[math-fun] The Hippasus Integers
The Heegner numbers are 1, 2, 3, 7, 11, 19, 43, 67, 163. The ℚ(√−1) numbers are known as Gaussian integers. The ℚ(√−3) numbers are known as Eisenstein integers. The ℚ(√−7) numbers are known as Kleinian integers. It's bugged me for awhile that ℚ(√−2) wasn't named. Today I decided that the obvious name was ... The ℚ(√−2) numbers are now known as Hippasus integers. Hippasus proved √2 was irrational. He was then murdered. Seemed like an apt name to use. I calculated the nine types of Heegner primes and plotted them out. http://community.wolfram.com/groups/-/m/t/965609 Anyone agree / disagree about calling ℚ(√−2) the Hippasus integers? Ed Pegg Jr
On 18/11/2016 02:43, Ed Pegg Jr wrote:
The Heegner numbers are 1, 2, 3, 7, 11, 19, 43, 67, 163.
The ℚ(√−1) numbers are known as Gaussian integers. The ℚ(√−3) numbers are known as Eisenstein integers. The ℚ(√−7) numbers are known as Kleinian integers.
It's bugged me for awhile that ℚ(√−2) wasn't named. Today I decided that the obvious name was ...
The ℚ(√−2) numbers are now known as Hippasus integers.
Hippasus proved √2 was irrational. He was then murdered. Seemed like an apt name to use.
Surely √2 and √−2 are quite different things, just as different as √2 and √3. -- g
I agree with Gareth about the important distinction between sqrt(2) and sqrt(-2). However, 'Hippasus integers' might still be appropriate, considering the fact that he didn't prove the irrationality of sqrt(2). (The Greeks had no concept of real number, only of length, area, etc.) Instead, he demonstrated that the sidelength and diagonal of a square are incommensurate; that is to say, there are no natural numbers p and q such that p*sidelength = q*diagonal. At no point did anyone ever consider the abstract constant (diagonal/sidelength), because the Greeks didn't have dimensionless reals. And now that we're interested in *lengths*, note that sqrt(-2) has the same length as the diagonal of a square of unit sidelength, so Hippasus *did* prove that those lengths are incommensurate. Best wishes, Adam P. Goucher
Sent: Friday, November 18, 2016 at 4:40 AM From: "Gareth McCaughan" <gareth.mccaughan@pobox.com> To: math-fun@mailman.xmission.com Subject: Re: [math-fun] The Hippasus Integers
On 18/11/2016 02:43, Ed Pegg Jr wrote:
The Heegner numbers are 1, 2, 3, 7, 11, 19, 43, 67, 163.
The ℚ(√−1) numbers are known as Gaussian integers. The ℚ(√−3) numbers are known as Eisenstein integers. The ℚ(√−7) numbers are known as Kleinian integers.
It's bugged me for awhile that ℚ(√−2) wasn't named. Today I decided that the obvious name was ...
The ℚ(√−2) numbers are now known as Hippasus integers.
Hippasus proved √2 was irrational. He was then murdered. Seemed like an apt name to use.
Surely √2 and √−2 are quite different things, just as different as √2 and √3.
-- g
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participants (3)
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Adam P. Goucher -
Ed Pegg Jr -
Gareth McCaughan