91Ó°ÊÓ

When we talk about dividing numbers, particularly negative numbers, it's essential to understand how signs affect the result. The division of two negative numbers results in a positive number. This might seem counterintuitive, but here's why: a negative number can be thought of as the opposite of a positive number. When you divide two 'opposites,' they essentially cancel each other out, leading to a positive result.

Let's take the example from the exercise where we divide \( -110 \) by \( -55 \). However counterintuitive it might seem, the division process itself remains the same as with positive numbers. You simply divide the absolute values of the numbers as normal, in this case, \( 110 ÷ 55 \), which equals \( 2 \). Once you've done that, you apply the rule of signs and conclude that the answer will be positive because both original numbers were negative. Hence, \( -110 ÷ -55 = +2 \).

Algebra often presents scenarios where you have to calculate with both positive and negative numbers, and knowing the rules of signs is indispensable. There's a simple set of rules to remember:

• A positive number times a positive number gives a positive result.
• A positive number times a negative number gives a negative result.
• A negative number times a negative number gives a positive result.
• The same rules apply to division.

The exercise provided is an application of these rules. With division, just like multiplication, when the signs are the same, the result is always positive. Conversely, when the signs are different, the result is negative. This is why in our division example of two negative numbers, we ended up with a positive result.

Arithmetic operations form the bedrock of mathematics, and they include addition, subtraction, multiplication, and division. These basic operations follow certain properties that help us to perform calculations systematically. For instance, the associative and commutative properties apply to addition and multiplication, but not to subtraction and division. Under these properties, numbers can be grouped or ordered differently without changing the result.

However, when dealing with negative numbers, the rules of signs become particularly important. They ensure that regardless of whether integers are positive or negative, operations are carried out correctly. Each operation follows particular rules, and when you're simplifying fractions involving negative numbers, you're usually dealing with division, which requires special attention to the signs involved, as we've discussed.