Pairing function - Wikipedia, the free encyclopedia
(1) i cant really understand why he is using the triangle number anyway
(2) why is t <= z ?
Pairing function - Wikipedia, the free encyclopedia
(1) i cant really understand why he is using the triangle number anyway
(2) why is t <= z ?
Hello dayscott
Welcome to Math Help Forum!Why is this true (1): $\displaystyle z = t+y$ ? Because$\displaystyle t = \frac{w(w+1)}{2}$Why is this true (2): $\displaystyle t \le z$ ? Because$\displaystyle = \frac{(x+y)(x+y+1)}{2}$, since $\displaystyle w = x+y$$\displaystyle \Rightarrow z =\frac{(x+y)(x+y+1)}{2}+y = t+y$$\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.
Grandad
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,but I can't see where the right-hand part $\displaystyle \frac{\sqrt{8z+1}-1}{2}<w+1 $ comes from.
and $\displaystyle t\le z$
$\displaystyle \Rightarrow w\le\frac{\sqrt{8z+1}-1}{2}$
Perhaps someone else may be able to help?
Grandad
we got an answer.. Talk:Pairing function - Wikipedia, the free encyclopedia
...which i still don't get - i can't even phrase a good question right now.