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When dealing with negative numbers, it’s essential to understand how they behave in arithmetic operations. Negative numbers are numbers less than zero and are represented with a minus sign (-) in front. For example, -5 is five units less than zero. These numbers can make calculations a bit tricky if we aren't careful.
In the problem \(25 - (-12) - 7 - (-2) + 9\), we encounter negative numbers inside parentheses. The double negative (e.g., \(-(-12)\)) can be simplified: two negatives make a positive. So, \(-(-x)\) is the same as +\(x\).
To simplify, change double negatives to positives:

• \(25 - (-12)\) turns into \(25 + 12\)
• \(-(-2)\) turns into \(+2\)

This transformation makes the entire expression friendlier to handle!

Addition and subtraction are fundamental arithmetic operations.
After converting double negatives to positives, we need to execute addition and subtraction correctly. In the expression \(25 + 12 - 7 + 2 + 9\), it’s best to follow a systematic approach:
Add and subtract numbers from left to right:

• First, add \(25 + 12\), which equals \(37\)
• Next, subtract \(7\) from \(37\), resulting in \(30\)
• Then, add \(2\) to \(30\), making it \(32\)
• Finally, add \(9\) to \(32\), reaching the final answer \(41\)

This careful step-by-step process ensures we don’t miss any details and arrive at the correct result.

It is crucial to follow the order of operations rules, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)). However, in our problem, parentheses play a critical role with negative numbers.
For an expression like \(25 - (-12) - 7 - (-2) + 9\), recognize that double negatives in parentheses simplify first. This is part of the 'Parentheses' rule in PEMDAS. Once simplified, the expression becomes \(25 + 12 - 7 + 2 + 9\).
Following addition and subtraction from left to right as per the 'A' and 'S' of PEMDAS:

• Combine \(25\) and \(12\) to get \(37\)
• Then subtract \(7\) to get \(30\)
• Add \(2\) to get \(32\)
• And finally add \(9\) to get \(41\)

Following these steps in order ensures accuracy and simplifies complex calculations.