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In a ternary number system, how are the $4$ arithmetic operations defined?

Bruna Gabrielly
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    The operations don't depend on the base chosen to represent the numbers for display in. I'm not sure I understand your question. – Daniel Fischer Jul 07 '13 at 16:07
  • $1+1=2,1+2=2+1=10,2+2=11$ and $1\cdot1=1, 1\cdot2=2\cdot1=2,2\cdot2=11$ – lab bhattacharjee Jul 07 '13 at 16:07
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    Just as they are in decimal, binary, or any other number system. The number system is just for representing the numbers. – Shaun Ault Jul 07 '13 at 16:07
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    The same way as always! The operations are immune to such trifles as presentation of the numbers. A slightly longer answer would involve the successor function and Peano axioms, but let's not go there. I rather guess that you want to know whether there will be changes in the algorithms for calculating the values of the operations. Not much. The single-digit multiplication table will be smaller, you carry w.r.t base three as opposed to ten et cetera. – Jyrki Lahtonen Jul 07 '13 at 16:09
  • @CameronBuie, thanks for pointing out timely, rectified. – lab bhattacharjee Jul 07 '13 at 16:12
  • @lab: No problem. – Cameron Buie Jul 07 '13 at 16:13
  • Ok, it's simple and all but I'm not getting, sorry. Could you explain it further, or maybe give me some reference? thanks – Bruna Gabrielly Jul 07 '13 at 16:33
  • For example in decimal notation we have $5+7=12$ and $5\cdot7=35$. In ternary notation the only thing that would change is the notation: $5=12_3$, $7=21_3$, $12=110_3$ and $35=1022_3$. So when we say that five times seven is stil thirty-five, we mean that $$12_3\cdot21_3=1022_3.$$ It doesn't matter, whether "five" is denoted $5$ or $12_3$. We mean the same quantity (the number of fingers in one hand) in both cases. Same with "seven" and "thirty-five". – Jyrki Lahtonen Jul 07 '13 at 16:38
  • @JyrkiLahtonen thanks – Bruna Gabrielly Jul 07 '13 at 17:00

1 Answers1

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Strictly speaking, one should call it the ternary numeral system. What is different from base-10 is not the numbers, but the numerals.

Addition, subtraction, multiplication, and division are not defined within a numeral system; they are defined independently of numeral systems.

One could ask, however, how to do arithmetic within a base-3 numeral system. There is an addition table: $$ \begin{array}{c|cc} 0 & 1 & 2 \\ \hline 1 & 2 & 10 \\ 2 & 10 & 11 \end{array} $$ and there is a multiplication table: $$ \begin{array}{c|cc} & 1 & 2 \\ \hline 1 & 1 & 2 \\ 2 & 2 & 11 \end{array} $$ Arithmetic is done the same way as in base 10, but using these tables rather than the ones you learned at your mother's knee.

There was a time, between some time in the '60s, I suspect, and maybe some time after 1990 or so, when math courses required of those who were to become elementary school teachers taught numeral ssytems in various different bases, I suspect because it was thought that it aided in understanding the theory behind the operations done in base 10. I think that was largely a mistake, but it could have been useful if they'd taken it a step further and asked students to do a bunch of arithmetic problems in base 8 or base 12, for this reason: It makes it look just as unfamiliar to them as it looks to elementary school pupils who are learning it for the first time. In other words, it shows them what arithmetic looks like to their pupils.

Michael Hardy
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