What Is Exponentiation?

Is there an intuitive definition of exponentiation?

In elementary school, we learned that

ab=aaaa(b  times)
where b is an integer.

Then later on this was expanded to include rational exponents, so that

abc=cab

From there we could evaluate decimal exponents like 43.24 by first converting to a fraction.

However, even after learning Euler’s Identity, I feel as though there is no discussion on what exponentiation really means. The definitions I found are either overly simplistic or unhelpfully complex. Once we stray from the land of rational powers into real powers in general, is there an intuitive definition or explanation of exponentiation?

I am thinking along the lines of, for example, 2π or 32 (or any other irrational power, really). What does this mean? Or, is there no real-world relationship?

To draw a parallel to multiplication:
If we consider the expression e5, I could tell you that this represents the area of a rectangle with side lengths e cm and 5 cm. Or maybe eπ is the cost of π kg of material that costs e dollars per kg.
Of course these quantities would not be exact, but the underlying intuition does not break down. The idea of repeated addition still holds, just that fractional parts of terms, rather than the entire number, are being added.

So does such an intuition for exponentiation exist? Or is this one of the many things we must accept with proof but not understanding?

This question stems from trying to understand complex exponents including Euler’s identity and 2i, but I realized that we must first understand reals before moving on the complex numbers.

Answer

My chief understanding of the exponential and the logarithm come from Spivak’s wonderful book Calculus. He devotes a chapter to the definitions of both.

Think of exponentiation as some abstract operation fa (a is just some index, but you’ll see why it’s there) that takes a natural number n and spits out a new number fa(n). You should think of fa(n)=an.

To match our usual notion of exponentiation, we want it to satisfy a few rules, most importantly fa(n+m)=fa(n)fa(m). Like how an+m=anam.

Now, we can extend this operation to the negative integers using this rule: take fa(n) to be 1/fa(n). then fa(0)=fa(nn)=fa(n)fa(n)=1, like how a0=1.

Then we can extend the operation to the rational numbers, by taking fa(n/m)=mfa(n). Like how an/m=man.

Now, from here we can look to extend fa to the real numbers. This takes more work than what’s happened up to now. The idea is that we want fa to satisfy the basic property of exponentiation: fa(x+y)=fa(x)fa(y). This way we know it agrees with usual exponentiation for natural numbers, integers, and rational numbers. But there are a million ways to extend fa while preserving this property, so how do we choose?

Answer: Require fa to be continuous.

This way, we also have a way to evaluate fa(x) for any real number x: take a sequence of rational numbers xn converging to x, then fa(x) is lim. This seems like a pretty reasonable property to require!

Now, actually constructing a function that does this is hard. It turns out it’s easier to define its inverse function, the logarithm \log(z), which is the area under the curve y=1/x from 1 to z for 0<z<\infty. Once you've defined the logarithm, you can define its inverse \exp(z) = e^z. You can then prove that it has all the properties of the exponential that we wanted, namely continuity and \exp(x+y)=\exp(x)\exp(y). From here you can change the base of the exponential: a^x = (e^{\log a})^x = e^{x\log a}.

To conclude: the real exponential function \exp is defined (in fact uniquely) to be a continuous function \mathbb{R}\to\mathbb{R} satisfying the identity \exp(x+y)=\exp(x)\exp(y) for all real x and y. One way to interpret it for real numbers is as a limit of exponentiating by rational approximations. Its inverse, the logarithm, can similarly be justified.

Finally, de Moivre's formula e^{ix} = \cos(x)+i\sin(x) is what happens when you take the Taylor series expansion of e^x and formally use it as its definition in the complex plane. This is more removed from intuition; it's really a bit of formal mathematical symbol-pushing.

Attribution
Source : Link , Question Author : baum , Answer Author : Gyu Eun Lee

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