Eli5: Why are exponents so common in scientific formulas? Why is it so common that numbers in nature are multiplied by themselves?
In: Mathematics
All living things procreating creates exponential growth. E.g. A turtle has 10 little turtles, then each 10 of those turtles have 10 turtles, and so on.
Any sort of relationship that depends on the shape or a geometry of a system or process will have exponents that represent different parts of geometry. In other words, if a relationship depends on area, you may see an x^2. If it depends on volume, you may see an x^3.
Some numbers affect a situation in two different ways.
Quick example, when you drop something its fall distance is 1/2at^2.
Time is squared because there are two ways it increases the fall distance: more time means the object is able to cover more distance, *and* more time means the object is able to pick up more speed from gravitational acceleration.
Second example, the light you get from a source is proportional to 1/d^2. The distance “counts twice” because the light is getting more spread out horizontally, and also more spread out vertically. As you get further away it’s like you’re taking the same amount of light and “spreading it thinner” over a bigger and bigger area.
It isn’t obvious to me how to do an explanation like that for every polynomial term, but…there probably is if you really know your shit and are really creative.
Exponential functions are not only useful to describe classical exponential behavior (multiplying numbers by itself a certain number of times). With the right mathematical tools (complex numbers) you can also use them to describe periodic/oscillating things like waves, pendulum movement, etc.
You could use sine and cosine functions for that, but that is often pretty tedious and using exponential functions (in a form like `exp(2π*i*t)`) is more elegant in many cases.
As wave-like things occur all over physics you will also encounter exponential functions all over physics: From pendulum movements, over light waves and optics, up to electronics and quantum mechanics.
Two main reasons that exponents come up are because something depends on the rate of change of something else, or on some kind of geometry.
In the case of uniform acceleration, the constant acceleration is the rate of change of the velocity over time. So the velocity depends on time. The velocity is the rate of change of the position over time, so the position picks up a further time dependence and behaves as t².
Geometry gives us things like the inverse square law. Imagine all of the light emitted by the sun in all directions. It expands outward in a spherical shell that increases in area as it gets farther away from the sun. Area increases as distance squared, so the intensity of the light *decreases* as distance squared because the light is evenly spread out over that expanding area. The same geometry applies to electrostatic and gravitational forces, which also fall off as 1/r².
When the rate of change of something is proportional to *itself*, you get an exponential relationship. You can see this in unconstrained population growth, e.g. of bacteria, where the number of new bacteria depends on the number that were there before and able to reproduce. You also see this in half-life behavior of radioactivity, where the number of particles decaying is proportional to how many particles you have, resulting in exponential decay.
In an oscillating spring or a pendulum, the acceleration (force) depends on the position, with a minus sign. You can translate that into a differential equation (an equation involving both a quantity and its rate of change), and the relationship you get out of that is a sine wave.
So you can get all kinds of mathematical relationships between quantities, depending on how they’re related physically.
The answer lies in how the numbers are used. Take, for instance, the area for a volume of something. For a cube, you multiply length times width times height. This is intuitive. Doing so gets you something like 2m long, 4m wide, 3m tall for a total of 24m^3.
This is NOT one measurement multiplied by itself three times, but three separate measurements all taken according to the same standard (a meter stick) and then combined (multiplied) to tell you the volume of the object.
The volume of a sphere is less straightforward, but results in (4/3)*pi*r^3. Why is the radius cubed here? Because the length, width, and height of the sphere all relate directly to the radius. In this special case, the three measurements are the same, but similar formulas that calculate the volume of an egg require three unique measurements to be taken of a spherical(ish) egg.
For a more advanced example, consider Power. Well, power is measured in watts. Watts are defined as being energy over time, or rather, the force required to move an object multiplied by the velocity of the object – if a device requires 10W to run, that is 10 Newtons of force constantly being applied to a 1kg object traveling at 1m/s.
Where do the exponents come from? Because Newtons themselves (force) is equal to the mass of an object times acceleration…
N=kg*m/s^2
So power can be rewritten as
P=F*v
P=(kg*m)/(s^2)*(m/s)
P=(kg*m^2)/s^3
Why so many exponents?
Because in order to measure the power being applied to the object, you need to know one mass, two lengths, and three separate measurements of time.
Sometimes, some measurements happen to be the same (like in the volume of a sphere). Other times, you need 6 different measurements to get power.
All other formulas are similar. Each measurement adds one variable. Similar variables are combined. That doesn’t mean the measurements are being multiplied by themself, only that the measurements are measuring similar things (lengths, times, masses, etc).
It’s because of how we measure units of stuff.
We represent terms with multiplication & division because it makes it so that we can make our units proportional to each other. If you hit something twice as hard as before, you’re either hitting it twice as fast, or making your fist twice as heavy (or a combination of the two). We also like to base our new units on preexisting ones, and that means as things get more complex, the same types of value start to get mixed in multiple times. Since repeated multiplication is exponentiation, we get exponents.
For example: Acceleration is m/s² because, it’s your increase/decrease in velocity over time. Since velocity is measured in meters per second, 1 unit of velocity per second = 1 meter per second, per second = 1m/s².
Then it just snow balls.
Force = Mass by Acceleration, so 1 newton of force is 1 Kilogram-Meter per Second per Second (kgm/s^(2))
Work is Force by Distance, so 1 joule = 1 Meter-Kilogram-Meter per Second per Second (kgm^(2)/s^(2))
Power is Work over Time, so 1 watt = 1 Kilogram-Meter-Squared per Sec per Sec per Sec (kgm^(2)/s^(3))
etc. etc. etc.
One big reason is that we live in three-dimensional space. When it comes to things like how forces act over distances, such as gravity, you can think of the total “force” being the same at every distance (ie, if you add all of the infinite force vectors at a given distance from a gravitational source, the sum total will be the same regardless of the distance). Since the strength at any distance is essentially proportional to the surface area of a sphere of that radius, the strength must decrease by the square of the distance. Similarly, any other effect or force in 3-dimensional space that has a single point of origin and acts in any direction will also decrease by the square of the distance.
Three general reasons:
1. Calculus. As you make derivatives and antiderivatives (say, distance, speed and acceleration), you often end up with exponents because that’s how calculus works. And the world seems to just work that way.
2. Because something behaves exponentially. One example is where the rate of change is proportional to the amount of the thing, for example population growth. The function that describes that mathematically is an exponential function. Another is diffusion (Arrhenius law), which increases exponentially with temperature.
3. Best fit methods. When you have a whole bunch of data you usually want to find a pattern it follows so that you can both describe it to others, and fill in the gaps where convenient. If you’re lucky, it’s linear. But more often than not you’re going to use an exponential 9r polynomial function to get the best fit. And those involve exponents. But it’s seriously just a choice in this case. You could do weird things with sines and cosines, or any other mathematical functions if you wanted. But exponental and polynomial functions are just easy to work with.
When a tree grows, it starts as a single stalk, and then splits into branches. Each branch then splits into multiple branches. In biology, things are exponential because things keep splitting.
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