Types of Integers With Examples and Number Line
The concept of integer in Maths is a foundation for arithmetic, algebra, and problem-solving in school studies. Understanding what is an integer in Maths helps students tackle exams, daily calculations, and logical reasoning with confidence. This topic is essential for Class 6, 7, 8, and competitive tests alike.
What Is an Integer in Maths?
An integer in Maths is any whole number, either positive, negative, or zero, that does not have any fractional or decimal part. Integers are used to count, measure, and compare quantities where only complete units are allowed. Real-life situations such as temperature, bank balances, and heights above or below sea level involve integers. The symbol for the set of integers is Z, which comes from the German word “Zahlen” meaning “numbers.”
The Set of Integers: Notation & List
The set of all integers is written as:
Z = { ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... }
This shows integers stretching from negative infinity to positive infinity, including zero.
Types of Integers
- Positive Integers: Numbers greater than 0 (e.g. 1, 2, 17, 100)
- Negative Integers: Numbers less than 0 (e.g. -1, -3, -52, -999)
- Zero: Neutral, neither negative nor positive (0)
All numbers in these groups are called integers. No decimals or fractions are allowed.
Key Differences: Integers vs. Whole & Natural Numbers
| Number Set | Includes | Examples |
|---|---|---|
| Natural Numbers (N) | Counting numbers from 1 onwards | 1, 2, 3, 4, 5, ... |
| Whole Numbers (W) | Natural numbers + zero | 0, 1, 2, 3, ... |
| Integers (Z) | Positive, negative numbers, and zero | ..., -3, -2, -1, 0, 1, 2, 3, ... |
Examples of Integers in Maths
- -7 (Negative integer)
- 0 (Zero is always an integer)
- 15 (Positive integer)
- -81, 45, -156 (Any countable negative or positive, including zero)
Non-examples: 4.5, -3.2, 1/2 are not integers because they include a fraction or decimal part.
Placing Integers on the Number Line
To visually understand integers, imagine a number line; zero is in the center, positive integers on the right, and negative integers on the left:
| ... -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 ... |
Every point to the right is larger; every point on the left is smaller.
Properties of Integers
- Closure: Sum, difference, or product of two integers is always an integer.
- Commutative: a + b = b + a and a × b = b × a
- Associative: a + (b + c) = (a + b) + c and a × (b × c) = (a × b) × c
- Identity: a + 0 = a and a × 1 = a
- Additive Inverse: a + (–a) = 0
These properties are tested often on school and board exams. For deep insights, read: Properties of Integers.
Arithmetic Rules for Integers
- Addition: Same signs: add values, keep sign. Different signs: subtract, keep sign of bigger value.
- Subtraction: Change the sign of second number, then add.
- Multiplication/Division: Same signs: answer is positive; different signs: answer is negative.
Practice integer operations using this integer calculator.
Step-by-Step Example: Integer Calculations
Let’s solve: What is (–7) + 4?
1. Find the absolute values: 7 and 42. Difference: 7 – 4 = 3
3. Larger value is 7 (–7), so result is –3
Final Answer: (–7) + 4 = –3
Try These Yourself
- List five integers between –10 and 5.
- Is 0.75 an integer?
- Find all negative integers between –8 and –1.
- Check if –19 is a whole number or just an integer.
Common Student Mistakes With Integers
- Thinking decimals or fractions (like 6.1, 3/2) can be integers – they are not.
- Mixing up sign conventions when adding or subtracting.
- Forgetting zero is an integer (it is, but not positive or negative).
- Believing integers can be percentages—they cannot.
Relation to Other Maths Topics
Learning what is an integer in Maths helps you master topics like number line representation, differences from whole numbers, and rational numbers. This strengthens foundational skills for algebra and arithmetic.
Quick Classroom Tips
Remember: All whole numbers are integers, but not all integers are whole numbers. On a number line, moving right means increasing, moving left means decreasing. Vedantu’s teachers often remind students, "No fractions or decimals for integers—just complete, signed numbers!"
We have explored what is an integer in Maths—it includes zero, positive and negative numbers, excludes fractions and decimals, and follows unique properties and rules. Practice regularly, and use Vedantu’s live classes for more clarity and speed tricks in exams!
FAQs on What Is an Integer in Maths?
1. What is an integer in mathematics?
An integer in mathematics is a whole number that can be either positive, negative, or zero. It does not include fractions or decimal parts. The set of integers is represented by the symbol Z and includes numbers like {..., -3, -2, -1, 0, 1, 2, 3, ...}.
2. What are the types of integers?
Integers are classified into three main types:
• Positive integers: Whole numbers greater than zero (e.g., 1, 2, 3...).
• Negative integers: Whole numbers less than zero (e.g., -1, -2, -3...).
• Zero (0): A neutral integer, neither positive nor negative.
3. Is zero an integer?
Yes, zero (0) is an integer. It's considered a neutral integer, neither positive nor negative.
4. Are fractions or decimals integers?
No, fractions and decimals are not integers. Integers are whole numbers without fractional or decimal parts.
5. What is the difference between integers and whole numbers?
Whole numbers include zero and all positive integers (0, 1, 2, 3...). Integers include whole numbers and their negative counterparts (..., -3, -2, -1, 0, 1, 2, 3, ...).
6. How do you add integers with different signs?
To add integers with different signs, find the difference between their absolute values. The result takes the sign of the integer with the larger absolute value. For example, (-8) + 3 = -5.
7. How do you subtract integers?
Subtracting an integer is the same as adding its additive inverse (opposite). For example, 5 - (-3) = 5 + 3 = 8.
8. What are the rules for multiplying integers?
When multiplying integers:
• Positive × Positive = Positive
• Positive × Negative = Negative
• Negative × Positive = Negative
• Negative × Negative = Positive
9. What are the rules for dividing integers?
The rules for dividing integers are the same as for multiplication:
• Positive ÷ Positive = Positive
• Positive ÷ Negative = Negative
• Negative ÷ Positive = Negative
• Negative ÷ Negative = Positive
10. What are some real-life examples of integers?
Integers are used to represent many real-world quantities, such as:
• Temperature (e.g., -5°C)
• Altitude (e.g., +1000 meters above sea level)
• Bank balances (e.g., -$50)
• Golf scores (e.g., -2 under par)
11. What is the additive inverse of an integer?
The additive inverse of an integer is the number with the opposite sign. When added together, an integer and its additive inverse equal zero. For example, the additive inverse of 5 is -5, because 5 + (-5) = 0.
12. Why is -2 greater than -9?
On a number line, numbers increase as you move to the right. Since -2 is to the right of -9, -2 is greater than -9. Negative numbers further from zero have smaller values.
