Key Points

Integers are the set of whole numbers and their opposites. They include positive integers ($1, 2, 3, \ldots$), negative integers ($\ldots, -3, -2, -1$), and zero ($0$).

Integers can be represented on a number line with $0$ at the center. Positive integers are to the right of $0$, and negative integers are to the left.

On a number line, a number to the right is always greater than a number to its left. Therefore, any positive integer is greater than any negative integer, and for negative numbers, $-2 > -5$.

Every integer has an opposite, called its additive inverse, and their sum is always zero. The additive inverse of $a$ is $-a$, so $a + (-a) = 0$. For example, the inverse of $7$ is $-7$.

To add two integers with the same sign, add their values (ignoring the signs) and keep the common sign. For example, $(+5) + (+2) = +7$ and $(-5) + (-2) = -7$.

To add two integers with different signs, subtract the smaller value from the larger value (ignoring the signs) and use the sign of the integer with the larger value. For example, $(+8) + (-3) = +5$ and $(-8) + (+3) = -5$.

Addition can be visualized as movement. Starting at the position of the first number, adding a positive integer means moving to the right, and adding a negative integer means moving to the left. For example, $-2 + 5$ means starting at $-2$ and moving $5$ units right to reach $3$.

The fundamental rule of subtraction is to add the additive inverse of the number being subtracted. The rule is $a - b = a + (-b)$. For example, $9 - 4$ is the same as $9 + (-4) = 5$.

Subtracting a negative integer is equivalent to adding its positive counterpart. The rule is $a - (-b) = a + b$. For example, $6 - (-2) = 6 + 2 = 8$.

Adding zero to any integer does not change its value: $a + 0 = a$. Subtracting zero from any integer also does not change its value: $a - 0 = a$. Subtracting an integer from zero results in its additive inverse: $0 - a = -a$.

Temperatures are a common application of integers. Temperatures above the freezing point ($0^{\circ}\text{C}$) are positive, and temperatures below freezing are negative, such as $-15^{\circ}\text{C}$.

Elevation is measured relative to sea level, which is considered $0$ m. A mountain's height is a positive integer (e.g., $+8848$ m), while the depth of a trench is a negative integer (e.g., $-11000$ m).

In banking, deposits or profits are represented by positive integers (credits), while withdrawals or losses are represented by negative integers (debits).