Recurring duodecimals fractions

Question

I get the idea about duodecimals from what I read till I reach the fractions point where:

$\frac{1}{8}=0.16$ instead of $0.15$

$\frac{1}{9}=0.14$ instead of $0.13333333$

$\frac{1}{5}=0.249797979797$ instead of $0.24$

Why is that happening I tried to convert the decimal representation into duodecimal and it came to prove me right on every case.

Thank you very much.

Answer 1

Note:

In every representation, you have: $\frac{1+1+1+1+1+1+1+1+1+1+1+1}{1+1}= 1+1+1+1+1+1$

Likewise, $\frac{\text{"one dozen"}}{\text{"two"}}=\text{"six"}$

In decimal, you have $\frac{12}{2}=6$

Notice though, in dozenal, the representation for "twelve" is not $12$, but is instead $10$.

In dozenal, you have $\frac{10}{2}=6$, just like all of the previous as this is in fact representing the same expression, and so similarly $\frac{1}{2}=0.6$ in dozenal.

You seem to have confused the fact that $\frac{1}{2}=0.5$ in decimal with the fact that $\frac{1}{2}=0.6$ in dozenal. That is to say "one half of one is five tenths" and "one half of one is six dozenths." It appears that you stopped thinking in dozenal halfway through the process of computing the arithmetic once you saw a fraction you thought you were familiar with. Indeed, $\frac{1}{8}=0.16$ in dozenal, i.e. "one eighth of one is a dozen and six gross-th's" or rather "one eighth of one is one and a half dozenths"

With $\frac{1}{8} = $" one and a half dozenths" you jumped to $1.5$ as meaning "one and a half" but again, $1.6$ is one and a half in this context. As such, we intend to use $0.16$ instead of $0.15$ to mean this quantity.

Answer 2

$\frac{1}{8} = 0.16_{12}$ is correct. From the definition of base 12, $0.16_{12} = \frac{1}{12} + \frac{6}{144} = \frac{2}{24} + \frac{1}{24} = \frac{3}{24} = \frac{1}{8}$.

Please give some working on how you obtained $0.15$ etc. so that we can help identify the mistake.

Answer 3

If you convert $1/8_{10}$ to base $12$, you might realize that $8$ divides $144_{10}=12_{10}^2$, so it will have a two digit terminating representation. $\frac 18=\frac {18}{144}_{10}=0.16_{12}$ because $16_{12}=18_{10}$. The others are similar. As you did not show where you got your values, we cannot see where the errors are.