What is an exponential object?

One interpretation for a power of natural numbers, such as 10^6, is that it is counting a certain collection of functions: in this case, the number of functions from a six-element set to a ten-element set. (One way to think about it is that 10^6 is the number of six-digit combinations from 000000 to 999999, and each combination corresponds to a function from \{1,\ldots,6\} to \{0,\dots,9\}: the first digit is the image of 1, the second digit is the image of 2, etc.)

In general, the number m^n is the number of functions from an n-element set to a m-element set. This is somewhat analogous to standard interpretations of other arithmetical operations:

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What is exponentiation?

Last time, we explored how to extend multiplication from an operation on positive integers to an operation on arbitrary real numbers. The three tools we had at our disposal were:

  1. Find a nice interpretation for the operation, and see how broadly it applies.
    That’s how we moved from multiplying positive integers to multiplying any real number by a positive integer.
  2. Write down the laws the operation satisfies, and try to use them to deduce extra cases where the operation must give a certain value.
    For example, the law of associativity let us deduce the result of multiplying by a rational number, and the law of distributivity gave us the result of multiplying by zero or a negative number.
  3. Use continuity, monotonicity, or other “nice” properties of functions to decide what the operation should give for even more values.
    Knowing how to multiply by rational numbers doesn’t tell you how to multiply by irrational ones, but there’s at most one way to do so that makes the multiplication operation continuous.

Today we’ll explore these ideas forĀ exponentiation instead of multiplication, and the story has a twist that finally explains why (a^2)^{1/2} doesn’t always equal a.

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