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I came across duodecimal (base 12) numbers.

In base 10 system 10/3 = 3.333.... i.e repeating decimal. But in base 12 system where "t" represents 10 of base 10 - t/3 = 3.4 which is a non repeating decimal.

I hope have done the above t/3 base 12 division correct.

Now, trying to grasp meaning and significance of recurring decimal. Does a recurring value as a result of 10/3 (in base 10) means we can never divide 10 into 3 parts with 100% accuracy ? But then we change number system to base 12 and t/3 now is not a recurring decimal.So now we can divide t into 3 equal parts accurately. But this is a contradiction.

Abhi
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  • If you denominator prime factors are the same as the prime factors of the base you decimal will terminate. Otherwise is will not. The denominators of the $2^n5^m$ will terminat in base $10$. The denominators of the form $2^n3^m$ will terminate in base $12$. – fleablood Jul 22 '18 at 06:03
  • True..but can you plz shed some light on the this --> "Now, trying to grasp meaning and significance of recurring decimal. Does a recurring value as a result of 10/3 (in base 10) means we can never divide 10 into 3 parts with 100% accuracy ? But then we change number system to base 12 and t/3 now is not a recurring decimal. So now we can divide t into 3 equal parts accurately. But this is a contradiction. – Abhi Jul 22 '18 at 12:05
  • Of course you can divide 10 into three parts with 100% accuracy. Each part is exactly $3 \frac 13$. You can't express it in base 10 decimal wiith 100% accuracy. But expressing something using a limited notation system does not mean it doesn't exist-- it just means the limited notation system is inadequate. And no-one ever claimed finite base 10 decimal system was adequate. (Now infinite base 10 decimal systems is another story....) – fleablood Jul 22 '18 at 15:41
  • You are correct my friend. Thanks ! – Abhi Jul 22 '18 at 15:49
  • The point of confusion is we introduce decimals with the implication they can express everything. Finite decimals can't. But they can get as precise as we wish with any specific degree of accuracy. And because they can get within any degree we can use infinite decimals to express anything. And then we teach decimals as though they can express everything. – fleablood Jul 22 '18 at 16:25

2 Answers2

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In base 12: $\frac {10}3 = 3\frac 13 = 3\frac {4}{12}= 3.4_{12}$.

In base 10: $\frac {10} 3 = 3\frac 13 = 3\frac {\frac {10}3}{3\frac {10}3} = 3 \frac {3\frac 13}{10} =3\frac 3{10} + \frac 1{30} = 3\frac 3{10}\frac 3{100} + \frac 1{300} = 3\frac 3{10}+ \frac 3{100} + \frac 3{1000} + ......= 3.333333....$.

In base $b$ a fraction $\frac nm$ will terminate if and only if you can get $\frac nm = \frac k{b^w}$ for some power $w$ of $b$.

In other word $m|b^w$.

In base $10$ to have $m|10^w = 2^w5^w$ we simply need to have $m$ to only have prime factors $2$ and $5$. Example $\frac {17}{40}$ and $40=2^3*5|10^3$ so $\frac {17}{40}*\frac {5^2}{5^2} = \frac {425}{1000} = .425$

In base $12$ to have $m|12^w = 2^{2w}3^w$ we simply need to have $m$ to only have prime factors $2$ and $3$. Example $\frac {23}{36}$ and $36=2^23^2|4^23^2$ so $\frac {23}{36}\frac 44 = \frac {23}{144} = \frac {23}{12^2} = .23$.

fleablood
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  • True..but can you plz shed some light on the this --> "Now, trying to grasp meaning and significance of recurring decimal. Does a recurring value as a result of 10/3 (in base 10) means we can never divide 10 into 3 parts with 100% accuracy ? But then we change number system to base 12 and t/3 now is not a recurring decimal. So now we can divide t into 3 equal parts accurately. But this is a contradiction. – Abhi Jul 22 '18 at 12:02
  • Why is that a contradiction? Suppose you went to a library and asked what movies they had and three of the movies were on dvd and one was on vhs. You only have a dvd play so you can't watch the fourth movie. But you buy an old VHS system and .... suddenly you can watch it. Is that a contradiction? – fleablood Jul 22 '18 at 15:46
  • Good example ! Thanks ! – Abhi Jul 22 '18 at 15:48
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It only sounds like a contradiction. Think of the problem of dividing $10$ by $3$ as trying to equally share $10$ cookies among $3$ children. You first give each child $3$ cookies, leaving $1$ left over. Now if you work base $10$, what you in effect do is cut that last cookie into ten (equal) pieces, and then try to mete them out equally: you'll again give each child $3$ pieces, and you'll again have one piece left over, so you'll again cut that piece into $10$ equal pieces, and so on and so forth.

On the other hand, if you work base $12$, you cut the leftover cookie into twelve equal pieces and try to mete them out equally. This time you succeed! Each child gets four of the twelve pieces, and you're done.

The only difference between the two distribution schemes is the arbitrary decision to cut any leftover portions of cookies into a particular number of pieces -- ten in the decimal scheme and twelve in the duodecimal. The most sensible approach here, of course, would be to work base three: cut the leftover cookie into three equal pieces, and give each child a piece.

Barry Cipra
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  • Thanks for the crystal clear explanation ! I too arrived at same conclusion but thought of getting a second opinion to confirm. – Abhi Jul 22 '18 at 12:52