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:
- 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.
- 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.
- 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 for exponentiation instead of multiplication, and the story has a twist that finally explains why doesn’t always equal .
Ordinarily, multiplication is interpreted as repeated addition:
But what does it mean to multiply two arbitrary real numbers, such as ?