Tertiary Arithmetics: Is it Possible?

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In summary, there is no arithmetic operation with three operands that cannot be calculated by a sequence of common binary and unary operations. This is because a pair of numbers can be encoded into a single number, allowing for a ternary function to be built using two binary functions. The encoding process involves taking digits from the two numbers and alternating them.
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Uke
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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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  • #2
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'
 

1. What is Tertiary Arithmetics?

Tertiary Arithmetics is a type of arithmetic system that uses three digits (0, 1, and 2) to represent numbers. It is a base-3 system, as opposed to the more commonly used base-10 system. This means that each place value is multiplied by a power of 3, rather than 10.

2. Is Tertiary Arithmetics widely used?

No, Tertiary Arithmetics is not widely used. It is primarily used in theoretical mathematics and computer science, as it has certain advantages in representing and manipulating data in binary systems. However, it is not commonly used in everyday calculations or in other fields of science or mathematics.

3. Is it possible for humans to learn and use Tertiary Arithmetics?

Yes, it is possible for humans to learn and use Tertiary Arithmetics. However, it may be more challenging to learn compared to other arithmetic systems, as it involves a different way of thinking and calculating. With practice, humans can become proficient in using Tertiary Arithmetics.

4. How is Tertiary Arithmetics related to binary and decimal systems?

Tertiary Arithmetics is related to binary and decimal systems in that it is also a positional notation system, where the value of a digit depends on its position in the number. However, Tertiary Arithmetics uses a different base (3) compared to binary (2) and decimal (10) systems. Additionally, Tertiary Arithmetics has certain similarities with binary systems, as both use only two digits (0 and 1) to represent numbers.

5. Are there any practical applications of Tertiary Arithmetics?

Yes, there are some practical applications of Tertiary Arithmetics. As mentioned earlier, it is commonly used in computer science and data representation. It can also be used in coding and decoding messages, as well as in advanced mathematical concepts such as modular arithmetic. However, it is not commonly used in everyday calculations or in other fields of science or mathematics.

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