# Thread: inverting the Pairing function

1. ## inverting the Pairing function

Pairing function - Wikipedia, the free encyclopedia

(1) i cant really understand why he is using the triangle number anyway

(2) why is t <= z ?

2. Hello dayscott

Welcome to Math Help Forum!
Originally Posted by dayscott
Pairing function - Wikipedia, the free encyclopedia

(1) i cant really understand why he is using the triangle number anyway

(2) why is t <= z ?
Why is this true (1): $\displaystyle z = t+y$ ? Because
$\displaystyle t = \frac{w(w+1)}{2}$
$\displaystyle = \frac{(x+y)(x+y+1)}{2}$, since $\displaystyle w = x+y$
$\displaystyle \Rightarrow z =\frac{(x+y)(x+y+1)}{2}+y = t+y$
Why is this true (2): $\displaystyle t \le z$ ? Because
$\displaystyle t=z-y$, and $\displaystyle y$ is a natural number.
As far as your question about triangle numbers is concerned, it is simply that the expression $\displaystyle \tfrac12w(w+1)$ happens to be the sum of the first $\displaystyle w$ natural numbers; which is the $\displaystyle w^{th}$ triangle number.

3. thx a lot !

another question - the red arrow marks the gap in my thinking ^^:

4. no one ? : /

5. Hello dayscott

I didn't post a reply earlier, because I can't see it either!

The LHS of the inequality is straightforward enough:
$\displaystyle w=\frac{\sqrt{8t+1}-1}{2}$, which is strictly increasing,

and $\displaystyle t\le z$

$\displaystyle \Rightarrow w\le\frac{\sqrt{8z+1}-1}{2}$
but I can't see where the right-hand part $\displaystyle \frac{\sqrt{8z+1}-1}{2}<w+1$ comes from.

Perhaps someone else may be able to help?