Numbers are an essential part of everyday life. We use them to count objects, measure distances, calculate money, record temperatures, and describe changes in quantities. However, not every situation can be explained using only positive numbers. For example, a temperature can fall below zero, a bank account can have a negative balance, and a person can move backward from a starting point. To represent these situations accurately, we use integers.
Integers include positive whole numbers, negative whole numbers, and zero. They help us describe quantities that increase or decrease, positions above or below a reference point, and values that can be greater or less than zero. Understanding integers is an important foundation for learning arithmetic, algebra, and many other areas of mathematics. In this article, we will learn what integers are, how positive and negative numbers work, how to compare integers, and how to perform basic mathematical operations using them.
What Are Integers?
Integers are numbers that include all positive whole numbers, all negative whole numbers, and zero. They do not include fractions or decimal numbers.
Positive integers are greater than zero, while negative integers are less than zero. Zero is neither positive nor negative.
For example, numbers such as 1, 2, 3, and 10 are positive integers. Numbers such as −1, −2, −3, and −10 are negative integers. The number 0 is also an integer.
Examples of integers:
Positive integers: 1, 2, 3, 4, 5, 10, 25
Negative integers: −1, −2, −3, −4, −5, −10, −25
Zero: 0
The complete set of integers extends infinitely in both directions. There is no largest positive integer and no smallest negative integer because we can always find another integer that is greater or smaller.
For instance, 100 is an integer, but 101 is greater. Similarly, −100 is an integer, but −101 is smaller.
Integers are commonly represented by the symbol ℤ.
Text block — Integer notation:
ℤ = {…, −3, −2, −1, 0, 1, 2, 3, …}
This notation indicates that integers continue indefinitely in both negative and positive directions.
Understanding Positive Numbers
Positive integers are whole numbers greater than zero. They represent quantities that are above zero, gains, increases, or movements in a positive direction.
In mathematics, a positive integer can be written with or without a plus sign. For example, +5 and 5 represent the same number.
Examples of positive integers:
+1
+4
+8
+15
+100
The plus sign is optional because a number without a negative sign is generally understood to be positive when discussing integers, unless the number is zero.
Real-Life Examples of Positive Integers
Positive integers appear in many everyday situations.
1. Temperature above zero
If the temperature is 12°C above zero, it can be represented as +12°C.
2. Money earned
If a person earns ₹500, the amount gained can be represented as +500 when describing a change in their money.
3. Floors above ground level
In a building, the third floor above ground level may be represented as +3 when the ground floor is designated as zero.
4. Increase in quantity
If a shop receives 20 additional books, the change in its stock is +20 books.
These examples show that positive integers are useful for representing gains, increases, and positions above a chosen reference point.
Understanding Negative Numbers
Negative integers are whole numbers less than zero. They are written with a minus sign before the number.
Examples include −1, −2, −5, −12, and −100.
The minus sign is important because it distinguishes a negative integer from its corresponding positive integer. For example, 7 and −7 have the same numerical distance from zero but represent opposite values.
Negative numbers are especially useful when describing quantities below a reference point, losses, decreases, or movement in the opposite direction.
Real-Life Examples of Negative Integers
1. Temperature below zero
If the temperature is 5°C below zero, it is written as −5°C.
2. Financial loss
If a business loses ₹800, the change in its financial position can be represented as −800.
3. Floors below ground level
A parking level two floors below ground may be represented as −2 if ground level is designated as zero.
4. Decrease in quantity
If the water level in a tank falls by 10 centimetres, the change can be represented as −10 cm.
Negative numbers do not always mean something is physically bad or undesirable. They simply describe a value in the negative direction relative to a reference point.
What Is Zero in Integers?
Zero is an important integer because it separates positive integers from negative integers.
Zero represents the absence of a positive or negative amount in many situations. For example, a temperature of 0°C is the point at which the Celsius scale separates positive temperatures from negative temperatures.
However, zero does not always mean that nothing physically exists. A temperature of 0°C still describes a measurable thermal condition.
Zero has three important properties:
Zero is an integer.
Zero is neither positive nor negative.
Zero is the reference point between positive and negative integers on a number line.
