What is an Integer? Complete Math Guide with Examples
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Quick Summary: Integers
- Definition: Integers are whole numbers (positive, negative, and zero)
- Includes: …, -3, -2, -1, 0, 1, 2, 3, …
- Does NOT include: Fractions, decimals, or mixed numbers
- Symbol: Represented by the letter ℤ (from German « Zahlen » = numbers)
- Examples: -5, 0, 7, -100, 42
⏱️ Read Time: 18 minutes
The Short Answer: What is an Integer?
If you’re a student in Montreal learning math, here’s what you need to know: An integer is any whole number, including positive numbers, negative numbers, and zero.
🎯 Key Concept: Integers are the numbers you count with (1, 2, 3…), plus their opposites (-1, -2, -3…), plus zero (0). No fractions or decimals allowed!
Simple Examples:
- Positive integers: 1, 2, 3, 4, 5, 10, 100, 1000…
- Negative integers: -1, -2, -3, -4, -5, -10, -100, -1000…
- Zero: 0 (neither positive nor negative)
NOT integers:
- 1.5 (decimal)
- ½ (fraction)
- 2¾ (mixed number)
- √2 (irrational number)
How Do You Visualize Integers on a Number Line?
The best way to visualize integers is on a number line:
… ← -5 ← -4 ← -3 ← -2 ← -1 ← 0 → 1 → 2 → 3 → 4 → 5 → …
← Negative integers | Zero | Positive integers →
Key points about the number line:
- Numbers to the right are greater
- Numbers to the left are smaller
- Zero is in the middle (neutral)
- The line extends infinitely in both directions
- Each integer is exactly 1 unit apart from its neighbors
What Are the Different Types of Integers?
1. Positive Integers (Natural Numbers)
Definition: Numbers greater than zero
Examples: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10…
Real-world use: Counting people at a Montreal Canadiens game, floors in a building, money earned
2. Negative Integers
Definition: Numbers less than zero
Examples: -1, -2, -3, -4, -5, -6, -7, -8, -9, -10…
Real-world use: Temperature in °C on a Montreal winter day, debt, basement levels, altitude below sea level
3. Zero
Definition: The neutral integer (neither positive nor negative)
Symbol: 0
Real-world use: Freezing point of water (0°C), starting point, balance point
How Do Integers Differ From Fractions and Decimals?
| Number Type | Description | Examples | Is it an Integer? |
|---|---|---|---|
| Integers | Whole numbers (positive, negative, zero) | -3, 0, 7, 42 | ✅ YES |
| Fractions | Parts of a whole | ½, ¾, -2/3 | ❌ NO |
| Decimals | Numbers with decimal points | 1.5, -0.25, 3.14 | ❌ NO |
| Mixed Numbers | Whole number + fraction | 2½, 3¼, -1⅔ | ❌ NO |
| Irrational Numbers | Cannot be written as fractions | π, √2, e | ❌ NO |
Real-World Examples: Integers in Montreal
Example 1: Temperature in Winter
Situation: Montreal winter temperatures
- -15°C: Integer (negative)
- 0°C: Integer (freezing point)
- 5°C: Integer (positive)
- -8.5°C: NOT an integer (decimal)
Example 2: Building Floors
Situation: Floors in Place Ville Marie downtown Montreal
- -2 (B2): Second basement level (negative integer)
- -1 (B1): First basement level (negative integer)
- 0 (Ground): Ground floor (zero)
- 1, 2, 3…: Upper floors (positive integers)
Example 3: Bank Account
Situation: Your savings account balance
- $500: Positive integer (you have money)
- $0: Zero (no money, no debt)
- -$50: Negative integer (you owe money/overdraft)
- $125.50: NOT an integer (has cents)
Example 4: Metro Stops
Situation: Counting stops on Montreal’s Orange Line
- 0 stops: You’re at your station
- +3 stops: Three stops ahead (positive direction)
- -2 stops: Two stops back (opposite direction)
Example 5: Time Zones
Situation: Time difference from Montreal (EST)
- Montreal: 0 (reference point)
- Vancouver: -3 hours (negative integer)
- Paris: +6 hours (positive integer)
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What Are the Key Properties of Integers?
Property 1: Closure
Rule: Adding or multiplying two integers always gives you another integer
Examples:
- 5 + 3 = 8 ✓ (integer)
- (-4) + (-2) = -6 ✓ (integer)
- 3 × (-5) = -15 ✓ (integer)
Property 2: Order Matters for Subtraction
Rule: a – b ≠ b – a (unless a = b)
Montreal example:
- Temperature change: 5°C – (-3°C) = 8°C (temperature rose)
- But: (-3°C) – 5°C = -8°C (temperature dropped)
Property 3: Division Doesn’t Always Give Integers
Rule: Dividing two integers might NOT give you an integer
Examples:
- 10 ÷ 2 = 5 ✓ (integer)
- 10 ÷ 3 = 3.333… ✗ (not an integer)
Property 4: Absolute Value
Rule: The absolute value of an integer is its distance from zero (always positive or zero)
Examples:
- |5| = 5
- |-5| = 5
- |0| = 0
How to Determine if a Number is an Integer
Follow these simple steps:
Step 1: Check for Decimals
If the number has a decimal point with non-zero digits after it, it’s NOT an integer.
