Can the Cantor pairing function be modified to support real numbers?

Question

I was looking at the cantor pairing function here https://en.wikipedia.org/wiki/Pairing_function and am quite amazed as to how it works very well for pairing and unpairing natural numbers.

So far, my test on natural numbers π(47, 32) work flawlessly but I have another special use case where I would want to use real numbers instead, for exampleπ(6036.154879072251, 21288). Is there a way to modify the function to allow support for real numbers?

Please do not add irrelevant tags back to your question. – Asaf Karagila Feb 02 '19 at 08:57 — Asaf Karagila, Feb 02 '19 at 08:57
@AsafKaragila Noted. Thanks – Abadi Eferu Feb 02 '19 at 09:01 — Abadi Eferu, Feb 02 '19 at 09:01

Henno Brandsma · Answer 1 · 2019-02-02T11:23:38.503

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The formula for the pairing function is given on that page.

$\pi(x,y)=\frac{1}{2}(x+y)(x+y+1) + y$

$\pi(0,1) =2$ and $\pi(-2,2) = 2$. So it already fails to be injective for $\mathbb{Z}^2$.

The are other ways to show that $\mathbb{R}$ and $\mathbb{R}^2$ are equipotent; it cannot be done by a simple algebraic formula like this..

edited Feb 02 '19 at 11:23

answered Feb 02 '19 at 10:20

Henno Brandsma

242,131

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To the proposer: A simple formula like the one for $\pi(x,y)$ is continuous but no bijection from $\Bbb R^2$ to $\Bbb R$ is continuous. – DanielWainfleet Feb 02 '19 at 10:26
Thanks all for the responses.In this regard, can you propose a better alternative to pairing two real numbers? – Abadi Eferu Feb 02 '19 at 10:39
@AbadiEferu this thread has some ideas. Or here for squares and intervals. Or wikipedia has a more set theoretic approach. GIYF. – Henno Brandsma Feb 02 '19 at 10:49
@Henno Brandsma, π(0,1)=2 is a valid pair but π(−2,2) isn't a valid pair since the pairing function is designed to work with only natural numbers. My aim is to know if the function can be modified or adjusted such that a positive real number pair π(0.567, 3) would yield a valid result on pairing and on unpairing.Thanks – Abadi Eferu Feb 02 '19 at 10:50
@AbadiEferu as Daniel said, no, this cannot be done. – Henno Brandsma Feb 02 '19 at 10:52
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I think you mean equipotent rather than idempotent. – Tobias Kildetoft Feb 02 '19 at 11:16
@TobiasKildetoft quite right, so edited. – Henno Brandsma Feb 02 '19 at 11:25

Can the Cantor pairing function be modified to support real numbers?

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