Explaining Scientific Notation

Date: 10/21/2002 at 12:51:37
From: Teaka
Subject: Scientfic Notation

Could you try to explain scientific notation to me?

Thank you.

Date: 10/21/2002 at 15:39:17
From: Doctor Ian
Subject: Re: Scientfic Notations

Hi Teaka,

There are actually only a few rules, but with lots of different 
applications.  

The first rule is this:

  If you have a product, a*b, you can divide a by something 
  if you multiply b by the same thing.  That is, 

     a*b = (a/c) * (b*c)

For example, 

     12*15 = (12/3) * (15*3)

Make sure that makes sense to you, because it's crucial to everything
else.

Now, let's start with a big number, like 

   219,425,000

I can write this as

   219,425,000 * 1

What if I divide the first number by 10, and multiply the second by
10? According to our rule, this shouldn't change the value of the product:

     219,425,000 * 1

  =   21,942,500 * 10

  =    2,194,250 * 100

  =      219,425 * 1000

And so on. Does that make sense? Now, what if I keep going until the 
number on the left is between 1 and 10?  I end up with 

  = 2.19425 * 100,000,000

Now, just to check that we haven't done anything dumb, we can make 
sure this is in the ballpark: the first number is about 2, and the 
second is a hundred million, so if we multiply them together, we get 
about two hundred million, which is about where we started. 

Sometimes we make the number on the right smaller instead of larger:

     0.000134 * 1

  =  0.00134  * 1/10

  =  0.0134   * 1/100

  =  0.134    * 1/1000
  
  =  1.34     * 1/10,000

Does this make sense?  

Now, it turns out that we have a very compact notation for these 
numbers on the right:

           n
  10^n = 10  = 1 (followed by n zeros)

For example, 

  10^0 =        1
  10^1 =       10
  10^2 =      100
  10^3 =    1,000
  10^4 =   10,000

For fractions, we have a similar rule:

  10^-1 =  1/10
  10^-2 =  1/100
  10^-3 =  1/1,000
  10^-4 =  1/10,000

Now, you might look at this and think: This is still a lot of rules!  
But in fact, there's probably nothing here that you couldn't have 
figured out if you understand what an exponent is. For example, 

  10^4 = 10 * 10 * 10 * 10

       =     100 * 10 * 10

       =         1000 * 10

       =             10000

That is, every time we multiply by 10, we add another zero on the 
right. That's _why_ the exponent matches the number of zeros. It's not 
a magic formula that someone made up. It's just something that follows 
from the way exponents are defined. 

In fact, the only thing that's new is the idea of splitting a number 
into the product of two numbers. And even that's not such a new idea, 
since you've probably been doing things like 

  12 = 3 * 4

for years. This is the same thing, except we're using tens.  

Anyway, this lets us write things very compactly:

  0.0000000000134         = 1.34 * 10^-11

  219,000,000,000,000,000 = 2.19 * 10^17

This is especially nice when we get big numbers like 

  6.02 * 10^23

(which is the number of water molecules in 18 grams of water), because 
we don't have to remember the name for the number (is this 6 zillion?  
6 jillion?). We can just use the exponent as a name. 

So that's all that's going on when we use scientific notation. In 
fact, there's just one rule that you have to follow: Start with

   something * 1

and multiply or divide by 10 until you get 

  (something between 1 and 10) * (some power of 10)  

It's true that there are lots of shortcuts that you can use, like 
counting zeros or moving decimal points around. However, if you try to 
memorize them without knowing why they work, you're likely to forget 
them (or worse, mix them up without realizing it) at the worst 
possible time. On the other hand, if you understand why they work, you 
don't have to bother memorizing them.

I hope this helps.  Write back if you'd like to talk more about this, 
or anything else. 

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