Random Stuff III. We didn't need this.

But since I'm posting here, I'll mine as well ask people why showing your work with basic math (basic to you, not me) is so important.

It depends on what you mean by basic math. I usually think in big steps. I try to get the intuition behind a problem first, and then make sure that each step in the reasoning actually works. When you break things down small enough, you know that something is obviously true and you don't need to show work for it anymore. If you're in fourth grade multiplying 56 by 47, you probably don't need to show by hand that 6 times 7 is 42 (by actually adding 6+6+6+6+6+6+6), because you're at the point where you've memorized that already. If you're in algebra you may not be expected to show how you got 15*3=45 because it's expected that you can do that easily at this point. If you're in calculus you won't need to show every step in solving an equation like 5x+12=2x-6 because it's assumed that you can do that easily.

After a certain point, there are diminishing returns for showing your work. There are all of the advantages Tables mentioned, but if you (and your audience) can do something so consistently that you're unlikely to make a mistake with it anyway, then writing down all the details takes more time than it will save you in the long run.

Of course, this really only applies to more computational math. In more proof-oriented math, you could say that the entire problem itself is to show your work. But even then, you generally assume that your audience can accept a lot of the basic details without you stating them, or else you at least refer them to other work that has worked out those details for you.