Addition and Subtraction of Integers (Rules and Examples)

Edited By Team Careers360 | Updated on Jul 02, 2025 05:27 PM IST

Positive integers are produced when two positive numbers are added as opposed to two negative integers, which results in a sum with a negative sign. However, adding two different signed integers will only yield subtraction, with the sign of the result matching that of the larger number. Integers can be increased or decreased in value using the operations of addition and subtraction. Whole numbers and negative numbers are both integers. An integer is any number displayed on a number line without a fractional component. The absolute value of an integer on a number line is the distance between a number and 0.

This Story also Contains

What Are Addition And Subtraction?
Rules For Addition And Subtraction
Rules For Addition
Rules For Subtraction
Properties Of Addition Of Integers
Properties Of Subtraction Of Integers
How To Add Integers On A Number Line?
How To Subtract Integers On A Number Line?
Points To Remember

What Are Addition And Subtraction?

The two main arithmetic operations in mathematics are addition and subtraction. In addition to these two operations, multiplication and division are the other two fundamental operations we learn in elementary math. The value added to the original value is represented by the addition. In addition to integers, rational and irrational numbers can also be added to and subtracted from. Both operations are therefore applicable to all real and complex numbers. Additionally, the same rules that apply to performing algebraic operations apply to adding and subtracting algebraic expressions.

Rules For Addition And Subtraction

Positive, negative, and zero-valued numbers that are not fractions are referred to as integers. There are universal rules for addition and subtraction. The integers that we add or subtract may be positive or negative. Thus, understanding the principles for positive and negative symbols is essential.

Rules For Addition

A positive integer is obtained when two positive integers are added. For instance, 10 + 6 =16.
When a positive number and a negative number are added, the operation and the output are determined by the sign of the larger number. For instance, 15 + (-20) = - 5.
A sum of integers with a negative sign is obtained when two negative numbers are added. For instance, (- 10) + (- 6) = - 16.
An integer when added to the inverse of itself, results in zero. For instance, 6 + (- 6) = 0.

Rules For Subtraction

Calculations are made easier by rewriting subtraction questions as additional questions. To accomplish this, change the subtraction sign to an addition sign. Take the inverse of the number that follows the sign after you have converted the sign. If the signs of the two numbers match, the absolute values are added, and the common sign is attached. If the signs of the two numbers differ, we calculate the difference between the absolute values and assign the larger number's sign to the output.

As an illustration, (- 2) - 4 = - 6

2 - 6 = - 4

Properties Of Addition Of Integers

Closure property - Any two integers added together to yield an integer. For instance, 6 and 4 are integers which when added result in an integer, that is, 10.
Commutative property - Any two integers added together have the same sum regardless of the sequence in which they are added. For instance, 2 + 3 = 3 + 2 = 5.
Associative property - When the sum of three or more numbers is calculated, the order in which the integers are grouped is irrelevant. For example, (2 + 3) + 5 = 2 + (3 + 5) = 10.
Additive identity - When you add zero to any number, the outcome is always an integer. The number zero is the additive identity. For instance, 0 + 6 = 6.
Additive inverse - When an integer is added to an integer, the result is always 0. The two opposite numbers are known as their additive inverses. For instance, 6 + (- 6) = 0.

NEET Highest Scoring Chapters & Topics

This ebook serves as a valuable study guide for NEET exams, specifically designed to assist students in light of recent changes and the removal of certain topics from the NEET exam.

Download E-book

Properties Of Subtraction Of Integers

Closure property - Any two integers are given, and the difference between them yields an integer. For instance, 6 and 4 are integers which when subtracted result in an integer, that is, 2.
Commutative property - When the order is reversed, the difference between any two provided integers changes. For instance, 6 - 4 = 2 but 4 - 6 = - 2.
Associative property - If there is a change in the grouping of three or more integers, the outcome of the subtraction technique will change. For instance, 8 - (3 - 6) = 11 but (8 - 3) - 6 = -1.

How To Add Integers On A Number Line?

We travel to the left side of the number line while adding a negative number.
We move to the right side of the number line while adding a positive number.

How To Subtract Integers On A Number Line?

The operation switches to addition after the subtraction fact is changed to an addition fact, allowing us to add numbers on a number line according to the same principles.
It's important to keep in mind that when adding a negative number, we move to the left side of the number line, and when adding a positive number, we move to the right side.

Points To Remember

When adding integers, maintain the same sign when the signs are the same.
Add the integers when the signs are different, but keep the sign of the absolute value with the higher number.
When subtracting integers, transform the subtraction sign to an addition sign and the second number's sign to the opposite.
If a number has no accompanying sign, we consider it to be positive.
Any fact that involves subtraction can be converted into an addition fact.
In an expression, negative integers are always included in brackets.
We go to the left side of the number line as we add a negative number.
We go to the right side of the number line as we add a positive number.

Frequently Asked Questions (FAQs)

1. Define an integer and its absolute value.

An integer is any number displayed on a number line without a fractional component. The absolute value of an integer on a number line is the distance between a number and 0.

2. State the rules for addition.

The various rules for addition are as follows:

A positive integer is obtained when two positive integers are added.
When a positive number and a negative number are added, the operation and the output are determined by the sign of the larger number.
A sum of integers with a negative sign is obtained when two negative numbers are added.
An integer when added to the inverse of itself, results in zero.

3. What is the commutative property for an addition?

Any two integers added together have the same sum regardless of the sequence in which they are added. For instance, 2 + 3 = 3 + 2 = 5.

4. What is the closure property for subtraction?

Any two integers are given, and the difference between them yields an integer. For instance, 6 and 4 are integers which when subtracted result in an integer, that is, 2.

