Why is the fundamental theorem of arithmetic special?

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The fundamental theorem of arithmetic is significant because it highlights the role of prime numbers as the essential building blocks of positive integers through multiplication, similar to how addition builds numbers by iterating +1. While some argue that multiplication is not inherently more fundamental than addition, it is crucial for constructing natural numbers multiplicatively. The theorem's historical roots trace back to Euclid, contributing to its established significance in mathematics. The distinction between operations like addition and multiplication may not warrant the term "fundamental" for one over the other, but both are essential for defining more complex operations. Ultimately, the naming conventions in mathematics serve to facilitate clear communication about these concepts.
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Why is it significant enough to be fundamental?

Some people say that it is fundamental because it establishes the importance of primes as the building blocks of positive integers, but I could just as easily 'build up' the positive integers just by simply iterating +1's starting from 0.
 
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Yes, you can build up all natural numbers by adding 1's. The primes do the same thing from the point of view of multiplication as 1 does from the point of view of addition. If you want to build up all the natural numbers using multiplication, you need to use primes.
 
In that case, why is multiplication so special? What makes it more special than addition, exponentiation, tetration, pentation, hexation, ... ?
 
japplepie said:
In that case, why is multiplication so special? What makes it more special than addition, exponentiation, tetration, pentation, hexation, ... ?
Multiplication is certainly not more "fundamental" than addition, but building up the natural numbers by adding 1's is apparently not difficult enough to call it a theorem. It may be how the natural numbers are defined. Multiplication and addition are the fundamental operations that are used to define the other operations.
 
If the question has to due with the fact why it has the name it has. I imagine this is a tradition bestowed upon it by the fact that the theorem has existed since Euclid. I'm sure there may exist other reasons. Anyway, I wouldn't get to hung up on the name. Sometimes we just name things so everyone knows what we're talking about.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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