How do you convert a ternary to a novenary?

Question

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?

score 2 · Accepted Answer · answered Jul 04 '16 at 17:41

2

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 $0.122101_3 = 0.571_9$.

answered Jul 04 '16 at 17:41

florence

12,819

Where did you get that expression for x? – Al jabra Jul 04 '16 at 18:09
@Aljabra It's the definition of a ternary/9-ary expansion. – florence Jul 04 '16 at 18:12
where did you get $si=3t_{2i−1}+t_{2i}$ from? – Al jabra Jul 04 '16 at 18:22
@Aljabra From your original setup. For $i = 1$ it's $s_1 = 3t_1 + t_2$, for $i = 2$ it's $s_2 = 3t_3 + t_4$, etc. – florence Jul 04 '16 at 18:34
makes sense thanks – Al jabra Jul 04 '16 at 18:34
I don't understand this part: ∑i=1∞3−(2i−1)t2i−1+3−2it2i = ∑i=1∞ti3−i – Romstar Jan 29 '21 at 12:11

How do you convert a ternary to a novenary?

1 Answers1