HOW TO AVOID CARELESS MIsTAkeS
This is a skill set in itself and some of the ways to avoid careless mistakes are helpful in becoming a better mathematician.
The following techniques can help:
The following techniques can help:
- Solve a problem in two or more ways. If you do that and get the same answer, the chances of having made a careless error go way down. This is also a good technique to learn more mathematics.
- If there is only one way to solve it, just solve it twice without referring to your first derivation when you do the second one. Less reliable than using a whole different method, but better than nothing.
- Make sure each step in your derivation is correct. Have a habit of "auditing" some of the steps so that you have a sense of how often you make a careless mistake.
- Dimensionality analysis as a sanity check for your final answer. This is more relevant to physics problems but applies to some math problems.
- Check corner cases. If your answer is a formula, make sure that for simple cases where you know the true answer, the formula agrees with it. Even if the problem you are solving isn't asking for a formula but a numerical answer, you can make it so, by considering the inputs of the problem as variables.
- Reverse your steps. This is especially easy for some calculus problems, which the original question focuses on. For example, if you calculate an indefinite integral, take the derivative of your answer and see if you get back to the original function.