Which general physical transformation to the number space does exponentiation represent?

Question

Addition and multiplication may be defined in two ways, one specific and one general:

Addition

• specific: addition is repeated incrementation.

This is specific and sub-optimal as while $2 + 4$ is defined, $2 + 1.3$ is not, as you cannot repeat an action $1.3$ times.

• general: addition is shifting the number by the number.

For example $2 + 1.3$: Keeping the position of the number line fixed, I move the number $2$ $1.3$ units in the positive direction (to the right). This makes physical sense and you can actually build the number spaces (lines for simplicity) and move them around.

For complex numbers, the imaginary component is just moving up and down instead of right and left for the real component.

Multiplication

• specific: multiplication is repeated addition

Same as above, how do I multiply by a decimal number?

• general: multiplication ($a * b$) is stretching the segment that goes from $0$ to $a$ until the segment from $0$ to $1$ becomes as long as $b$

Again, this works for decimals and makes physical sense as you can imagine taking a rubber segment and pulling the extremes apart to stretch it and compare it to the original number line.

Multiplying by an imaginary is rotating instead, that you can also easily imagine doing (or actually do) to a plane.

Exponentiation

• specific: exponentiation is repeated multiplication

Same as above, how do I exponentiate by a decimal number?

• general: ???

• • What is a physical representation of exponentiation that gives a clear yet rigorous and general description for it?
• • Could you also talk about exponentiation by complex numbers?
• • In other words, if I gave you a rubber complex plane (or more simply, real number line) and asked you to raise a number to the power of another number, what would you do?
• • How would you deform it to represent the operation in an intuitive way?

Answer 1

There are actually two answers to your question, because exponentiation is the first hyperoperation that is not commutative. Addition and multiplication are commutative, so they can only be curried in one way. To explain what I mean better, look at multiplication. We have $f(x, y) = x \times y$. From this, we can define "multiplication by $x$" as $m(x) = \lambda y \rightarrow f(x, y)$ i.e. it is the function that takes a $y$ and maps it to $x \times y$. But we could also have chosen $m(x) = \lambda y \rightarrow f(y, x)$. However, since multiplication is commutative, it doesn't matter which we choose. It's the same for addition.

OK, so what about exponentiation? Here, there is a big difference: exponentiation is not commutative. This may be obvious, but I'll give an example anyway: $3^2 = 9 \neq 8 = 2^3$. Right, so we've established that commutativity is lost. So how does this change things? Well, it means that you now have to consider two functions. Setting a constant $a$, we can either look at the function $p(x) = x^a$ or $g(x) = a^x$. Generally speaking, the latter is called an exponential and the former a polynomial, but both use exponentiation. The difference is that in a polynomial, the base varies and the exponent remains fixed, whereas in the latter it is the base that remains fixed while the exponent varies. So to answer your question, we need to consider these two completely different functions. To keep things simple, I'll just focus on the real line.

Polynomial function: Consider the function $p(x) = x^a$. What does it do to the real line? We'll focus just on the positive real line: the negative real line is very interesting in its own right, but we end up with complex values for non-integer $a$, so to keep things simple I'll omit discussion of this for now. Right, so for this function to be interesting we have $a \neq 0$, otherwise we just have $p(x) = x^0 = 1$; in other words, all points get squashed onto the point $1$. If $a$ is negative, then we must consider first what the function $\lambda x \rightarrow \frac{1}{x}$ does, so we'll leave this too for now. So focus on a positive $a$. What does it do? There are three cases to consider:

• $a=1$ is boring. It just leaves all points untouched.
• $0 < a < 1$ can be visualised as follows. Imagine the point $1$ as a sort of gravity well whose strength increases as $a$ gets smaller. All points get "sucked in" towards $1$, from both the right and the left hand side. The smaller $a$ is, the more severe this "sucking" is, until at $a = 0$, all points get sucked into a singularity. At the other extreme, $a=1$ gives us the boring case above, where there is zero gravity.
• $a > 1$ can be visualised similarly to the above, but with all points being repelled away from $1$. Just think, as we were turning the switch from $a < 1$ to $a = 1$, gravity was ever decreasing, then it went out completely, and now as we increase $a$ above $1$, we reverse the direction of the force. The larger $a$ gets, the further the points get pushed away from $1$. But note that no points less than $1$ get pushed onto the negative axis: they just bunch up closer and closer to $0$.

