The Fundamental Theorem of Arithmetic is easy enough to understand, saying that every integer greater than 1 is either prime or is the product of a unique combination of prime numbers.

What I don't understand is why this is "fundamental." This may have massively important implications in number theory and cryptography and whatever else, but in terms of arithmetic, which I think of as adding, subtracting, multiplying, and dividing, it doesn't really actually seem to have that much importance....I don't see why it should be so fundamental.

Can someone explain its importance, or why it isn't called perhaps the "Fundamental Theorem of Number Theory"?

I would expect the Fundamental Theorem of Arithmetic to be something.

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    $\begingroup$ Relevant. $\endgroup$ – Zubin Mukerjee Apr 10 '15 at 13:53
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    $\begingroup$ Number Theory is the "modern" name of Arithmetic. $\endgroup$ – Cla Apr 10 '15 at 14:03
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    $\begingroup$ The word arithmetic comes from the Greek word arithmos, meaning number. There is an entire book in the Bible called Numbers. When Jews translated the Torah into Greek around the third century BC (see Septuagint), the name of that book was Arithmoi. $\endgroup$ – Lucian Apr 10 '15 at 17:18
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    $\begingroup$ @Lucian: To add on the conspiracy here, the Hebrew word for "theory" in the mathematical context (e.g. set theory) is the same word as the Torah. So the old testament had an entire book about number theory. $\endgroup$ – Asaf Karagila Apr 11 '15 at 12:08
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    $\begingroup$ @AsafKaragila: I know ! :-) There's an entire scene in Darren Aronofsky's 1998 movie Pi about it. ;-) $\endgroup$ – Lucian Apr 11 '15 at 19:03

Because Arithmetic is another name for Number Theory.

Unique factorization was used widely for ages without anyone bothering to prove it or even feeling any need for a proof. It was Gauss that recognized this and finally proved it in Disquisitiones Arithmeticae in 1801.

The Fundamental Theorem of Arithmetic is also important because it does not hold in all number rings (that is, rings of integers of an algebraic number field). Attempts to understand this led to the important development of ideal numbers by Kummer and Dedekind and the birth of algebraic number theory and modern algebra.

  • $\begingroup$ See also math.stackexchange.com/a/83987/589. $\endgroup$ – lhf Apr 10 '15 at 14:16
  • $\begingroup$ Your final paragraph totally lost me. Not sufficiently versed in terminology. But thank you nonetheless. $\endgroup$ – temporary_user_name Apr 10 '15 at 14:22
  • $\begingroup$ @Aerovistae Consider for example numbers of the form $a + b \sqrt{-5}$ in $\mathbb{Z}[\sqrt{-5}]$. Some of these numbers have more than one factorization. But the ideals in this domain can be factored uniquely as products of prime ideals. $\endgroup$ – Robert Soupe Apr 11 '15 at 1:19
  • $\begingroup$ @Aerovistae, for further and simpler examples of failure of unique factorization see the first half of these slides. $\endgroup$ – lhf Apr 11 '15 at 1:51

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