For overambitious apes like us, adding integers is the easiest thing in the world. Take one berry, add another, and you have two. Each remains separate, you can lay them in a row and count them one by one, each distinct thing adding up to a group of distinct things.
Other things in math are less like berries. Add two real numbers, like pi and the square root of two, and you get another real number, bigger than the first two, something you can write in an infinite messy decimal. You know in principle you can separate it out again (subtract pi, get the square root of two), but you can’t just stare at it and see the parts. This is less like adding berries, and more like adding fluids. Pour some water in to some other water, and you certainly have more water. You don’t have “two waters”, though, and you can’t tell which part started as which.
Some things in math look like berries, but are really like fluids. Take a polynomial, say . It looks like three types of things, like three berries: five , six , and eight . Add another polynomial, and the illusion continues: add and you get . You’ve just added more , more , more , like adding more strawberries, blueberries, and raspberries.
But those berries were a choice you made, and not the only one. You can rewrite that first polynomial, for example saying . That’s the same thing, you can check. But now it looks like five , negative four , and seven . It’s different numbers of different things, blackberries or gooseberries or something. And you can do this in many ways, infinitely many in fact. The polynomial isn’t really a collection of berries, for all it looked like one. It’s much more like a fluid, a big sloshing mess you can pour into buckets of different sizes. (Technically, it’s a vector space. Your berries were a basis.)
Even smart, advanced students can get tripped up on this. You can be used to treating polynomials as a fluid, and forget that directions in space are a fluid, one you can rotate as you please. If you’re used to directions in space, you’ll get tripped up by something else. You’ll find that types of particles can be more fluid than berry, the question of which quark is which not as simple as how many strawberries and blueberries you have. The laws of physics themselves are much more like a fluid, which should make sense if you take a moment, because they are made of equations, and equations are like a fluid.
So my fellow overambitious apes, do be careful. Not many things are like berries in the end. A whole lot are like fluids.
If I add one Laguerre polynomial to any other Laguerre polynomial, is the summation unique?
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