For example, in a bank account, a balance of zero means there is no money remaining in the account balance. On a number line, zero separates the negative integers on the left from the positive integers on the right.
Understanding zero makes it easier to compare integers and perform calculations involving positive and negative numbers.
Integers on a Number Line
A number line is a straight line on which numbers are arranged in order. It helps us understand the positions of integers and compare their values.
Zero is placed at the centre of the integer sequence shown in a typical number line. Positive integers are placed to the right of zero, while negative integers are placed to the left.
Text block — Number line representation:
−5 — −4 — −3 — −2 — −1 — 0 — 1 — 2 — 3 — 4 — 5
Each consecutive integer is separated by an equal distance.
The number line follows a simple rule: moving to the right increases the value, while moving to the left decreases the value.
For example, 3 is greater than 1 because 3 is located to the right of 1. Similarly, −2 is greater than −5 because −2 is located to the right of −5.
How to Read a Number Line
To read a number line correctly, follow these steps:
Locate zero as the reference point.
Move right to find positive integers.
Move left to find negative integers.
Compare the positions of the numbers to determine which is greater or smaller.
For example, if you compare −3 and 2, the number 2 is to the right of −3. Therefore, 2 is greater than −3.
A number line is also helpful when learning addition and subtraction because these operations can be understood as movements in different directions.
Comparing Positive and Negative Integers
Comparing integers means determining whether one integer is greater than, less than, or equal to another integer.
The symbols used for comparison are:
Greater than: >
Less than: <
Equal to: =
The position of an integer on the number line determines its relationship with another integer.
Rules for Comparing Integers
Rule 1: Every positive integer is greater than zero.
For example:
Text block — Comparison:
5 > 0
Rule 2: Every negative integer is less than zero.
For example:
Text block — Comparison:
−4 < 0
Rule 3: Every positive integer is greater than every negative integer.
For example:
Text block — Comparison:
3 > −7
Rule 4: Among positive integers, the integer with the greater numerical value is greater.
For example:
Text block — Comparison:
9 > 4
Rule 5: Among negative integers, the integer closer to zero is greater.
For example:
Text block — Comparison:
−2 > −8
This last rule can initially seem confusing. Although 8 is greater than 2 when considering positive numbers, −8 is smaller than −2. On the number line, −2 lies to the right of −8.
Arranging Integers in Ascending and Descending Order
Ascending order means arranging numbers from smallest to largest. Descending order means arranging them from largest to smallest.
Consider the integers −4, 3, 0, −1, and 5.
Text block — Ascending order:
−4, −1, 0, 3, 5
Text block — Descending order:
5, 3, 0, −1, −4
Arranging integers in order is useful in temperature comparisons, financial calculations, measurements, and mathematical problem-solving.
Absolute Value of an Integer
The absolute value of an integer is its distance from zero on the number line, without considering direction. Absolute value is always zero or positive.
The absolute value symbol consists of two vertical bars placed around a number.
For example, the absolute value of 5 is 5, and the absolute value of −5 is also 5. Both numbers are five units away from zero.
Text block — Absolute value notation:
|5| = 5
|−5| = 5
|0| = 0
Absolute value describes distance rather than direction. This is useful when determining how far a temperature is from zero or how much a value differs from a reference point.
For example, if the temperature is −8°C, its absolute value is 8. This means the temperature is eight degrees below zero, not eight degrees above zero.
Remember that absolute value does not change a negative temperature into a positive temperature in the original situation. It only describes the distance from zero.
Basic Mathematical Operations with Integers
Integers can be added, subtracted, multiplied, and divided. However, positive and negative signs affect the results of these operations.
Understanding the rules for each operation helps prevent common calculation errors.
Addition of Integers
When adding integers with the same sign, add their numerical values and keep the common sign.
For example, adding two positive integers gives a positive result.
Text block — Addition of positive integers:
(+4) + (+3) = +7
Similarly, adding two negative integers gives a negative result.
Text block — Addition of negative integers:
(−4) + (−3) = −7
When adding integers with different signs, subtract the smaller absolute value from the larger absolute value. The result takes the sign of the integer with the greater absolute value.
