Comparing and Ordering Integers

Once you understand what integers are and where they sit on the number line, the next step is learning to compare them. With positive numbers, bigger is bigger: $8$ is obviously greater than $3$. But negative numbers can trip you up. Is $-2$ greater than $-7$? Is $-10$ less than $5$? Working through these comparisons carefully now will prevent sign errors later when you start computing with integers.

The Number Line Rule

The number line gives you a simple, reliable rule for comparing any two integers:

The number farther to the right is greater. The number farther to the left is lesser.

This rule works for every combination of positive, negative, and zero.

Comparing Two Integers

We use inequality symbols to express comparisons:

• $a > b$ means ”$a$ is greater than $b$”
• $a < b$ means ”$a$ is less than $b$”
• $a = b$ means ”$a$ is equal to $b$“

Key Patterns

1. Any positive integer is greater than any negative integer.

   • $3 > -100$ (3 is to the right of $-100$ on the number line)

2. Any positive integer is greater than zero.

   • $1 > 0$

3. Any negative integer is less than zero.

   • $-5 < 0$

4. For two negative integers, the one closer to zero is greater.

   • $-2 > -7$ (because $-2$ is to the right of $-7$)

That fourth pattern is the one that causes the most confusion. Think of it this way: $-2$ is a small debt, and $-7$ is a larger debt. Being $2 in debt is “better” (greater) than being $7 in debt.

Worked Examples

Example 1: Comparing Two Negatives

Compare $-4$ and $-9$.

On the number line, $-4$ is to the right of $-9$. Therefore:

$-4 > -9$

Alternatively, $-4$ is closer to zero than $-9$, so $-4$ is greater.

Example 2: Comparing a Positive and a Negative

Compare $-15$ and $6$.

Every negative number is less than every positive number:

$-15 < 6$

Example 3: Comparing with Zero

Compare $-3$ and $0$.

Zero is greater than every negative number:

$-3 < 0$

Example 4: Ordering a Set from Least to Greatest

Arrange the following integers from least to greatest: $5, -2, 0, -8, 3, -1$.

Step 1: Find the most negative (leftmost on the number line): $-8$.

Step 2: Continue leftward to rightward: $-8, -2, -1, 0, 3, 5$.

$-8 < -2 < -1 < 0 < 3 < 5$

Example 5: Ordering from Greatest to Least

Arrange from greatest to least: $-12, 7, -3, 0, -7, 4$.

Start with the rightmost on the number line and work left:

$7 > 4 > 0 > -3 > -7 > -12$

Real-World Comparisons

Temperature

Three cities recorded the following overnight low temperatures: City A at $-14°\text{F}$, City B at $-6°\text{F}$, and City C at $2°\text{F}$.

Which city was coldest? Which was warmest?

Ordering from least to greatest: $-14 < -6 < 2$.

• Coldest: City A ($-14°\text{F}$)
• Warmest: City C ($2°\text{F}$)

Elevation

Consider these locations and their elevations relative to sea level:

Ordering from lowest to highest elevation:

$-430 < -86 < -2 < 1{,}609$

The Dead Sea shore is the lowest point, and Denver is the highest.

Tips for Avoiding Mistakes

• Do not confuse “bigger negative” with “greater.” The number $-100$ has a bigger absolute value than $-3$, but $-100 < -3$. Farther from zero in the negative direction means less, not more.
• Draw a quick number line if you are unsure. Place the numbers on it and read left to right for least to greatest.
• Use real-world thinking. If $-20°\text{F}$ and $-5°\text{F}$ are temperatures, which day was warmer? The $-5°\text{F}$ day, because $-5 > -20$.

Practice Problems

Problem 1: Insert $>$, $<$, or $=$ : $\quad -6 \;\square\; -1$

Problem 2: Insert $>$, $<$, or $=$ : $\quad 0 \;\square\; -4$

Problem 3: Arrange from least to greatest: $3, -5, -1, 7, 0, -3$

Problem 4: A freezer is set to $-18°\text{C}$ and a refrigerator is set to $4°\text{C}$. Which is colder, and by how many degrees?

Problem 5: Arrange from greatest to least: $-20, -2, 15, -15, 0, 8$

Key Takeaways

• On the number line, the number farther to the right is always greater.
• Any positive number is greater than any negative number, and both are compared to zero accordingly.
• When comparing two negative numbers, the one closer to zero (with the smaller absolute value) is greater: $-3 > -10$.
• To order integers, visualize them on a number line and read from left to right (least to greatest) or right to left (greatest to least).
• Real-world contexts like temperature and elevation make integer comparisons intuitive: warmer temperatures and higher elevations correspond to greater integers.

Return to Arithmetic for more foundational math topics.