Scientific Notation

Date: 10/02/98 at 13:42:51
From: Craig H.
Subject: Scientific Notation

Dr. Math,

Your Web site is very handy. I have a question for you. How do you 
write a number in scientific notation? 

Thanks,
Craig Hauke 

Date: 10/02/98 at 17:17:07
From: Doctor Rick
Subject: Re: Scientific Notation

Thanks for the compliment, Craig. We are glad to be of help. I will 
try to help you understand scientific notation.

Scientific notation starts with the fact that when you multiply 10 by 
itself some number of times, you get 1 with that many zeros. You 
probably know that we use this notation for multiplying a number by 
itself, but I will review it for you:

    1
  10  = 10

    2
  10  = 10 * 10 = 100

    3
  10  = 10 * 10 * 10 = 1000

    4
  10  = 10 * 10 * 10 * 10 = 10000
  ...

We call it "10 raised to the 4th power," or "10 to the 4th power," or 
just "10 to the 4th."

The next fact is that if you multiply a number by 1 with some number 
of zeros, you move the decimal point to the right that many places:

  2.5 * 10000 = 
   |____
        |
  2 5000.

This means that the scientific notation for 25000 is 2.5 * 10^4. (I 
will use "^" from now on for the exponent instead of putting the "4" 
above the line - it's easier to write.)

To write 25000 in scientific notation, you just move the decimal point 
to the left until there is only one digit to the left of it. (You will 
have a number between 1 and 10.) Then count the places you moved the 
decimal point, and that becomes the exponent (the number above the 
line, after the 10).

The next thing to think about is what happens if we have a number less 
than one, for instance 0.0025. This can be written:

             25           -4
  0.0025 = ----- = 25 * 10
           10000

So we could write 0.0025 as 25 * 10^(-4), but in true scientific 
notation, the first number is always between 1 and 10. We have to move 
the decimal point one place to the left, which means adding 1 to the 
exponent:

  0.0025 = 2.5 * 10^-3

We could have done this in one step by using this rule: If you have to 
move the decimal point right to get one non-zero digit to the left of 
it, count the places and use a negative exponent.

Once more, all together:

1. Move the decimal point, left or right, until there is one non-zero 
   digit to the left of the new decimal point. Count the places you 
   moved it.

2. Multiply the number by 10 raised to this power. Make the power 
   positive if you moved the decimal point left, and negative if you 
   moved the decimal point right.

More examples:

  5280 = 5.28 * 10^3          0.000705 = 7.05 * 10^(-4)

  5 280.                      0.0007 05
    ___| 3 places left         |____     4 places right
   |                                |
  5.280                            7.05

There is more to learn, about using scientific notation - for instance, 
how do you multiply:

  (5.28 * 10^3) * (7.05 * 10^(-4))   ?

If you have questions about this, please ask again!

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/