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When we talk about negative numbers, we're referring to numbers that are less than zero. These numbers are often represented with a minus sign (-) in front of them. Negative numbers are used to denote a value that is opposite in magnitude to their positive counterparts but in the direction opposite to the positive direction on the number line.

For example, in the exercise \( -45 + 45 \) we see \( -45 \) as a negative number. It is 45 units to the left of zero on the number line. This concept is essential in various branches of mathematics and real-life situations, such as measuring temperatures below freezing or indicating debt. While using negative numbers, especially in addition, it's crucial to pay attention to their signs, as they determine the direction in which you move on the number line.

The absolute value of a number is the distance of that number from zero on the number line, without considering the direction. It's always a non-negative number. You can think of absolute value as the 'numerical value' of a number regardless of its sign. It is denoted by two vertical lines surrounding the number, like this: \( |x| \).

For instance, the absolute value of -45 is 45, and the absolute value of 45 is also 45. As we can see from the exercise \( |{-45}| = 45 \) and \( |45| = 45 \). When adding integers, especially one positive and one negative with the same absolute values, they cancel each other out because they are equal distances from zero but in opposite directions.

The addition of opposite numbers refers to adding two numbers that are the same distance from zero on the number line but are on opposite sides. Essentially, one is positive and the other is negative. When you add opposite numbers, the result is always zero. This is because the value of one number 'cancels out' or 'neutralizes' the value of the other.

In our exercise example, \( -45 \) is the opposite of \( 45 \). When they are added together as \( -45 + 45 \), they cancel each other out, resulting in zero. This principle is a fundamental concept in mathematics and forms the basis for solving many types of algebraic problems. It's important to recognize pairs of opposite numbers, as it simplifies the process of calculation in addition and subtraction.