The exponential function is a self-multiplication operation where is multiplied by itself number of times. However, things get peculiar when we try to determine what is . What is nothing multiplied by itself zero times? Does that sentence even make sense? Surely, zero multiplied by itself any number of times is zero, right? However, isn’t any number raised to zero equal to one?
Until the 19th century, most mathematicians agreed, for reasons we will discuss below, that is equal to 1. However, doubt seeped in when Cauchy, a gem of a mathematician, grouped and in the same column, which he referred to as “indeterminate forms”. So, what then is its value?
Is it equal to 1?
A clever student may cite this proof to prove this claim:
For , .
The student has proved it very slyly, for while the proof appears to be consistent, only someone scrupulous will tell you that it actually isn’t: after the substitution, we are dividing zero by zero, an operation that is meaningless or undefined. While it cannot be proved algebraically, can it be proved analytically?
Consider this trend:
The trend limns the function . Surely, the trend suggests that must be equal to 1?
However, this is not true. The function approaches as the value of approaches , but approaching a value is different from reaching and being exactly it. This does not “prove” that In fact, the minute we substitute for , the proof becomes self-defeating.
This, in fact, represents a neat proof for why does not equal , but !
It is undefined
There is no consensus today about whether the value is or ; one can prove both with equally compelling arguments. Some mathematicians have accepted that it has no definite value, but instead depends on the context in which it is being used. However, there are other mathematicians who repudiate this convenience and are adamant that the value is 1. This is because certain rules require the equation to be necessarily assigned to . For instance, the derivative of the exponential function becomes invalid if isn’t considered to be equal to .
There seems to be no answer either algebraically or analytically. We’ve only run into contradictions, just like one does when trying to determine what is zero divided by zero. The essential contradiction being: zero raised to any positive number must be zero, but any positive number raised to zero must be one; we cannot have it both ways. Mathematicians describe these problems as either meaningless or undefined.