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
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.
Latest Answers