Text block — Addition of integers with different signs:
(+8) + (−3) = +5
(−9) + (+4) = −5
For the first example, the positive value has the greater absolute value, so the result is positive. In the second example, the negative value has the greater absolute value, so the result is negative.
If two integers have equal absolute values but opposite signs, their sum is zero.
Text block — Opposite integers:
(+6) + (−6) = 0
Subtraction of Integers
Subtraction can be understood as adding the opposite of a number. This means that subtracting a positive integer is equivalent to adding a negative integer, while subtracting a negative integer is equivalent to adding a positive integer.
Text block — Subtraction rule:
a − b = a + (−b)
Here, a and b represent integers.
For example:
Text block — Subtraction examples:
7 − 3 = 4
5 − 8 = −3
6 − (−2) = 6 + 2 = 8
−4 − 3 = −4 + (−3) = −7
The rule for subtracting a negative number is especially important. When a negative integer is subtracted, its opposite is added.
For example, subtracting −2 from 6 gives 8, not 4, because the operation becomes 6 + 2.
Multiplication of Integers
Multiplication follows a sign rule. If both integers have the same sign, their product is positive. If they have opposite signs, their product is negative.
Text block — Multiplication sign rules:
Positive × Positive = Positive
Negative × Negative = Positive
Positive × Negative = Negative
Negative × Positive = Negative
Examples:
Text block — Multiplication examples:
4 × 3 = 12
(−4) × (−3) = 12
(−4) × 3 = −12
4 × (−3) = −12
These rules apply regardless of the numerical values involved. For example, multiplying −7 by −5 gives 35, while multiplying −7 by 5 gives −35.
Division of Integers
Division follows the same sign rules as multiplication, provided the divisor is not zero.
If both integers have the same sign, the quotient is positive. If they have different signs, the quotient is negative.
Text block — Division examples:
12 ÷ 3 = 4
(−12) ÷ (−3) = 4
(−12) ÷ 3 = −4
12 ÷ (−3) = −4
An important rule is that division by zero is undefined. For example, 5 ÷ 0 does not produce a valid numerical result.
Also, the quotient of two integers is not always an integer. For example, 5 ÷ 2 = 2.5. Although both 5 and 2 are integers, their quotient is not an integer.
Properties of Integers
Integers follow several mathematical properties that make calculations easier and more systematic.
Closure Property
Integers are closed under addition, subtraction, and multiplication. This means that adding, subtracting, or multiplying any two integers always produces another integer.
Examples:
Text block — Closure examples:
3 + (−5) = −2
4 − 9 = −5
(−3) × 2 = −6
However, integers are not closed under division because dividing two integers can produce a fraction or decimal.
For example:
Text block — Division example:
5 ÷ 2 = 2.5
Since 2.5 is not an integer, division does not always produce an integer.
Commutative Property
The commutative property states that changing the order of two numbers does not change the result of addition or multiplication.
Text block — Commutative property:
a + b = b + a
a × b = b × a
For example, 4 + (−2) and (−2) + 4 both equal 2.
However, subtraction and division are not generally commutative. For example, 7 − 3 is not equal to 3 − 7.
Associative Property
The associative property states that changing the grouping of integers does not change the result of addition or multiplication.
Text block — Associative property:
(a + b) + c = a + (b + c)
(a × b) × c = a × (b × c)
For example, (2 + 3) + (−4) equals 1, and 2 + [3 + (−4)] also equals 1.
This property is useful when simplifying expressions containing several integers.
Additive Identity and Additive Inverse
Zero is the additive identity because adding zero to any integer leaves the integer unchanged.
Text block — Additive identity:
a + 0 = a
Every integer also has an additive inverse. The additive inverse is the number that produces zero when added to the original integer.
For example:
Text block — Additive inverse:
5 + (−5) = 0
−8 + 8 = 0
The additive inverse of 5 is −5, and the additive inverse of −8 is 8.
Difference Between Integers, Whole Numbers, and Natural Numbers
Integers are related to natural numbers and whole numbers, but these number sets are not identical.
Natural numbers are commonly used for counting, such as 1, 2, 3, 4, and so on. Some mathematical conventions include zero among the natural numbers, so the definition can vary depending on the context.
