Definition of Integers with Examples and Properties
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 Mathematics
1. What is an integer in maths?
An integer is a whole number that can be positive, negative, or zero, without any fractions or decimals. Integers include:
- Positive numbers: 1, 2, 3, 4...
- Zero: 0
- Negative numbers: -1, -2, -3, -4...
2. What are examples of integers?
Examples of integers are numbers with no decimal or fractional part. Common examples include:
- -5 (negative integer)
- 0 (neither positive nor negative)
- 8 (positive integer)
3. Is 0 an integer?
Yes, 0 is an integer because it is a whole number with no decimal or fraction. Zero is important because it separates positive integers (1, 2, 3…) from negative integers (-1, -2, -3…). It is neither positive nor negative, but it still belongs to the set of integers.
4. What is the difference between integers and whole numbers?
The main difference is that integers include negative numbers, while whole numbers do not.
- Whole numbers: 0, 1, 2, 3, 4…
- Integers: …-3, -2, -1, 0, 1, 2, 3…
5. What is the difference between integers and natural numbers?
The difference is that natural numbers are counting numbers, while integers also include zero and negative numbers.
- Natural numbers: 1, 2, 3, 4…
- Integers: …-2, -1, 0, 1, 2…
6. How do you add integers?
To add integers, follow the sign rules for positive and negative numbers.
- If both integers have the same sign, add them and keep the sign. Example: -3 + (-2) = -5
- If the signs are different, subtract the smaller absolute value from the larger and keep the sign of the larger number. Example: 7 + (-4) = 3
7. How do you subtract integers?
To subtract integers, change subtraction into addition of the opposite number. Use the rule: a − b = a + (−b). Example:
- 5 − 3 = 5 + (−3) = 2
- 4 − (−2) = 4 + 2 = 6
8. How do you multiply and divide integers?
When you multiply or divide integers, use sign rules.
- Same signs → Positive result: (−4) × (−2) = 8
- Different signs → Negative result: (6) ÷ (−3) = -2
9. What are the properties of integers?
Integers follow important mathematical properties such as closure, commutative, associative, and distributive properties.
- Closure: The sum or product of integers is always an integer.
- Commutative: a + b = b + a
- Associative: (a + b) + c = a + (b + c)
- Distributive: a(b + c) = ab + ac
10. Where are integers used in real life?
Integers are used in real life to represent quantities above and below zero. Common examples include:
- Temperature: -5°C, 10°C
- Bank balance: -$50 (debt), $200 (credit)
- Elevations: -10 meters (below sea level), 500 meters (above sea level)
