What are Integers? Definition, Examples, and Common Doubts
The concept of integers plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. Every student—whether learning addition, solving equations, or preparing for competitive exams—must be quick and accurate with integers for overall maths success.
What Is an Integer?
An integer in maths is any whole number that can be positive, negative, or zero. Integers do not include fractions or decimals—just complete numbers like -3, 0, 7, or 102. You’ll find this concept applied in topics such as number systems, comparing values, arithmetic operations, and even word problems.
- Positive integers: 1, 2, 3, ...
- Negative integers: -1, -2, -3, ...
- Zero (0): neither positive nor negative
The set of integers is often written as:
Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
Here, ‘Z’ comes from the German word “Zahlen,” meaning numbers.
Key Properties and Rules of Integers
Here are some important properties and operational rules related to integers, which you will use very often:
- Addition and Subtraction: Adding numbers with the same sign keeps that sign; adding numbers of different signs is like subtraction—take the sign of the bigger absolute value.
- Multiplication and Division: Two integers with the same sign make a positive result; different signs make it negative.
- Closure: Integers are closed under addition, subtraction, and multiplication (but not division).
- Associative and Commutative Properties: The way you group or order integers in addition or multiplication doesn’t change the answer.
- Additive Identity: 0 is the only integer that, when added, does not change the other number.
These rules are the backbone for all calculations involving integers—from basic sums to complex algebra in higher grades.
Integers on a Number Line
To compare and visualize integers, the number line is your best friend. Here, zero sits in the middle, positive numbers extend to the right, and negative numbers go left:
-4 -3 -2 -1 0 1 2 3 4
(Left means smaller; right means bigger)
This helps when solving problems such as, “What is greater: -5 or -2?” (Answer: -2, since it is to the right of -5).
Difference: Integers vs Whole Numbers vs Real Numbers
| Type | Definition | Example |
|---|---|---|
| Integers (Z) | All whole numbers including negatives and zero | -5, 0, 3, 12 |
| Whole Numbers (W) | All natural numbers and zero (no negatives) | 0, 4, 17 |
| Real Numbers (R) | Integers plus fractions, decimals, irrationals | -3, 0, 1.2, √2 |
Common Errors with Integers
- Confusing the sign when adding or subtracting (like (-3)+5; answer is 2, not -8).
- Forgetting that zero is an integer.
- Mixing up whole numbers and integers.
- Assuming decimals and fractions can be integers (they can’t).
- Using wrong signs when multiplying/dividing.
Step-by-Step: Solved Examples with Integers
Example 1: Add -8 and 3
1. Start with the numbers: -8 and 32. Since the signs are different, subtract the smaller absolute value from the bigger one: 8 - 3 = 5
3. The bigger number is -8, so keep the negative sign: Answer = -5
Example 2: Multiply -4 × -6
1. Multiply as usual: 4 × 6 = 242. Both numbers are negative, so result is positive: Answer = 24
Speed Trick: Integer Addition/Subtraction
When adding or subtracting quickly in exams, stack numbers by sign:
Add all positives together, add all negatives, then add the two sums. For example:
-5 + 7 – 4 + 3 = (7+3) + (-5-4) = 10 + (-9) = 1
This method avoids mistakes with back-and-forth signs! Such strategies are often shared in Vedantu’s live maths classes.
Where and How Are Integers Used?
Integers in real life: Temperatures below zero (like –10°C), banking (profits and losses), elevator floors (B2 is -2), and coding/programming (data types, loop counts). In number systems, they are a bridge to more advanced maths like algebra and rational numbers.
Try These Yourself
- List five consecutive integers between -3 and 3.
- Is 0 an integer? Is -100 an integer?
- Mark -2, 0, 2 on a number line.
- Find the sum of -7 and 12.
Relation to Other Maths Concepts
The topic of integers connects with many other maths concepts:
- Even and Odd Numbers: Every integer is either even or odd.
- Whole Numbers: These are a subset of integers.
