Mean

Not sure this will answer your question, but it caused me to think: is the arithmetic mean the position that minimizes the distance between itself and all the data? One can treat this as a simple minimization problem, and indeed solving for that unknown position recovers the arithmetic mean. Neat. More on reddit.com

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Greg Egan in one of his stories wrote something to the effect of mathematics being a study or catalogue of everything that is not self-contradictory. A similar thought was presented by Michio Kaku, that mathematics is "the set of all possible self-consistent structures". One may want to add that these "everythings" and structures should be mathematically interesting, which returns us to the question of what mathematics is. Some other descriptions of mathematics that I like, which aren't necessarily entirely satisfactory, but are nonetheless compelling: mathematics is the study of a priori truths, using rigorous deductive logic; mathematics is the study of abstract phenomena that recur in otherwise unrelated situations. I say "[not] entirely satisfactory" because these descriptions may not capture everything we think about maths, or subjectively they may leave too much up to interpretation. But at least they're suggestive to people who are already familiar with mathematics at large. More on reddit.com

It's unlikely that the parts of the questions that say: "Define as" or "Define to be" are asking you to do anything. I know it sounds like they're telling you to do something, but what those phrases really mean is that they have created a name for something and are telling you what that name means. For example, if I said "Define the function f(x) to equal 1 when x > 0 and 0 when x <=0", then there's nothing for you to do. I'm basically telling you how to interpret expressions such as f(3). When you see f(3), I've just told you that f(x) = 1 when x > 0, therefore f(3) = 1. If I said define a sequence a(n) so that a(0) = 1 and a(n+1)=a(n)+3, I've just told you how to interpret things like a(1), a(2), a(3). So I'm not telling you to actually do anything, I'm just giving you information so that you can understand the next things that I write. We don't really do this in English because we don't really need to be constantly coming up with new names and symbols for things. But if I said to you that from now on when I say the word "ploof", I mean a "fluffy dog", then that would be the same as the mathematic directive: "Define 'ploof' to mean 'fluffy dog'". Note: I'm just guessing here. I could very well be that a question is asking you to come up with your own definitions. I'd have to see the entire question and surrouding context to be sure. More on reddit.com