# Why is it called the Fundamental Theorem of Arithmetic?

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.

• – Zubin Mukerjee Apr 10 '15 at 13:53
• Number Theory is the "modern" name of Arithmetic. – Cla Apr 10 '15 at 14:03
• 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. – Lucian Apr 10 '15 at 17:18
• @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. – Asaf Karagila Apr 11 '15 at 12:08
• @AsafKaragila: I know ! :-) There's an entire scene in Darren Aronofsky's 1998 movie Pi about it. ;-) – Lucian Apr 11 '15 at 19:03

• @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. – Robert Soupe Apr 11 '15 at 1:19