How Do You Check Your Computations?

Question

I am already a grad student, but sadly I still have problems with careless computations. In my most recent mid-term (multivariable analysis), I lost 21 points out of 100 because of calculation errors and other cringe-worthy mistakes. If you ask me to explain proofs, I don't believe there will be a problem, but I have no "feels" for computing results. A lot of times, when I have an error, I just don't see it.

I admit too that sometimes I detest calculations and just want to get over with them. Unlike learning a proof, there is no understanding in calculating a result. I still remember learning Taylor series in Calculus -- hated it. After tons of calculations, I had no idea if I was right or wrong, did not know how to check it, nor understand why I was doing a series expansion in the first place. (I got every question in my homework wrong because I could not keep count on the index. The grader gave me 1 point for turning in something ....)

Anyway, there is no use whining; a wrong answer is a wrong answer. Also, I cannot afford to lose 20 points on each exam. What self-check mechanism do other MSE users have?

Answer 1

I have one method for you.But this is only to check if multiplications(that you do by hand) are correct.I mean, it will only help a quick check whether or not you have done the simple additions and products in a long multiplication correctly.Say, you have

$$a_1a_2\dots a_n\\\times b_1b_2\dots b_m \\ -----\\c_1c_2\dots c_p$$

$a_i,b_j \ \&\ c_k$ represent the digits of the multiplicand, multiplier and the answer respectively.

The rule is :

Define $A = \Sigma a_i$. If $A\ge10$, redefine $A=\Sigma$ Digits of previous $A$ ; and continue redefining $A$ until $A$ is some number below $10$.

Similarly define and (if necessary) redefine $B$ to a number below $10$. And same for $C$.

Now let $D=A\times B$. Redefine $D$ (similarly) as above to a number below $10$.

If $D=C$ , your answer is correct.

Please don't ask for a proof or reference. It's a method I gathered from a primary teacher and has worked well for me.