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Negative numbers are numbers that are less than zero, often depicted with a minus sign (-) in front. These numbers are positioned on the left side of the number line, opposite to positive numbers. They are used to represent things like debts or temperatures below zero. For example, if you owe $10, you can express this as -10 dollars.
When working with negative numbers, it's important to understand that they reverse the direction on the number line. This means subtracting a negative number actually results in addition. For example, \(-(-5)\) becomes \(+5\).
Understanding negative numbers is crucial because they often appear in equations and real-life situations, such as calculating temperature changes or financial losses.

Absolute values refer to the distance of a number from zero on the number line, disregarding the direction. It is always a non-negative number. The absolute value of a number \(x\) is denoted by \(|x|\). For instance, the absolute values of -7 and 7 are both 7, because both are 7 units away from zero.
To find the absolute value of a negative number, simply remove the negative sign. For example, the absolute value of \(-30\) is \(30\).
When dividing integers, it's often helpful to first consider their absolute values to simplify the calculation. For example, in dividing \(-30\) by \(-10\), consider the absolute values: \(30\) divided by \(10\) gives \(3\).
Remember, the absolute value represents magnitude without regard for direction, which simplifies the process of calculations involving divisors and dividends.

Sign rules are essential in performing operations with integers. They help determine the sign of a result when multiplying or dividing two numbers.

• If both numbers have the same sign (both positive or both negative), the result is positive.
• If the numbers have different signs (one positive and one negative), the result is negative.

Let's apply this to division: dividing two negative numbers, like \(-30\) by \(-10\), results in a positive \(3\) because both numbers share the same negative sign. Conversely, dividing a negative number by a positive one, or vice versa, results in a negative number.
Understanding and applying sign rules can help you quickly determine whether a quotient, product, or sum will be positive or negative with confidence. This simplifies otherwise complex calculations, making it easier to solve problems accurately.