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Comment by jordanpg | Hacker News Reader

Agree with this and would also add: in addition to being able to do algebra fast and have it be second-nature, at higher levels it helps if you are able to do it 100% accurately. 5 nines isn't enough.

I didn't appreciate this until I was a graduate physics student and was frequently frustrated by minor algebra errors in long, complex problems.

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What's important isn't being able to do it right quickly the first time. It's having a strong habit of checking your work as you go, both at a low level and a high level. (Kinda like unit testing and integration testing...)

Fast arithmetic and algebra are fairly particular skills. As one progresses into new world of math, the calculations and operations may be completely different from what's been practiced, making the fast math less useful. But a good habit for careful progress is completely general!

I've known a lot of mathematicians, some of whom were really good, and five nines (one-in-one hundred thousand) is an implausibly high standard for humans to reach. I'm certainly nowhere near there.

What's worked for me is to develop an intuition about how things 'should' look, so that if I do make a mistake I'll get a 'hey, that's not right' feeling a couple steps later.

Just to expand on this, when I worked as a grader in college I noted on engineering test keys the professor's answers were always roughly three short lines of handwritten equation. But the student's answer section were often totally darken with pencil they wrote so much. The heuristic was definitely the longer the answer the more wrong it was.

This is something I've taken to heart professionally too. If the equations are getting out of hand it's either wrong or I need some simplifying assumptions (like that cow needs to be spherical). Otherwise one just can't keep track of it and reason about it.

If I need a more precise answer, that's what computer numerics is for.

Perfect accuracy is an insane goal. Much better is to develop the skill of finding and correcting your mistakes. Part of this is doing frequent "sanity checks" - for instance, if you are computing an area, and you get the answer -3, probably you made an algebra mistake somewhere earlier, and you should find and fix it before going on (some ways of checking your work also help you pinpoint where the mistake occurred, others are less helpful for that). Another way to check your work is to try plugging in numerical values for all of the variables and going through the arithmetic to check all of the steps (if you need to check your arithmetic, try something like "casting out 9s").

Learning to find and correct your mistakes will at first make it take ten times as long to solve any problem as just scribbling your way to an answer. But if you practice it, you will discover two things:

- with practice, the overhead of finding and correcting mistakes drops significantly,

- the feedback you get from noticing your mistakes helps you avoid making those mistakes in the future. Eventually mistakes become extremely rare, and you can go through a very long computation without making a mistake, even if you don't bother checking for them any more!

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