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Thread: 2 @ 2 = 4

  1. #1 2 @ 2 = 4 
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    Does anybody else find this fascinating--that if both operands are 2 for any "growth" operator, then the result will always be 4. By a "growth" operator, I mean +, x, power, etc.--and operation whose result is greater than its operands (as opposed to "shrinking" operators like -, /, square root, etc.).

    2 + 2 = 4
    2 x 2 = 4
    2 ^ 2 = 4
    2 # 2 = 4 (where we could suppose x # y means x raised to itself y times).

    Wouldn't this trend go on indefinitely? Couldn't we say that 2 @ 2 = 4 where @ is any "growth" operator whatever?

    And wouldn't a similar rule apply where 4 % 2 = 2 where % is any "shrinking" operator whatever?

    NOTE: I'm aware that I've defined "growth" operations poorly--for example, 10 x 0.5 = 5 (5 is hardly greater than 10)--but I think it's good enough to get the idea across that +, x, ^ seem to be one group of operators that have a common feature in their results, and similarly for -, /, square root.


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  3. #2  
    Forum Junior JoshuaL's Avatar
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    You could limit the idea to natural numbers to solve issues like 10 * 0.5, since 0.5 is not a natural number. I can't really say more than that since I'm not a mathematician, but I like the idea. Does anyone else know of other operations to test this out on?


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  4. #3  
    Forum Junior epidecus's Avatar
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    Yes... depending on what you mean. The trend is indefinite with respect to what? You haven't really given an adequate set of operations for us to observe. I'm going to do this in an intuitive, non-rigorous, informal, please-don't-yell-at-me-mathematicians kind of way.

    Consider the common binary operations, and let's restrict the operands to integers only. This allows us to define an ordered sequence for the operations in which each is an iteration of the previous function. for is the binary growth operation in which and are the operands. denotes what "level" the operation is in the iterative sequence.

    : Addition
    : Multiplication
    : Exponentiation
    ... and so forth.

    For sake of a shorter answer, I'll only show the iterative equality where one of the operands is 2 (for exponentiation and higher operations, the base remains arbitrary with the corresponding hyper-power being 2)...

    Multiplication :
    Exponentiation :
    Tetration :
    ... and so forth.

    Knowing this, from an intuitive view, we see that the operation for any level can be "broken down" to a lower and lower level until we reach . So no matter what "level" of operation we are at, 2@2 will equal 4. And we can see this thanks to the iterative definition.
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  5. #4  
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    Quote Originally Posted by epidecus View Post
    Yes... depending on what you mean. The trend is indefinite with respect to what? You haven't really given an adequate set of operations for us to observe. I'm going to do this in an intuitive, non-rigorous, informal, please-don't-yell-at-me-mathematicians kind of way.

    Consider the common binary operations, and let's restrict the operands to integers only. This allows us to define an ordered sequence for the operations in which each is an iteration of the previous function. for is the binary growth operation in which and are the operands. denotes what "level" the operation is in the iterative sequence.

    : Addition
    : Multiplication
    : Exponentiation
    ... and so forth.

    For sake of a shorter answer, I'll only show the iterative equality where one of the operands is 2 (for exponentiation and higher operations, the base remains arbitrary with the corresponding hyper-power being 2)...

    Multiplication :
    Exponentiation :
    Tetration :
    ... and so forth.

    Knowing this, from an intuitive view, we see that the operation for any level can be "broken down" to a lower and lower level until we reach . So no matter what "level" of operation we are at, 2@2 will equal 4. And we can see this thanks to the iterative definition.
    wow, you really broke that thing down
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  6. #5  
    Forum Junior epidecus's Avatar
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    I see what you did there
    Last edited by epidecus; October 23rd, 2012 at 03:58 PM.
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