Big Number Multiplication

Date: 05/04/97 at 20:33:23
From: Victor Yoon
Subject: Multiplication

Dr. Math,

What does 4000*500*32*768*4 equal?

Date: 05/05/97 at 03:21:26
From: Doctor Mike
Subject: Re: Multiplication

Hi Victor,  

I'm just going to do this on my calculator, but in a way that will not 
give such a big number that it will cause an overflow. 

I am going to factor out some powers of ten so as to make the numbers 
a little smaller for the calculator. That comes out as:
   
      4 * 1000 * 5 * 100 * 32 * 768 * 4  =  

      4 * 5 * 32 * 768 * 4 * 1000 * 100  
  
I will ask the calculator to find 4*5*32*768*4 and then I will add 5 
zeros onto the end.  That is because 1000*100 is 10 to the 5th power, 
and each time you multiply by 10, it adds a zero onto the right end of 
the product. 

Actually, I'm just seeing that the 4*5 at the start equals 20, so I am 
going to re-write this calculation as follows:
    
      20 * 32 * 768 * 4 * 1000 * 100  =  

      2 * 32 * 768 * 4 * 10 * 1000 * 100 
  
I will ask the calculator to find 2*32*768*4 and then I will add 6 
zeros onto the end. That is because 10*1000*100 is 10 to the 6th 
power.  Actually, there is one more thing I want to do before I pull 
out the calculator. If there is something I can do in my head before 
I start to push the keys, I like to do that, since sometimes the keys 
stick and I want to minimize the chance for a mistake. 

I know, and you probably do too, that as long as all you're doing is 
multiplication, you can multiply numbers together in any order, so I'm 
going to re-arrange them and do some stuff in my head. It comes out 
like this: 

      2 * 4 * 32 * 768 * 10 * 1000 * 100  =  
 
      8 * 32 * 768 * 10 * 1000 * 100  
  
Now I use the calculator to find that 8*32*768 = 196608 so the full 
answer is 196608000000.  

If you write this with commas it looks like 196,608,000,000, which is 
a number between 196 billion and 197 billion.  By the way, what were 
you counting?  Or was this a big problem you just thought of?   

-Doctor Mike,  The Math Forum
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