Exponential function: We now explore $g(x) = a^x$. Again, I'll ignore negative $a$ in order to avoid stepping into the domain of complex numbers. I'll also avoid $0^x$ because this is just $0$ everywhere except at $0$, where it's $1$, so it's not really a smooth transformation. That leaves us with a positive value for $a$. Furthermore, we'll just consider $a>1$, because $a=1$ is the boring constant function, and $0 < a < 1$ gives the mirror image of some function $a > 1$, so we just study the latter and note the symmetry. So what does $a^x$ look like over the real number line, for some $a > 1$?

• For $x < 0$, we are looking at the entire negative real axis. This entire axis gets squished onto the open interval $(0, 1)$. No value gets mapped to $0$. We can imagine this as the exponential function being an incredibly strong vacuum cleaner capable of sucking up the entire infinitely long negative real axis into the open unit interval.

• $a^0 = 1$, i.e. the exponential function always maps the value $0$ onto $1$. This is interesting, because it is independent of the value $a$. It is as if the mapping from $0$ to $1$ is more concrete than the rest, which vary with $a$.

• For $x > 0$ the exponential function stretches the real number line in a non-uniform way. There is a subtle but important difference to multiplication. With multiplication, you stretch all points uniformly i.e. you make all distances scale up by the amount $a$. With exponentiation, you stretch more and more the further to the right you go. That is, points get stretched further and further apart the further out to the right you go. Slightly more precisely, imagine multiplying $x + d$ by $a$. You first stretch $x$ out by $a$, then relative to that point, you translate out (add) by $d$, and then stretch it by $a$. Note that the stretching of $d$ by $a$ is independent of $x$ i.e. you only stretch the $d$, not the rest. This is simply the distributivity law for addition and multiplication. Exponentiating $a$ to the $x + d$ is different. You first exponentiate $x^a$, then you stretch this by $x^d$. In other words, the amount by which you stretch in the exponential function, is relative to how much you've already stretched, whereas with multiplication, the stretching is constant.

There's a lot more I could have covered in this answer, particularly extending this reasoning to the complex domain and giving more precise mathematical definitions. However I chose instead to focus in on some special cases in detail, and visualising where things go, rather than trying to cover everything. In light of this there is plenty of scope for others to answer this question and cover these additional concepts. I see Ante Paladin has already made some progress towards this end.

Answer 2

I am certainly not an expert and here I will only give some of my thoughts on the questions you raised and I will do it in the form of a story that will, I hope, touch at least partially every of the questions you raised on the general principles of exponentiation when we are exponentiating in the field of real and complex numbers.

First, I hope that you are aware that the exponential function (let us say which is defined over the real numbers) have its Taylor series representation which is $\exp(x)=e^x=\sum_{n=0}^{\infty}\frac {x^n}{n!}$.

Now, we will write exactly the same representation of the exponential function in the following way $e^x=\lim_{n \to \infty}\sum_{i=0}^{n}\frac{x^i}{i!}$.

So, basically, exponential function $e^x$ is really, viewed in this way, just the limit of the sequence of polynomials $P_n(x)=\sum_{i=0}^{n}\frac{x^i}{i!}$.

So, if you would really like to know in what way does exactly the exponential function deforms the real line then the answer is, clearly, dependent on what piece of the real line do you want to investigate, because the exponential function will not do the same thing to some interval $(a,b)$ as it will do to some other interval $(c,d)$ if we have that $a\neq c$ and $b\neq d$.

So, where do we find the reason for such behavior of this function?

The reason is in that that the exponential function is the limit of the sequence of polynomials, and, if we would like to understand what exactly does the exponential function do to some interval $(a,b)$ we could investigate it in such a way that we investigate what every member of the sequence of polynomials which converge to an exponential function do to interval $(a,b)$ and, as the greater is the degree of the polynomial we should have the better picture what will exactly happen in the limit, or, in other words, what will exponential function do to some interval.

