The World of Exponents


We all use calculators, spreadsheets, etc. for doing various kinds of calculations like addition, multiplication, division and subtraction. Now let us consider the operation of multiplication. When we multiply same number we are able to represent it in terms of exponents. For example:

2 \times 2 = 2^2
Thus exponents naturally creep into our mathematical notations, because of their short-hand notation and convenience. Now let us consider exponents as an independent operation (like we created multiplication out of repeated addition). Thus:
a^n = \underbrace{a \times a \times \ldots \times a}_\text{n times} \qquad \text{where,} \quad n \quad \text{is a natural number \&} \quad a \quad \text{is an integer.}

Further we say that:

a^{-n} = \underbrace{\frac{1}{a} \times \frac{1}{a} \times \ldots \times \frac{1}{a}}_\text{n times} \qquad \text{where,}\quad n \quad \text{is a natural number \&} \quad a \quad \text{is a non-zero integer.}

Now after handling negative and positive integers we can define for zero:

a^0 = 1 \qquad \text{where,}\quad a \quad \text{is a non-zero integer.}
0^n = 0 \qquad \text{where,}\quad n \quad \text{is a positive integer.}
\text{in fact} \quad 0^n = \text{Not Defined} = \infty \quad \text{if n is a negative integer}

Here I am leaving one case, namely 0^0 which has a very long and controversial history. We will come back to it after some time.

Now since we know other Number Systems let us try to generalize this exponent notation into what we call exponential function as:

f(x) = a^x \qquad \text{where, a is a fixed non-negative real number and x is any real number}

Now let us consider few examples to understand this function; fix a=2, then:

f(2) = 2^2 = 4
f(3) = 2^3 = 8
f(1.5) = 2^{1.5} = 2^{\frac{3}{2}} = 2\sqrt{2}
f(1.765) = 2^{1.765} = ?
f(\sqrt{2}) = 2^{\sqrt{2}} = ??

But if we play with base i.e. a, it isn’t that much interesting for me, for example, set a=\sqrt{2} this is equivalent to a=2^{\frac{1}{2}}

But calculators can calculate all these values easily. HOW? They have inbuilt logarithmic tables!!

Now let us return to our most weird case of generalization:

What is the value of 0^0?

This is a very old question and many big-shots have tried to answer this question, and here I try to provide a big-picture of their attempts:

  •  Euler argues for 0^0 = 1 since a^0 = 1 for a\neq 0. The controversy raged throughout the nineteenth century.
  • Cauchy had listed 0^0 together with other expressions like \frac{0}{0} and \infty - \infty in a table of undefined forms.

But in book Concrete Mathematics [p.162] authors remarked:

We must define x^0 = 1 for all x, if the binomial theorem is to be valid when x = 0, y = 0 and x = -y. The theorem is too important to be arbitrarily restricted. By contrast, the function 0^x is quite unimportant.

But Alex Lopez-Ortiz firmly says:

But no, no, ten thousand times no! Anybody who wants the binomial theorem (x + y)^n = \sum_{k = 0}^n \binom{n}{k} x^k y^{n-k} to hold for at least one non-negative integer n must believe that 0^0 = 1, for we can plug in x = 0 and y = 1 to get 1 on the left and 0^0 on the right.