What You Will Learn

After working through this page you will be able to compare any two integers using the symbols <<<, >>>, and ===, and you will know how to arrange a set of integers in ascending or descending order. You will also understand why negative numbers that seem "bigger" (like −100) are actually smaller than those that seem "tiny" (like −1).

Theory

The number line is your best tool

A number line is a straight line where integers are placed in order. Numbers increase as you move to the right and decrease as you move to the left:

⟵−6−5−4−3−2−10123456⟶
Key rule: An integer is greater than any integer to its left and less than any integer to its right.

Comparing two integers

Use the inequality symbols:

Symbol Meaning
<<< less than
>>> greater than
=== equal to

Comparing rules at a glance:

  1. Every positive integer is greater than every negative integer: 1 > −100.
  2. Every positive integer is greater than zero, and every negative integer is less than zero.
  3. For two positive integers, the one with the larger absolute value is greater: 8 > 3.
  4. For two negative integers, the one closer to zero (smaller absolute value) is greater: −2 > −7.

Rule 4 is the one most students find tricky. Think of it this way: −2 is only 2 steps below zero, while −7 is 7 steps below zero, so −2 is higher up on the number line.

Absolute value and comparison

The absolute value of an integer is its distance from zero on the number line:

∣−5∣=5 and ∣5∣=5

Absolute value is always non-negative. It tells you the magnitude, but not the direction. When comparing two negative integers, the one with the smaller absolute value is the greater integer:

∣−3∣=3 and ∣−8∣=8

Since 3 < 8, we know −3 > −8.

Ordering integers

Ascending order (least to greatest) means arranging from leftmost to rightmost on the number line.

Descending order (greatest to least) goes from rightmost to leftmost.

Strategy for ordering a set of integers:

  1. Separate the numbers into negative, zero, and positive groups.
  2. Order the negatives: the one with the largest absolute value is the smallest (farthest left).
  3. Place zero next (if present).
  4. Order the positives normally.
  5. Combine the groups.

Example: Order {3,−7,0,−2,5} from least to greatest.

  • Negatives (largest absolute value first): −7,−2
  • Zero: 0
  • Positives: 3,5
  • Result: −7,−2,0,3,5

Comparing integers in real-world contexts

Comparing integers is not just abstract math. Consider:

  • Temperature: −15°C is colder than −3°C because −15 < −3.
  • Elevation: A valley at −50 m is lower than a shore at 0 m because −50 < 0.
  • Bank balance: A debt of $200 (−200) is worse than a debt of $50 (−50) because −200 < −50.

Worked Examples

Example 1: Comparing two negative integers (easy)

Problem: Insert <<<, >>>, or === between −4 and −9.

Step 1: Find the absolute values. ∣−4∣=4 and ∣−9∣=9.

Step 2: Both are negative. The one with the smaller absolute value is greater. 4 < 9 ⟹ −4 > −9.

Answer: −4 > −9.

Example 2: Comparing a negative and a positive integer (easy)

Problem: Insert <<<, >>>, or === between −12 and 3.

Step 1: One is negative and the other is positive. Every positive integer is greater than every negative integer.

Answer: −12 < 3.

Example 3: Ordering five integers in ascending order (medium)

Problem: Arrange 8,−3,−11,0,5 from least to greatest.

Step 1: Identify the negatives, zero, and positives.

  • Negatives: −11,−3
  • Zero: 0
  • Positives: 5

Step 2: Order the negatives (largest absolute value = smallest value). −11 < −3.

Step 3: Combine all groups.
Answer: −11,−3,0,5.

Example 4: Ordering with duplicates in descending order (medium)

Problem: Arrange −5,2,−5,7,0,−1 from greatest to least.

Step 1: In ascending order first: −5,−5,−1,0,2,7.

Step 2: Reverse for descending order.

Answer: 7,2,0,−1,−5,−5.

Example 5: Real-world comparison (challenging)

Problem: During a science experiment, rank the sensors from coldest to warmest: Sensor A: −8°C, Sensor B: −3°C, Sensor C: 2°C, Sensor D: −8°C.

Step 1: Order the temperatures from least to greatest. −8 < −8 < −3 < 2.

Step 2: Map back to sensors.

  • Coldest (tied): Sensor A and Sensor D (both −8°C)
  • Next: Sensor B (−3°C)
  • Warmest: Sensor C (2°C)

Answer: Coldest to warmest: A = D, B, C.

Common Mistakes

Mistake 1: Thinking a "bigger" negative number is greater.
✅ −10 < −2.

Mistake 2: Confusing the direction of <<< and >>>. ✅ 5 > 3.

Mistake 3: Forgetting zero when ordering.
✅ −4,−1,0,3,7.

Practice Problems

  1. Insert <<<, >>>, or ===: −6 □ −1.
  2. Insert <<<, >>>, or ===: −15 □ 0.
  3. Order from least to greatest: 4,−9,1,−3,0,−9.
  4. Order from greatest to least: −20,15,−7,0,8,−20.
  5. Coldest and warmest temperature during a week.

Solutions: 1. −6 < −1
2. −15 < 0
3. −9,−9,−3,0,1,4
4. 15,8,0,−7,−20
5. Coldest: −5°C, Warmest: 3°C.

Summary

  • On a number line, numbers increase to the right and decrease to the left.
  • Every positive integer is greater than every negative integer, and both are compared to zero.
  • For two negative integers, the one closer to zero is greater: −2 > −7.
  • Ascending order = least to greatest.
  • Descending order = greatest to least.