Inventing Mathematics
A hunt for a new operation
Introduction
In school, we are taught a couple of basic operations: addition, subtraction, multiplication, division, and maybe exponentiation. Only later we find out that really, subtraction is addition by a negative number and division is multiplication by a fraction.
That leaves us with addition, multiplication, and exponentiation.
But of course, the story doesn’t stop there because there are relations between the operations.
For example, addition and multiplication have an interesting relationship via the distributive law:
- a â‹… (b + c) = a â‹… b + a â‹… c.
So multiplication distributes over addition. But actually, multiplication and exponentiation have a similar relationship in that exponentiation distributes over multiplication.
- (a â‹… b)^c = a^c â‹… b^c
These relations show that the relationship between exponentiation and multiplication is similar to the relationship between multiplication and addition (the operations themselves are of course very different) and exponentiation is not commutative i.e. a^b is not necessarily the same as b^a but multiplication and addition are commutative.
