Lesson 4: Number Line, Sequences

New Concept

Integers include all of the counting numbers as well as their opposites—their negatives—and the number zero. They are represented on the number line.

What’s next

You'll now apply this concept with worked examples on comparing integers and visualizing addition and subtraction on the number line.

Property

A number line is used to arrange numbers in order. The zero point is the origin. Numbers to the right are positive (greater than zero), and numbers to the left are negative (less than zero). Integers are whole numbers, their opposites, and zero, marked by tick marks.

Examples

• To order $-2, 0, 1$, you find their places on the line, which shows the correct order is $-2, 0, 1$.
• To compare $-5$ and $3$, notice $-5$ is to the left of $3$, so you write $-5 < 3$.
• To solve $3 + 2$, you start at 0, move 3 units right, and then move 2 more units right, landing on $5$.

Explanation

Think of a number line as a giant map for numbers. Zero is your starting point, or 'origin.' Heading right takes you to the land of positive numbers, which get bigger with every step. Heading left leads you to the chilly negative numbers, which get smaller. It’s the ultimate tool for seeing which number is the king of the hill!

Property

A sequence is an ordered list of terms that follows a certain pattern or rule.

Examples

• In the sequence $5, 10, 15, 20, ...$ the rule is to add 5 to the previous term to find the next.
• The sequence $1, 4, 9, 16, ...$ is a list of perfect squares, following the rule $k = n \cdot n$. The next term is $5 \cdot 5 = 25$.
• Using the formula $k = 2n$, the first four terms are $2, 4, 6, 8$ (by substituting $n=1, 2, 3, 4$).

Explanation

A sequence is like a secret code where numbers follow a hidden rule. Your mission, should you choose to accept it, is to become a pattern detective! By figuring out the rule—whether it's adding, multiplying, or something trickier—you can predict all the future numbers in the line and look like a math magician.

Property

In an arithmetic sequence, the same number is added to each term to find the next term. The numbers in the sequence are equally spaced on a number line.

Examples

• The sequence of odd numbers $1, 3, 5, 7, ...$ is arithmetic because you consistently add 2.
• The sequence $20, 15, 10, 5, ...$ is arithmetic because the rule is to add $-5$ (or subtract 5) each time.
• To find the next term of $100, 110, 120, ...$, you see the rule is 'add 10', so the next term is $120 + 10 = 130$.

Explanation

Imagine you're climbing a staircase where every single step is the exact same height. That's an arithmetic sequence! You just keep adding the same number over and over again, making it a steady and predictable climb. This constant step, called the common difference, is the secret to finding any term in the sequence, no matter how far along.