B Older student going back to basics

AI Thread Summary
Improving general math skills involves revisiting foundational concepts, particularly the importance of BODMAS in operations. The discussion highlights the significance of parentheses in mathematical expressions, as they clarify the order of operations and prevent ambiguity. Squaring a negative number requires proper notation, such as using brackets to indicate that the negative sign is part of the base being squared. The conversation also touches on the interpretation of mathematical expressions, emphasizing the need for clarity in notation to avoid miscalculations. Overall, understanding these principles enhances comprehension and accuracy in mathematical problem-solving.
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General operations query
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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?
 
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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
 
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##.
 
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.
 
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.
 
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