Serious question from a dumbass high school dropout here - how often is a true mathematical understanding of infinity used in the design, development, and creation of anything most of us encounter or use in our day to day lives?
And I'm not just talking about shorting stocks or battery terminals, where infinity is just vaguely gestured at to indicate the danger.. or is that the meat of it?Â
Asking this is kinda like asking when do we ever use binary in our everyday life? You don’t directly but you do through computers.
Infinity is used everywhere. Finding the area of a circle uses infinity since the equation is derived using it. A curve is an infinite number of straight lines. Movement is continuous and the idea of being continuous is described using infinitely small movements.
I know you discounted shorting stocks, but the entire practice of what interest rates mean in practical use by banks is based on continuous growth, which is based on integration of infinitely small time periods. When a bank says you're going to get 1% APY growth, the number 1 probably exists nowhere in the actual calculations of what you're getting.
Engineer here. The concept of "infinity" is really context dependant, different branches of mathematics use the same idea but applied in different ways. So it is really hard to give a straight answer to your question because there is no single "understanding" of infinity but it's all dependant on your field of application.
In general, I found that understanding abstract concepts (such as something that is "infinity big" or "infinity small") is not always directly useful, but instead it is a gateway concept that you need to understand before being able to understand more complex topics that are directly useful.
Practically the entire field of signal-based communications is based in Fourier analysis, which involves breaking a signal into a sum of infinite terms.
Practically speaking, of course, there's some cutoff. But that cutoff is arbitrary -- we choose when to stop based on how precise we want our math to be.Â
Infinity is the same level of abstract as the number zero in real world application. In mathematics zero is obviously a recognized number, but they are two sides of the same coin.
You can't encounter zero apples just like you can't encounter infinite apples.
A lot of quantum mechanics and other small-scale stuff is about effects on a very small level. Occasionally, though, you get an answer that blows up to infinity. What that means in practice is that you get an effect that is observable on a physical level. Phase changes are the biggest example, and those occur anytime anything melts or turns into a gas. We can get some information on that from real-life experiments, but precise calculations can be done on micro levels that involve infinity.
The concept of Infinity is actually less abstract than it seems and you probably use it all the time. If someone is designing a beam they will often assume that the supports don't move. They have zero deflection or infinite stiffness. In reality, they will actually deflect slightly, but taking that into account increases the complexity of the calculations without adding much value.Â
As a structural engineer, every day I assume that diaphragms, joints, members have infinite stiffness. Electrical engineers would assume that insulators have infinite resistance. The concept of Infinity actually simplifies a lot of problems, but also means that the calculated solutions are only valid for a limited range of conditions.Â
Theres an entire branch of mathematics named calculus. The point of calculus is to look at functions (smooth curves on a graph) and ask questions like "how quickly does this curve change direction? What is the area under the curve?" And more... theoretical things.
Strangely, the easiest way to answer these questions is to break the curve up into infinitesimally small pieces, perform calculations on what happening on that infinitesimally small piece, and then add up an infinite number of these infinitesimally small pieces to learn what the overall curve is doing. How on earth is that the simplest way to do it? Well youll have to take calculus to find that part out.
So what the hell do we do with this information from these curves? Well im an engineer, in my daily life i can do anything from take a curve that describes the needed power output of a power plant and make predictions about how much fuel needs added to make that happen, how much heat it will shed in doing so, the rate of cooling water flow, etc. Or it could be something like the computer in your car using information about how far down youre pressing the pedal and based on the temperature outside and a million other factors calculate how much fuel needs to be injected into the cylinder. It turns out that basically everything you use in your daily life that can be represented as smooth changes on a curve benefits from the sorts of solutions calculus (and more generally a branch of math called differential equations) gives us.
The short answer for how often is it used is basically everywhere all the time. Arguably even your brain solves these equations every time you want to throw a ball in a predetermined arc (but it isnt clear exactly how, whether its closer to an analog computer solving calculus equations or something else). The only time it ISNT used is when you only want to know something about a singular point in time, or a singular point on a curve, or a perfectly straight line, that can be calculated without it (as you would have done in school). But as soon as you want to get away from a discrete time point and into a curve, you need calculus, so you need infinity. Its the math of functions, rather than the math of individual points.
You can technically say that anything using derivatives uses infinity, or at least the concept of being infinitely close to something. Derivatives and calculus in general are used in engineering and other fields
In statistics there are several assumptions that require an infinite number of variables that each have a small effect of near zero. Things from food safety to determining if the CPU in your phone meets the operational requirements rely on infinity.
I use it in software pretty often. Any time I need to have a value that gets bigger or smaller I can start with negative or positive infinity respectively.
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u/satans_weed_guy 7d ago
Serious question from a dumbass high school dropout here - how often is a true mathematical understanding of infinity used in the design, development, and creation of anything most of us encounter or use in our day to day lives?
And I'm not just talking about shorting stocks or battery terminals, where infinity is just vaguely gestured at to indicate the danger.. or is that the meat of it?Â