When viewed in this way, the physical transformation which is achieved by applying the exponential function on the real line or on some interval is just one out of infinite number of transformations of the real line, because every function which converges to its Taylor series will represent some transformation, and, because every such function is the limit of the sequence of polynomials, and because two polynomials of different degree do two different transformations it could be that some general trivial description of what, in particular, does the exponentiation do to the real line, will not be achievable.

Now, in the complex numbers case, the situation is, from certain perspective, maybe not so esoteric at all.

We have that in the real numbers case the exponential function is equal to its own Taylor series, so we have $e^x=\sum_{n=0}^{\infty}\frac {x^n}{n!}$.

Now, what is the most natural way to define exponential function over the complex numbers?

Well, if we write instead of the real variable $x$ the complex variable $z$ and on the both sides of the expression $e^x=\sum_{n=0}^{\infty}\frac {x^n}{n!}$ do that change we arrive at the expression $e^z=\sum_{n=0}^{\infty}\frac {z^n}{n!}$.

You could ask what did this change of variable do?

Well, the change is in that that we now have as input the point $z=x+yi$ which is the point in the plane and we have as an output the point $e^z=q+wi$ which is also the point in the plane.

And as the real exponential function was the limit of the sequence of polynomials so is complex exponential function, and as the real exponential function deformed subsets of the line the complex exponential function will deform subsets of the plane and in the complex numbers case also will the polynomials of different degree do different change of form, or deformation, of some subset, and the story in the complex numbers case when talking about physical interpretation of the exponentiation are similar when talking about what will the exponentiation of some part of the plane do to that part of the plane? It will do different things which depend on what exactly part of the plane did we choose to investigate.

It could/should be important to be aware that since exponential function both in the real and complex case is limit of the sequence of polynomials that the change of form of some part of the domain set, be it part of the real line in the real numbers case or part of the plane in the complex numbers case, can be approximable to whatever precision we want by some polynomial in the sequence of polynomials that has exponential function as its limit, and so it could be that you should investigate polynomials to gain better understanding of the functions which are limit of them.

I hope that some things are now clearer to you.

Answer 3

I found this Complex Function Plotter to be indispensable in helping me visualize the general physical transformation applied to the complex plane by $f(x)=x^n$. Basically, it expands AND rotates space.

For real values of n, in terms of expansion, starting points are exploded away from the perimeter of the unit circle. Space inside the unit circle (e.g. positive reals between 0 and 1) squeezes toward the origin; space outside of it exploded (e.g. positive reals greater than 1). At the same time, points rotate away from the positive real line. The further a point is from the positive real line, the greater the rotation (just like points expand outward at a distance proportional to their starting distance from the unit circle). The greater n is above 1, the greater these effects. At n=2, this means a starting point like -2 (which starts at max rotational distance from the positive real line) gets rotated all the way around (as it expands) to 4. At n=3, negative reals get rotated 360 degrees and land on the negative real line again (e.g. $-2^3=-8$).

On the other hand, as n decreases from 1 toward 0, the opposite happens: starting points are simultaneously imploded toward the perimeter of the unit circle and rotationally sucked toward the positive real line. At n=0, this process reaches its logical conclusion: every starting point has been sucked toward the one point that is both on the unit circle and on the positive real line (it's 1!).

When n is negative, everything I've described still holds, but the entire space is inverted: meaning that everything that starts outside the unit circle inverts inside of it, and everything that starts inside it reverts outside. It's a simple concept but difficult to picture, especially while trying to picture the effects of the expansion and rotation I've already described. So it's hard to put, for example, $f(x)=x^{-0.5}$ into plain terms.

Explaining what general transformation the complex part of n applies is even tougher: it's a kind of spiral effect. Because rotational displacement seems to vary with both rotational distance from the positive real axis and distance from the perimeter of the unit circle, the space is transformed into a spiral reminiscent of a conch shell.