What does cheating mean in math terms?

Answer 1

"Cheating" is a colloquialism often used by mathematicians whenever they utilize previously derived results that are too advanced mathematically for their audience to understand in order to solve a problem. In reality, the usage of these results is perfectly valid since they have all been mathematically proven somewhere down the line. A better term might be "exploiting," but even that suggests an abuse of mathematical rules, which in actuality, is not happening.

For example, consider the trivial differential equation, dy/dx = x. To solve this, a math teacher will usually "cheat" by multiplying both sides of the equation by dx and then integrating. Well, the legality of multiplying something by an infinitesimally small number can't really be proven until one studies the mathematical discipline of real analysis, which is far more advanced than differential equations. A math professor certainly isn't going to open up that can of worms in a differential equation class, so instead, they just "cheat."