# 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.