

Subject:
how do I define multiplication of real numbers in terms of addition?
Category: Science > Math Asked by: mxnmatchga List Price: $10.00 
Posted:
04 Jul 2006 21:24 PDT
Expires: 08 Jul 2006 16:04 PDT Question ID: 743424 
This is a continuation of my question at: http://answers.google.com/answers/threadview?id=743129 I tried going through: http://ioannis.virtualcomposer2000.com/math/exponents4.html But I got lost in the 3rd section. From the summary at the end it looks to me like I can define my method g the same way as before except that I add a special case when 0 < y < 1. For instance, g(1,2,4.5) = 2 + 2 + 2 + 2 + g(1,2,0.5) I just need to know how to define that last part in terms of addition. I'm betting that if I do that then it'll work all the way up. It looks like it involves limits on some sort of sequence, but I can't figure out what it should be.  
 
 


There is no answer at this time. 

Subject:
Re: how do I define multiplication of real numbers in terms of addition?
From: saem_aeroga on 06 Jul 2006 06:46 PDT 
To address the original question, along with the first clarification, the problem appears to be the stumbling block of multiplying by ten (or division). In this case, take the number you wish to multiply by ten and ADD it to itself 9 more times. eg: 0.5*10 = 0.5+0.5+0.5+... Now you have avoided the decimal problem. You can construct a similar operation for division. Example: (2.5)*(3.5) = (25/10)*(35/10) = 25*35/100 = 25*35*(.01) So we can write 25*35*.01 in terms of adding 25*35, 0.01's together. Its easy now. I instructed someone to write this code for a general case and it does indeed work only using + operations. Your example: 2*4.5 = 20*45*.01 = (0.1+0.1+0.1+0.1+(roughly 20*45 more additions). I fear that with large answers there may be numerical error. Can you devise a different method to overcome these problems? 
Subject:
Re: how do I define multiplication of real numbers in terms of addition?
From: mxnmatchga on 06 Jul 2006 12:54 PDT 
That will work for multiplication, but I wanted this in a general way. In my clarification I proposed: g(a,x,y) = g(a, g(a,x,y*k), g(a+1,y*k,1) ) For a=2 (exponentiation) and k=10 we get: g(2,x,y) = g(2, g(2,x,y*10), g(3,y*10,1) ) = root(x^(y*10), y*10) (where root(m,n) is the nth root of m) For a=1 (multiplication) and k=10 we get: g(1,x,y) = g(1, g(1,x,y*10), g(2,y*10,1) ) = x*y*10 / y*10 For a=0 (addition) and k=10 we get: g(0,x,y) = g(0, g(0,x,y*10), g(1,y*10,1) ) = x + y*10  y*10 In other words, we multiply y by 10 and then use it as y and then undo undo the multiplication by 10 in such a way that it works, not just for multiplication, but for any operation g(a,x,y). That's pretty much what you described, but I don't know how to do g(a,x,1) for any operation where a > 3. 
Subject:
Re: how do I define multiplication of real numbers in terms of addition?
From: berkeleychocolatega on 06 Jul 2006 18:24 PDT 
In a specific context one can prove that multiplication cannot be defined in terms of addition. The context is that of the natural numbers and "firstorder predicate logic" in which the only quantification allowed is over the natural numbers. For a reference see Enderton, "A Mathematical Introduction to Logic", Corollary 32G which follows from the theorem that a set of natural numbers is definable in terms of 0, successor operation, <, and + iff it is eventually periodic. This does not settle whether one can define * using higherorder quantification and/or over the real numbers instead of just the natural numbers. 
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