Tertiary Arithmetics: Is it Possible?

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SUMMARY

The discussion centers on the feasibility of performing arithmetic operations with three operands, termed "ternary" operations. It concludes that any ternary function can be constructed using a sequence of binary and unary operations by encoding two numbers into a single number and then applying a binary function to this encoded number and a third operand. An example of encoding is provided, where the numbers 12345 and 678 are combined into 1020364758. The term 'ternary' is emphasized as the correct terminology for three-argument functions.

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Is there any arithmetic operation with three operands (or arguments), such that it cannot be calculated by a sequence of common binary and unary operations? This is not a homework problem or anything like that, I am just curious.
 
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What precisely do you mean by 'arithmetic' here? Anyways, if you're simply talking about functions, then the answer is no, because you can encode a pair of numbers into a single number, and build your ternary function from the two binary functions:

1. Encode the first two numbers into a single number
2. Take the output of (1) and the third number, unpack (1) and compute the ternary function


An example of how to do the encoding would be to alternate taking digits from your two numbers. For example,

encode(12345, 678) = 1020364758

Incidentally, the word is 'ternary'
 

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