Number - Integers and the Number Line

Integers are a set of numbers that include all positive whole numbers ($1, 2, 3, \dots$), their negative opposites ($-1, -2, -3, \dots$), and zero ($0$). On a horizontal number line, zero acts as the central origin point, with positive numbers extending to the right and negative numbers extending to the left. Integers do not include fractions or decimals.

The number line is a visual representation where every point corresponds to a specific number. Numbers increase in value as you move from left to right. For example, even though the digit $5$ is larger than $2$, the integer $-5$ is smaller than $-2$ because $-5$ is located further to the left on the number line.

Opposites, also known as additive inverses, are pairs of numbers that are exactly the same distance from zero but in different directions. For instance, $+8$ and $-8$ are opposites. Visually, if you imagine the number line as a piece of paper folded at the $0$ mark, the positive numbers would land exactly on their negative counterparts.

Absolute value represents the magnitude or distance of an integer from zero, regardless of its direction. It is denoted by vertical bars, such as $|-5| = 5$. Visually, the absolute value is the 'length' of the jump from $0$ to the number; since length cannot be negative, $|-10|$ and $|10|$ both equal $10$.

Comparing and ordering integers involves using inequality symbols like $<$ (less than) and $>$ (greater than). On a horizontal number line, the number that is positioned to the right of another number is always the greater value. Therefore, $0 > -100$ because zero is further to the right than negative one hundred.

Integers are used to represent opposite real-world situations using a vertical or horizontal scale. Positive integers represent gains (+$20$ dollars), rises in temperature (+$15^\circ C$), or elevation above sea level. Negative integers represent losses (-$10$ dollars), drops in temperature (-$5^\circ C$), or depth below sea level.

The sum of any integer and its opposite is always zero, expressed as $a + (-a) = 0$. This is visually represented as moving $a$ units in one direction away from zero and then moving the exact same number of units back toward the starting point at $0$.