Lesson 1: Number Line: Comparing and Ordering Integers

New Concept

Welcome to a new way of learning math! We will build a solid foundation by introducing concepts one step at a time and practicing them together.

What’s next

To begin, we'll explore the number line. Next, you’ll use it to understand, compare, and order the foundational numbers of math: the integers.

Property

Integers include all counting numbers, their opposites, and zero. They can be positive, negative, or zero, and are shown as tick marks on a number line.

Examples

• The integer $-4$ is located 4 units to the left of 0.
• The set of integers is shown as: $\{..., -2, -1, 0, 1, 2, ...\}$
• The set of whole numbers is shown as: $\{0, 1, 2, 3, ...\}$

Explanation

Think of a number line as a thermometer. Positive numbers are the warm temperatures above zero, and negative numbers are the freezing temperatures below. Zero is the starting point! An integer that is not a whole number is any negative number, like $-10$.

Property

The absolute value of a number is its distance from the origin on a number line. It is always positive because distance cannot be negative. For example, $|-5| = 5$.

Examples

• $|-8| = 8$
• $|12| = 12$
• If $|n| = 4$, then $n$ can be either $4$ or $-4$.

Explanation

Absolute value is like asking, “how many steps from zero are you?” The direction doesn’t matter, only the count. That's why the answer is always a positive number or zero. You can't take negative three steps!

Property

To compare integers, find their positions on a number line. Numbers increase in value as you move from left to right. The symbol's small end ($<$ or $>$) always points to the smaller number.

Examples

• Since $-1$ is to the right of $-3$, we write $-1 > -3$.
• Since $-5$ is to the left of $2$, we write $-5 < 2$.
• $|-5| > |3|$ because this comparison simplifies to $5 > 3$.

Explanation

The number line doesn't lie! Any number to the right is greater than any number to its left. Think of it like a race: the runner at position $-2$ is ahead of the runner at position $-5$.