Whole numbers include zero and all positive integers. They do not include negative integers.
Integers include negative whole numbers, zero, and positive whole numbers.
Text block — Number set examples:
Natural numbers: 1, 2, 3, 4, 5, …
Whole numbers: 0, 1, 2, 3, 4, 5, …
Integers: …, −3, −2, −1, 0, 1, 2, 3, …
The main difference is that integers include negative numbers, while whole numbers do not. This makes integers suitable for representing quantities that can fall below zero.
Fractions such as 1/2 and decimal numbers such as 3.75 are not integers.
Applications of Integers in Everyday Life
Integers are not limited to mathematical exercises. They are used in many real-world situations where quantities must be described relative to a reference point.
Temperature Measurement
Weather reports frequently use positive and negative temperatures. A temperature of 15°C is above zero, while −5°C is below zero. Comparing these values helps people understand differences in weather conditions.
Banking and Finance
Integers can represent gains and losses in financial calculations. A deposit of ₹1,000 can be represented as +1,000, while a loss of ₹300 can be represented as −300 when recording changes in a balance.
Actual account balances may also involve decimal values because of currency subdivisions and interest calculations.
Elevation and Depth
Geographers use positive and negative values to describe positions relative to sea level. Locations above sea level can be represented by positive elevations, while depths below a chosen sea-level reference can be represented by negative values.
Sports and Games
In some sports and games, scores or point differences can be positive or negative. A team with a point difference of +8 has a positive difference, while a team with −3 has a negative difference.
Science and Technology
Integers are used in computer programming, temperature measurements, data processing, and mathematical models. For example, a program might use positive integers to count items and negative integers to represent changes in a value.
These applications demonstrate why understanding integers is useful beyond the classroom.
Common Mistakes to Avoid When Working with Integers
Learners often make a few predictable mistakes when working with positive and negative numbers. Recognizing these errors makes integer calculations easier.
1. Assuming that a larger negative sign means a greater value
A common mistake is thinking that −9 is greater than −3 because 9 is greater than 3. In reality, −9 is smaller than −3 because it lies farther to the left on the number line.
2. Treating zero as positive or negative
Zero is neither positive nor negative. It is the boundary between positive and negative integers.
3. Forgetting the sign when adding negative integers
When adding two negative integers, add their absolute values and keep the negative sign.
For example:
Text block — Correct calculation:
−5 + (−4) = −9
4. Confusing subtraction of a negative number
Subtracting a negative number is equivalent to adding its opposite.
For example:
Text block — Correct calculation:
8 − (−3) = 11
5. Applying multiplication rules incorrectly
Two integers with the same sign produce a positive product, while integers with different signs produce a negative product.
Practising these rules with simple examples and number-line diagrams can help reduce errors.
Conclusion
Integers are an important part of mathematics because they allow us to represent positive values, negative values, and zero using a single number system. Positive integers describe values greater than zero, negative integers describe values below zero, and zero serves as the dividing point between them. A number line makes it easier to understand their positions, compare their values, and visualize mathematical operations.
By learning the rules for addition, subtraction, multiplication, and division, we can solve integer problems accurately. Understanding absolute value and the properties of integers also provides a strong foundation for more advanced mathematical topics. Since integers are used in temperature measurement, finance, elevation, computing, and everyday calculations, mastering them is a valuable step toward developing confidence in mathematics.
FAQs
1. What are integers in mathematics?
Integers are numbers that include all positive whole numbers, all negative whole numbers, and zero. They do not include fractions or decimal numbers. Examples of integers are −5, −2, 0, 3, and 10. Integers are useful for representing quantities above or below a reference point, such as temperatures, financial gains and losses, and elevations. They are commonly represented by the symbol ℤ. Understanding integers helps learners develop a strong foundation in arithmetic and algebra because many mathematical operations and equations involve positive and negative numbers.
2. What is the difference between positive and negative integers?
Positive integers are whole numbers greater than zero, while negative integers are whole numbers less than zero. Positive integers are written with a plus sign or without a sign, such as +5 or 5. Negative integers always have a minus sign, such as −5. On a number line, positive integers appear to the right of zero, while negative integers appear to the left. Positive integers can represent gains or increases, whereas negative integers can represent losses or decreases. Both types are important for solving mathematical problems and describing real-life situations involving changes in quantities.
