Arithmetic operations in ternary number system In a ternary number system, how are the $4$ arithmetic operations defined?  
 A: 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.
