How are complex mathematical equations “invented”? As in, how do you come up with the formulas and know definitively the answers to them are correct?
In: Mathematics
A lot of them are compositions of known parts that do specific things. For example, you might multiply a variable times a periodic damping term.
You take a lot of measurements. You plot the data. You look for trends, and you find the equations that describe those trends. You then look at what the equation says about measurements you haven’t taken. Those are predictions. Then you go do an experiment that lets you take that measurement and see if it agrees with the prediction. If it does, then you have support for your equation. If it doesn’t, then you have more data to refine your equation. Eventually you get to where you’ve got an equation that only has support and nobody finds a contradiction. At that point, it’s presumed to be correct. But it’s never definitively known that it’s correct, at any time a contradictory observation can nullify it (or require further refinement).
You can sometimes just guess. Say the first few are 1, 2, 4, 8, 16, then 32 might be a good guess. But this can lead astray, because if you continue with 31 then this also does something: draw a circle, a few points, all the lines between them, and count the number of areas.
A second approach comes from guessing or even figuring out the underlying mechanism. If I want the area under a curve then integration does exactly that, and we can show that it has certain properties that are really useful to actually calculate it.
Third, sometimes the formula is just approximation.. Then there are algorithms to find a “simple” formula that goes as close to the data-points as possible. That’s what we often do in natural sciences, especially if we have lots of data.
Lastly, one can also combine all the above. Guess a formula (1st approach) and a mechanism (2nd one), pick what kind of formula you look for and optimize (3rd), and prove/verify/check/do more experiments. Which of the last ones depends on what you do. Proper mathematics has actual proofs from pure logic (and axioms); sciences have to deal with reality and its inaccuracies.
Depends on what your goal is. If it’s math research (“pure math”), then it’s one step at a time. You start with things already known to be true, and slowly transform them until you get something brand new.
If your goal is predicting or modeling a part of the physical world, you’re comparing the math to real world data, then tweaking it based on the errors you observe.
Mathematics is a set of rules, and as long as you follow the rules you know that the result must be correct. This is called a proof.
Normally you don’t just “come up with” some equations. Most of the time you have some question you want to answer, and after putting together all the bits that form the question, and following some rules to simplify and/or solve it, you get the equation you need at the end.
Math is just a language. People who speak the language well can describe many things with it. Given the ‘grammar’ is without exceptions (the rules are completely consistent), it is relatively easy to read the formulas and find mistakes.
Every system of math is built on a set of rules. If you have an idea, then that idea must fit within those rules. If you can follow the rules and prove that your idea is correct, then you have created proof.
This is the case for simple addition all the way up to abstract algebra.
Equations are typically *derived* rather than invented. The all ubiquitous **equals** sign being the most important feature; firmly stating that what is on one side of it, is exactly the same as what is on the other side. Use proper substitutions and prior defined manipulations to determine new, and sometimes, easier to interpret equations. This is mathematics. Much of the other stuff I’m reading on here is physics.
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