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Dividing negative numbers may initially seem confusing, but it follows a consistent rule that makes the process straightforward once understood. When you divide a negative number by another negative number, the division rule states that two negatives will give a positive result. This might appear counterintuitive at first, because we often associate negatives with subtraction or loss. However, division with negative numbers operates differently.

For example, if you have \( -27 \div (-3) \) like in our exercise, you are essentially asking how many times \( -3 \) fits into \( -27. \) To make sense of this, you can imagine that each \( -3 \) added together gets you closer to zero from \( -27 \), and it takes 9 instances of \( -3 \) to reach zero. Thus, the quotient here is 9. It’s essential to remember that the division of two negative numbers results in a positive quotient.

The absolute value of a number can be thought of as its distance from zero on a number line, without considering which direction from zero it is. Absolute value is always non-negative, because distance itself is a non-negative attribute. It is denoted by two vertical lines, like this: \( |x| \). This means that \( |5| \)= 5 and \( |-5| \)= 5; both are five units away from zero.

In the context of our division problem, we consider the absolute values of both numbers. \( |-27| = 27 \) and \( |-3| = 3. \) When you divide these absolute values \( 27 \div 3 \) you get 9, which is the absolute number of times \( -3 \) fits into \( -27. \) Since both original numbers were negative, the rule of negative number division applies, confirming that the quotient is a positive 9.

Inverse operations are pairs of operations that reverse the effects of each other, such as addition and subtraction, or multiplication and division. When dealing with division, multiplying by the reciprocal (or inverse) of the divisor will result in the original number (also known as identity element for multiplication, which is 1). Therefore, division and multiplication are inverse operations.

Applying this concept to our problem: if you have a number \( -27 \) and you divide it by \( -3 \) to get 9, you can check your work by multiplying 9 by \( -3 \). If the operations are truly inverse to each other, multiplying should bring you back to \( -27 \) which it does \( (9 \times -3 = -27) \). Whenever you're unsure about your division of negative numbers, you can use this inverse operation as a reliable check.