• A discussion with T.A.

  • Generalization of operations such as addition, multiplication, exponentiation, etc. in arithmetic

  • From there, I wonder if we can define operations before addition (n<1).

    • Similar to extending the power function to exponential function.
      • The exponential function extends the power function to real numbers. image
  • It seems good to find some broad laws and then narrow them down.

    • First, as an intuition, ack(a,b,n) > ack(a,b,n+1).
      • Multiplying 3 by 3 is bigger than adding 3 and 3, right?
      • I want to prove this.
      • By establishing such laws as a basis, we can extend the definition.
  • https://math.stackexchange.com/questions/35598/why-are-addition-and-multiplication-commutative-but-not-exponentiation

    • In the case of a^b, the roles of a and b are completely different, so it’s not commutative, right?
      • I feel like I had already come to this conclusion myself.
  • A three-dimensional graph as f(a,b)=c

    • Addition

      • image
    • Multiplication

      • image
    • Exponentiation

      • image image
    • When the commutative property holds, a graph that is symmetric about the diagonal of the xy plane is formed.