3. Is zero a positive integer or a negative integer?
Zero is neither a positive integer nor a negative integer. It is an integer that separates positive numbers from negative numbers on the number line. Positive integers are greater than zero, while negative integers are less than zero. For example, 4 is positive, and −4 is negative, but 0 has neither classification. Zero can represent a starting point, a neutral value, or the absence of a particular quantity. It also plays an important role in arithmetic because adding zero to any integer leaves that integer unchanged. Therefore, zero is an essential part of the integer number system.
4. How can integers be represented on a number line?
Integers can be represented on a number line by placing zero at a reference point and arranging the other integers at equal distances on either side. Positive integers appear to the right of zero, while negative integers appear to the left. For example, −3 is three units to the left of zero, and 3 is three units to the right. Moving right increases the numerical value, while moving left decreases it. A number line helps learners compare integers, understand their order, and perform addition and subtraction. It is particularly useful when visualizing calculations involving positive and negative numbers.
5. Which integer is greater, −3 or −8?
The integer −3 is greater than −8 because −3 is located to the right of −8 on the number line. When comparing negative integers, the number closer to zero is always greater. Although 8 is greater than 3 when comparing positive numbers, the relationship reverses for their negative counterparts. For example, −2 is greater than −7, and −5 is greater than −9. Remember that negative integers represent values below zero, so moving farther to the left makes a number smaller. Understanding this rule helps prevent common mistakes when comparing and arranging negative numbers.
6. What is the absolute value of an integer?
The absolute value of an integer is its distance from zero on the number line, regardless of direction. It is always zero or positive. Absolute value is represented by two vertical bars around a number. For example, the absolute value of 6 is 6, and the absolute value of −6 is also 6. Both numbers are six units away from zero. The absolute value of zero is zero. This concept is useful in mathematics when calculating distances, comparing differences, and understanding the magnitude of positive and negative values. It measures distance rather than indicating whether a number is positive or negative.
7. What are the rules for adding positive and negative integers?
When adding integers with the same sign, add their absolute values and keep the common sign. For example, 4 + 3 = 7, while −4 + (−3) = −7. When adding integers with different signs, subtract the smaller absolute value from the larger one and keep the sign of the number with the greater absolute value. For example, 8 + (−3) = 5, while −9 + 4 = −5. If two integers have equal absolute values but opposite signs, their sum is zero. Practising these rules makes integer addition easier and more accurate.
8. What happens when you subtract a negative integer?
Subtracting a negative integer is equivalent to adding its positive opposite. This rule is important when performing calculations involving negative numbers. For example, 7 − (−3) becomes 7 + 3, which equals 10. Similarly, 5 − (−4) equals 9. The reason is that subtracting a quantity changes the direction of the operation, so removing a negative value increases the result. However, this does not mean that every subtraction produces a positive answer. The final result depends on the values involved. Understanding this rule helps learners simplify integer expressions and avoid common errors in arithmetic.
9. Are integers closed under addition, subtraction, multiplication, and division?
Integers are closed under addition, subtraction, and multiplication because performing any of these operations on two integers always produces another integer. For example, 5 + (−2) = 3, 4 − 9 = −5, and (−3) × 2 = −6. However, integers are not closed under division because the result may be a fraction or decimal. For example, 5 ÷ 2 = 2.5, which is not an integer. Division by zero is also undefined. Understanding closure helps learners identify which mathematical operations always keep their results within the integer number system.
10. Where are integers used in everyday life?
Integers are used in many everyday situations that involve positive and negative values. Weather reports use negative integers to describe temperatures below zero and positive integers for temperatures above zero. In finance, positive values can represent gains, while negative values can represent losses or changes in a balance. Elevations can be expressed relative to sea level, with negative values representing locations below the reference level. Integers are also useful in sports scoring, computer programming, and mathematical calculations. Learning integers helps people interpret numerical information accurately and understand changes in quantities, positions, and measurements in real-world situations.
