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Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught in secondary education and includes the study of symbols and the rules for manipulating these symbols to solve problems.

When dealing with negative numbers in algebra, understanding the fundamental rules is crucial. The expression \( \frac{-144}{-12} \) presents a perfect example of how elementary algebra applies these rules. In this case, the symbols \( -144 \) and \( -12 \) represent negative numbers. Elementary algebra tells us that we can manipulate these numbers using division, while following specific arithmetic rules related to negative numbers.

Arithmetic operations refer to the basic mathematical operations that consist of addition, subtraction, multiplication, and division. These are the building blocks for more advanced mathematics.

Division, the operation in question, is essentially the process of determining how many times one number is contained within another. But when we enter into the realm of negative numbers, students often become confused. To clear up this confusion, it's important to remember that dividing a negative by a negative gives a positive result because the two negative signs cancel each other out. Thus, the operation \( \frac{-144}{-12} \) simplifies to dividing the absolute values of the numbers, giving us a positive answer.

Division rules are the guidelines followed to correctly divide numbers. These rules become especially handy when dealing with positive and negative numbers in arithmetic operations.

One key division rule to remember is: when you divide two numbers that both have the same sign, whether negative or positive, the result is always positive. Conversely, if you divide two numbers with different signs, the result is always negative.

Therefore, when we divide \( -144 \) by \( -12 \) as seen in the expression \( \frac{-144}{-12} \), we apply this rule and know our result will be positive. After dividing the absolute values, 144 by 12, we affirm the rule by concluding the result as a positive 12. These rules streamline the process of division and help in maintaining consistency and understanding in arithmetic operations.