How to denote computing parts of a larger formula, without having to rewrite this formula over and over?

Question

When I was a school kid, we used to have to compute "formulas" like this one:

$$2+5+3+3+6+4=?$$

We were supposed to write down every single computation we were making, so the full solution that was expected from us looked like this:

$$2+5+3+3+6+4=7+3+3+6+4=10+3+6+4=13+6+4=19+4=23$$

This included rewriting the formula over and over. I didn't like it at all. So I resorted to simply write the particular computations I was making:

$$2+5=7+3=10+3=13+6=19+4=23$$

So much more compact, isn't it? But, of course, my teachers were not impressed. They told me this was incorrect and I was supposed not to do my computations this way. I was groaning and bemoaning: to me, that requirement seemed most artificial.

Well, I now understand how incorrect such "notation" is. But still, even now, I may write my computations when I do them by hand. Rewriting the whole formula over and over is simply too much hassle.

But what is the standard solution to this kind of problems? I somehow doubt mathematicians have a likening of rewriting the whole possibly complex formulas over and over?

Answer 1

Your summation $$ 2+5+3+3+6+4=7+3+3+6+4=10+3+6+4=13+6+4=19+4=23 $$ IMHO might be written as $$ 2+5+\dotsb =7+3+ \dotsb=10+3 + \dotsb =13+6+ \dotsb =19+4=23 $$ to keep the meaning of "=", i.e. that the left and right sides of each equation $L = R$ are equal.

Answer 2

I wouldn't disagree with

$$((((2+5=7)+3=10)+3=13)+6=19)+4=23$$

though this is not common practice. Logicians would wonder why you are adding truth values to numbers.

Answer 3

Simply use braces: $$ \underbrace{\underbrace{2+5}_{=7} + 3}_{=10} + \dots $$

Answer 4

You could use associativity and write $$ (2+5)+(3+3)+(6+4)=7+6+10=\cdots, $$ or maybe even $$ \underbrace{\underbrace{\underbrace{\underbrace{\underbrace{2+5}_{=7}+3}_{=10}+3}_{=13}+6}_{=19}+4}_{=23}. $$