- 7.5 → NOT an integer ✗
- 7.0 → IS an integer ✓ (same as 7)
Step 2: Check for Fractions
If the number is written as a fraction (other than fractions that equal whole numbers), it’s NOT an integer.
- ½ → NOT an integer ✗
- 8/4 → IS an integer ✓ (equals 2)
Step 3: Check if it’s a Whole Number
Can you count to it without using parts or decimals? Then it’s an integer!
- -10, -9, -8… -1, 0, 1, 2, 3… → All integers ✓
Practice Problems: Identifying Integers
Test your understanding with these 15 practice problems! Solutions are provided below.
Basic Identification (1-5)
- Is -7 an integer?
- Is 0 an integer?
- Is 3.14 an integer?
- Is ½ an integer?
- Is 100 an integer?
Real-World Scenarios (6-10)
- Montreal temperature is -12°C. Is -12 an integer?
- You owe your friend $5.50. Is 5.50 an integer?
- The elevator is on floor -2 (basement). Is -2 an integer?
- A recipe calls for 2¾ cups of flour. Is 2¾ an integer?
- The Montreal Canadiens won 3-0. Are both 3 and 0 integers?
Advanced Problems (11-15)
- Is 16/4 an integer?
- Is -0.5 an integer?
- Is √16 an integer?
- Is √15 an integer?
- List all integers between -3 and 3 (inclusive).
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Solutions to Practice Problems
Basic Identification Solutions (1-5)
1. YES → -7 is a negative integer ✓
2. YES → 0 is an integer (neither positive nor negative) ✓
3. NO → 3.14 has decimals, not an integer ✗
4. NO → ½ is a fraction, not an integer ✗
5. YES → 100 is a positive integer ✓
Real-World Scenarios Solutions (6-10)
6. YES → -12 is an integer (negative whole number) ✓
7. NO → 5.50 has cents (decimals), not an integer ✗
8. YES → -2 is an integer (basement floors use negative integers) ✓
9. NO → 2¾ is a mixed number, not an integer ✗
10. YES → Both 3 and 0 are integers ✓
Advanced Problems Solutions (11-15)
11. YES → 16/4 = 4, which is an integer ✓
12. NO → -0.5 has decimals, not an integer ✗
13. YES → √16 = 4, which is an integer ✓
14. NO → √15 ≈ 3.872…, not an integer ✗
15. -3, -2, -1, 0, 1, 2, 3 → Seven integers total ✓
Common Mistakes to Avoid
❌ Mistake #1: Thinking zero is not an integer
Wrong: « Zero isn’t a number, so it can’t be an integer »
Right: Zero IS an integer! It’s neither positive nor negative, but it’s still an integer ✓
Montreal example: 0°C (freezing point) is an integer temperature
❌ Mistake #2: Confusing integers with positive numbers only
Wrong: « Integers are only 1, 2, 3, 4, 5… »
Right: Integers include negatives too: …, -3, -2, -1, 0, 1, 2, 3, … ✓
Memory trick: INTEGER = IN + NEGATIVE (includes negatives!)
❌ Mistake #3: Thinking 5.0 is not an integer
Wrong: « 5.0 has a decimal point, so it’s not an integer »
Right: 5.0 = 5, which IS an integer! The .0 doesn’t change the value ✓
Rule: If ALL digits after the decimal are zeros, it’s still an integer
❌ Mistake #4: Thinking fractions that equal whole numbers aren’t integers
Wrong: « 8/4 is a fraction, so it’s not an integer »
Right: 8/4 = 2, which IS an integer! ✓
Rule: If a fraction simplifies to a whole number, it represents an integer
❌ Mistake #5: Confusing integers with counting numbers
Wrong: « Integers and counting numbers are the same thing »
Right: Counting numbers (1, 2, 3…) are only POSITIVE integers. Integers also include 0 and negatives ✓
Comparison: Counting numbers ⊂ Integers (subset relationship)
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Frequently Asked Questions (FAQ)
1. What is an integer?
Answer: An integer is any whole number, including positive numbers (1, 2, 3…), negative numbers (-1, -2, -3…), and zero (0). Integers do NOT include fractions, decimals, or mixed numbers.
2. Is 0 an integer?
Answer: Yes! Zero (0) is an integer. It’s neither positive nor negative, but it’s still classified as an integer. On the number line, zero is the center point between positive and negative integers.
3. Are negative numbers integers?
Answer: Yes! Negative whole numbers like -1, -2, -3, -10, -100 are all integers. Integers include positive numbers, negative numbers, and zero.
4. Is a fraction an integer?
Answer: Most fractions are NOT integers. However, if a fraction equals a whole number (like 8/4 = 2), then it represents an integer. Fractions like ½, ¾, or 5/3 are not integers because they don’t equal whole numbers.
