Older student going back to basics

[logicandtruth]

TL;DR Summary

General operations query

I am trying to imporove my general maths and revisting my pre-college text books. For the above example question, my answer was 4 which is not correct as it is -4. The only difference I can see in my working is the first operation 3a^2. Here I simply did 3 x -2^2 which is -12 but in the answer working in blue font it shows (-2)^2 in brackets. This of course changes the answer due to BODMAS as 3(-2)^2 is 12 as -2 x -2 = 4. But is it correct to simply inser a bracket when the question does not show one?

 

[anuttarasammyak]

I see your point is "what is $a^2$ where a=-2".
$a^2$=a x a = -2 x -2 = 4
Here "x-" is ambiguous : multiply or subtract ? So in multiplication we put minus number in bracket.
$a^2$=a x a = -2 x (-2) = 4

 

[Ibix]

To phrase it another way, squaring a number means multiplying it by itself. By taking the minus outside the square, you've changed the meaning to "the square of the absolute value of the number, then multiplied by -1 if the original number was negative".

Another way to look at it is that $a^2=(a)^2$. So therefore the square of $-2$ must be $(-2)^2$.

 

[jbriggs444]

Coming from a computer science background, I see this in a bit of a different light -- as a question about parsing. In the absence of BODMAS, the grammar of mathematics is ambiguous.

When we make the substitution for "a" with "-2", we are replacing a one character string with a two character string. If we want the new expression to parse the same way as the old expression, we need to arrange for the "-2" to be treated as a single element.

So we parenthesize the "-2". The grammar of mathematics treats parenthesized sub-expressions as separate entities, so our goal is achieved.

But yes, the simple answer is that $a = (a)$ is a valid rule of mathematical expression evaluation.

 

[logicandtruth]

Thank you all very much for your responses. It is now much more clearer and I now understand the treatment of the operation with the parenthesis.