 9. Since the decimal moved to the left, the sign of the number for the power of 10 (ten) will be positive. The decimal was moved six places to the left so the value of the power will be +6.

Rewrite the number as a power of 10 (ten) using the new number and adding the x 10 portion as is shown below.

1.234567 x 10

Next simply add the power as an exponent to the 10 (ten).

1.234567 x 10+6

or simply

1.234567 x 106

For another example:

0.000001234567

Since scientific notation requires the number in front of the decimal to be between 1 and 9, the decimal must be moved to the right. Let's see how many places the decimal must move.

  One (1) place  00.00001234567
 
  Two (2) places 000.0001234567

Three (3) places 0000.001234567

 Four (4) places 00000.01234567

 Five (5) places 000000.1234567
 
  Six (6) places 0000001.234567
Now the number in the position to the left of the decimal has a value between 1 and 9. Since the decimal moved to the right, the sign of the number for the power of 10 (ten) will be negative. The decimal was moved six places to the right so the value of the power will be -6.

Rewrite the number as a power of 10 (ten) using the new number and adding the x 10 portion as is shown below.

1.234567 x 10

Next simply add the power as an exponent to the 10 (ten).

1.234567 x 10-6

When converting to scientific notation, KEEP ALL the original digits. Digits are not dropped in scientific notation. (See the paper on Significant figures to learn about dropping digits and rounding.)

Try the following numbers for practice in changing numbers into the scientific notation format.

       A) 987.88    B) 0.00203     C) 1.005       D) 15432 
       E) 0.0003    F) 111.1       G) 0.00000001  H) 2.135