5. Is a decimal an integer?
Answer: Decimals are generally NOT integers. However, if all digits after the decimal point are zeros (like 5.0 or -3.00), then it equals a whole number and IS an integer.
6. What is the symbol for integers?
Answer: The symbol for the set of integers is ℤ (a bold or blackboard bold letter Z). This comes from the German word « Zahlen » which means « numbers. »
7. What’s the difference between integers and whole numbers?
Answer: Whole numbers include 0, 1, 2, 3, 4… (only zero and positive numbers). Integers include negative numbers too: …, -3, -2, -1, 0, 1, 2, 3, …
8. Can an integer be a decimal?
Answer: Only if the decimal equals a whole number! For example, 7.0 is an integer (it equals 7), but 7.5 is NOT an integer.
9. Is every whole number an integer?
Answer: Yes! Every whole number (0, 1, 2, 3…) is an integer. But not every integer is a whole number (because integers also include negative numbers).
10. What are consecutive integers?
Answer: Consecutive integers are integers that follow each other in order, with a difference of 1. For example: 5, 6, 7, 8 or -3, -2, -1, 0, 1.
Study Tips for Montreal Students
📚 For Elementary Students (Grades 5-6)
- Use the number line: Draw a number line and practice finding integers
- Real-world connections: Think of temperature, floors in buildings, money
- Memory trick: « Integers are IN-cluding negatives! »
- Practice daily: Identify integers in your Montreal environment (metro stops, temperatures)
📊 For Secondary 1 Students
- Master operations: Practice adding, subtracting, multiplying, and dividing integers
- Understand absolute value: Know that |-5| = 5
- Graph practice: Plot integers on coordinate planes
- Word problems: Solve real-world problems using integers
🔢 For Secondary 2 Students
- Advanced concepts: Work with integer exponents and roots
- Number theory: Study divisibility rules, prime numbers, factors
- Algebra prep: Use integers in algebraic expressions
- Exam strategies: Review past Quebec Math exam questions
Quick Reference Guide
🎯 Integer Quick Reference
| Question | Answer |
|---|---|
| « What ARE integers? » | Whole numbers: positive, negative, and zero |
| « What are NOT integers? » | Fractions, decimals, mixed numbers |
| « Is 0 an integer? » | YES (neutral integer) |
| « Are negatives integers? » | YES (-1, -2, -3…) |
| « Symbol for integers » | ℤ |
Memory Tricks:
- INTEGER = INcludes negatives
- No FRACTions, no DECImals, just whole!
- Think: Counting + Opposites + Zero
Quick Test:
- Has a decimal? → Probably NOT an integer (unless .000…)
- Is a fraction? → Probably NOT an integer (unless it equals a whole number)
- Can you count to it? → Probably IS an integer!
Real-World Applications in Montreal
🌡️ Temperature & Weather
Montreal’s winter temperatures regularly use negative integers. When the weather forecast says « -15°C, » that’s an integer representing how cold it is!
🏢 Building Levels
Downtown Montreal buildings like Place Ville Marie use negative integers for basement levels (B1 = -1, B2 = -2) and positive integers for floors above ground.
💰 Banking & Finance
Your bank account balance uses integers when dealing with whole dollar amounts. Positive integers show savings, negative integers show debt or overdraft.
🚇 Metro System
The STM metro system uses integers to count stops. If you’re 3 stops away, that’s +3 in one direction or -3 in the opposite direction from your starting point.
🏒 Sports Scores
The Montreal Canadiens’ scores are integers! A 4-2 win uses the integers 4 and 2. Goal differential (+5 or -3) also uses integers.
📍 Elevation & Geography
Mont Royal is 233 meters above sea level (positive integer). The St. Lawrence River’s depth uses negative integers to show depth below water level.
⏰ Time Zones
Montreal (EST) is the reference point (0). Time zones use integers: Vancouver is -3 hours, London is +5 hours, Tokyo is +14 hours.
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- ✅ Number systems & integer operations
- ✅ Positive and negative numbers
- ✅ Number line mastery
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Conclusion: You’re Now an Integer Expert!
Congratulations! You now understand what integers are, how to identify them, and why they’re important in mathematics and everyday life.
Key Takeaways:
- ✅ Integers = whole numbers (positive, negative, and zero)
- ✅ Zero IS an integer (neither positive nor negative)
- ✅ Negatives ARE integers (-1, -2, -3…)
- ✅ Fractions and decimals are NOT integers (usually)
- ✅ Symbol: ℤ
Whether you’re learning basic number systems in elementary school or preparing for algebra in secondary school, mastering integers is essential for math success!
💡 Pro Tip: Practice finding integers everywhere in Montreal – temperature displays, building floors, metro stops, bank statements. The more you see integers in real life, the better you’ll understand them!
Need help with integers or other math concepts? Our Montreal tutors specialize in number systems, operations, and building strong math foundations. Book your free evaluation today!