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Negative integers are numbers less than zero, found to the left of zero on the number line. They represent a lack, loss, or deficiency and are commonly encountered in financial debts, temperature drops, or elevations below sea level. For example, -(3), -(22), and -(100) are all negative integers representing values less than zero.

When working with negative integers, especially in subtraction and addition, it's crucial to understand their rules. Subtraction of a negative number is the same as adding its positive counterpart, which can sometimes be counterintuitive. For instance, subtracting -5 is the same as adding +5. This concept is important as it helps simplify and solve many arithmetic and algebraic problems.

Adding integers, whether positive or negative, follows specific rules based on their signs. When the signs are the same, you add their absolute values and keep the common sign. However, when the signs are different, you find the difference between their absolute values and keep the sign of the larger absolute value.

To illustrate, consider the addition of two negative integers such as -13 and -28. Following our rule for same-signed integers, we add their absolute values (which are 13 and 28) and keep the negative sign, resulting in -41. This is consistent with our understanding that combining two losses leads to a greater loss, which in number terms is a larger negative integer.

Visualizing Integer Addition

Imagine pulling in two directions on a number line; if you pull in the same negative direction, you go further from zero. This is analogous to adding two negative integers.

The absolute value of a number refers to its distance from zero on a number line, regardless of direction. It's always a non-negative number because distance cannot be negative. The absolute value of a number 'n' is denoted as |n|. For instance, the absolute values of -5 and 5 are both 5, as both are five units away from zero on the number line. So in our previous example, |−13|=13 and |−28|=28.

Understanding absolute value is essential when performing operations with integers, such as our subtraction by equivalent addition example. To add -13 and -28, we assess their absolute values and use the additive rule for integers with the same sign. Combining absolute values and then applying the appropriate sign leads us to the correct answer, maintaining coherence with the geometric interpretation of these numbers on the number line.