Say I have the ternary expansion $$0_3.t_1t_2...t_{2n-1}t_{2n}...$$
When converted into a novenary I am told it equals
$$0_9.(3t_1 +t_2)(3t_3+t_4)...(3t_{2n-1}+t_{2n})...$$
I am not sure how to get this.
How do you get from the ternary to the novenary?
How do you get this novenary representation?
Using your notation, $$x = \sum_{i=1}^{\infty}t_i3^{-i} = \sum_{i=1}^{\infty}s_i9^{-i}$$ Where $0.s_1s_2s_3...$ is the 9-ary expansion. Simply substituting $s_i = 3t_{2i-1}+t_{2i}$ into the right sum gives us $$\sum_{i=1}^{\infty}(3t_{2i-1}+t_{2i})9^{-i} = \sum_{i=1}^{\infty}(3t_{2i-1}+t_{2i})3^{-2i} = \sum_{i=1}^{\infty}3^{-(2i-1)}t_{2i-1}+3^{-2i}t_{2i} = $$ $$\sum_{i=1}^{\infty}t_i3^{-i}$$ The equality checks out, and so this is a valid way to calculate the 9-ary expansion from the 3-ary expansion.
As an example, let's look at how you would convert $0.122101_3$ into 9-ary. $12_3 = 5_9$, $21_3=7_9$, $01_3=1_9$, and so $0.122101_3 = 0.571_9$.