- Rational and Irrational Numbers: Integers are rational numbers, but not all rationals are integers.
Understanding these links helps build strong problem-solving skills for higher levels.
Classroom Tip
To easily remember integer operations, think of the number line as an elevator: Going right (positive) means ‘up’, going left (negative) means ‘down’. Zero is always the ground floor! Vedantu teachers often use this analogy for clear demonstrations in class.
We explored integers—from definition, rules, mistakes, examples, real-life uses, and their relationship with other maths topics. Practice with more sums and worksheets on Vedantu to get faster and more confident with integers in every exam and real situation.
FAQs on Integers Explained With Rules, Properties & Examples
1. What is an integer in mathematics? Please provide examples.
An integer is a whole number that can be positive, negative, or zero; it cannot be a fraction, a decimal, or a percentage. Integers are the set of numbers that include all the whole numbers and their negative counterparts. For example, -15, -3, 0, 1, 5, and 198 are all integers. Numbers like 3.14, 1/2, and -0.5 are not integers because they have fractional or decimal parts.
2. What are the main types of integers?
Integers can be classified into three main types based on their value:
- Positive Integers: These are all the whole numbers greater than zero (1, 2, 3, 4, ...). They are also known as natural numbers.
- Negative Integers: These are all the numbers less than zero that are the opposites of the positive integers (-1, -2, -3, -4, ...).
- Zero: The number 0 is an integer that is considered neutral; it is neither positive nor negative.
3. What are the key properties of integers under addition and multiplication?
Integers follow several key properties for the operations of addition and multiplication:
- Closure Property: The sum or product of any two integers is always an integer. (e.g., 5 + (-2) = 3; 5 × (-2) = -10).
- Commutative Property: The order in which two integers are added or multiplied does not change the result. (e.g., a + b = b + a; a × b = b × a).
- Associative Property: The way integers are grouped when adding or multiplying three or more integers does not change the result. (e.g., (a + b) + c = a + (b + c)).
- Distributive Property: Multiplication of an integer distributes over the addition of two other integers. (e.g., a × (b + c) = a × b + a × c).
- Identity Property: The additive identity is 0 (since a + 0 = a), and the multiplicative identity is 1 (since a × 1 = a).
4. Where can we see examples of integers in real life?
Integers are essential for describing situations involving values above and below a central point. Common real-world examples include:
- Temperature: Temperatures can be positive (like 25°C) or negative (like -5°C).
- Finance: Bank account transactions use positive integers for deposits (credits) and negative integers for withdrawals (debits).
- Elevation: Geographical locations are measured in relation to sea level. A mountain's height is a positive integer (e.g., 8,848 meters), while a point below sea level is a negative integer (e.g., -420 meters).
- Games and Sports: Goal differences, scores, or yardage gains/losses in sports are represented using integers.
5. What is the difference between whole numbers and integers?
The key difference is that the set of integers includes negative numbers, while the set of whole numbers does not. Whole numbers are all the positive numbers starting from zero: {0, 1, 2, 3, ...}. Integers include all these whole numbers plus their negative counterparts: {..., -3, -2, -1, 0, 1, 2, 3, ...}. Therefore, every whole number is an integer, but not every integer is a whole number.
6. How do signs affect the result when multiplying or dividing integers?
The rules for signs in multiplication and division are straightforward:
- If the two integers have the same sign (both positive or both negative), the result is always positive. Example: (-5) × (-3) = 15.
- If the two integers have different signs (one positive and one negative), the result is always negative. Example: (10) ÷ (-2) = -5.
This consistency is a fundamental rule for integer operations.
7. Why is zero considered an integer but is neither positive nor negative?
Zero is a special integer that acts as the origin or a dividing point on the number line. It is neither positive nor negative because it represents a state of 'nothing' or 'no value'. Positive integers (1, 2, 3...) lie to its right, representing quantities, while negative integers (-1, -2, -3...) lie to its left, representing deficits or opposites. Because it is the boundary between them, it does not belong to either group.