5. How to add integers on a number line?

The steps to be followed in order to add integers on a number line are as follows:

We travel to the left side of the number line while adding a negative number.
We move to the right side of the number line while adding a positive number.

6. How does adding two negative integers differ from adding two positive integers?

When adding two negative integers, the result is always negative and has a greater absolute value than either of the original numbers. For example, (-3) + (-4) = -7. With positive integers, the result is positive and also larger than the original numbers. The key is to understand that adding negatives means moving further left on the number line.

7. What role does the number line play in understanding integer addition and subtraction?

The number line is a visual tool that helps in understanding integer operations:

8. How can you mentally calculate -15 + 23 quickly?

To mentally calculate -15 + 23:

9. What's the importance of the number zero in integer operations?

Zero plays a crucial role in integer operations:

10. How does understanding integer operations help in algebra?

Understanding integer operations is fundamental to algebra because:

11. How can the number line help visualize addition and subtraction of integers?

The number line is a powerful tool for visualizing integer operations. Addition moves to the right, subtraction to the left. Positive numbers move right, negative numbers move left. For example, to add 5 + (-3), start at 5 and move 3 units left, landing on 2. This visual representation helps understand the direction and magnitude of integer operations.

12. What's the difference between -5 + 3 and 3 - 5?

While -5 + 3 and 3 - 5 both equal -2, they represent different thought processes. -5 + 3 starts at -5 and moves 3 units right on the number line. 3 - 5 starts at 3 and moves 5 units left. Understanding these differences helps in problem-solving and equation manipulation.

13. How does the concept of opposites relate to integer addition and subtraction?

Opposites in integers are numbers that are the same distance from zero but on opposite sides of the number line. Adding opposites always results in zero (e.g., 5 + (-5) = 0). In subtraction, subtracting a number is the same as adding its opposite (e.g., 7 - 3 = 7 + (-3)). This concept is fundamental to understanding integer operations.

14. Why is subtracting a larger number from a smaller one possible with integers?

Subtracting a larger number from a smaller one is possible with integers because we can represent numbers less than zero. For example, 3 - 7 = -4. This operation isn't possible with just natural numbers, showcasing how integers expand our mathematical capabilities to represent real-world scenarios like debt or temperatures below freezing.

15. Why is it important to keep track of signs when working with integers?

Keeping track of signs is crucial because they determine the direction and nature of the result. Mishandling signs can lead to errors in calculations and misinterpretation of real-world situations. For example, confusing -5 + 3 with 5 + 3 leads to very different outcomes. Proper sign management is essential for accurate problem-solving.

16. What's the relationship between addition and subtraction of integers?

Addition and subtraction of integers are inverse operations. Subtracting an integer is the same as adding its opposite. For example, 8 - 3 is the same as 8 + (-3). This relationship helps in solving equations and understanding the connection between these operations on the number line.

17. Why is -(-5) equal to 5?

The negative of a negative number is positive because it's like reversing a reversal. Think of it as changing direction twice: if you face north and turn around twice, you're facing north again. Mathematically, it's because multiplying two negative numbers results in a positive number.

18. How can you determine the sign of the result when adding or subtracting integers?

To determine the sign of the result:

19. Why does subtracting a negative number give the same result as adding its positive counterpart?

Subtracting a negative number is equivalent to adding its positive counterpart because it's like removing a debt. For instance, 5 - (-3) is the same as 5 + 3 = 8. Think of it as canceling out the negative effect, which results in a positive outcome.

20. How can you use real-life examples to understand integer operations?

Real-life examples make integer operations more tangible:

21. How does the commutative property apply to addition of integers?

The commutative property states that the order of addends doesn't affect the sum. This applies to integers as well. For example, 3 + (-5) = (-5) + 3 = -2. This property helps simplify calculations and understand the relationship between positive and negative numbers in addition.

22. How can you explain the concept of "zero pairs" in integer addition?

Zero pairs are pairs of integers that sum to zero, like 5 and -5. They're useful in simplifying complex additions. For instance, in (-2) + 5 + (-5) + 2, we can identify (5 + -5) as a zero pair and (2 + -2) as another. Removing these leaves us with 0, the final sum. This concept helps in understanding how positive and negative numbers interact.

23. How does the associative property work with integer addition?

The associative property states that the grouping of addends doesn't affect the sum. With integers, (a + b) + c = a + (b + c), regardless of whether the numbers are positive or negative. For example, (2 + (-3)) + 5 = 2 + ((-3) + 5) = 4. This property is useful in mental math and algebraic manipulations.

24. What's the relationship between addition, subtraction, and the concept of opposites in integers?

The relationship between addition, subtraction, and opposites in integers is fundamental:

25. Why is it important to understand the difference between -5 + 3 and -(5 + 3)?

Understanding the difference between -5 + 3 and -(5 + 3) is crucial because:

26. Why doesn't subtraction of integers follow the commutative property?

Subtraction is not commutative because changing the order of numbers in subtraction changes the result. For example, 5 - 3 ≠ 3 - 5. This is because subtraction is not a symmetric operation; it matters which number is being subtracted from which. Understanding this helps avoid common mistakes in integer calculations.

27. What's the difference between 5 - (-3) and 5 + 3?

While 5 - (-3) and 5 + 3 both equal 8, they represent different concepts. 5 - (-3) means subtracting a negative 3, which is equivalent to adding 3. 5 + 3 is straightforward addition. Understanding this difference is crucial for interpreting and solving more complex equations involving negative numbers.

28. How can you use the concept of debt to explain negative numbers in addition and subtraction?

The concept of debt is an excellent way to explain negative numbers:

29. Why is it that (-a) - (-b) = -a + b?

The expression (-a) - (-b) = -a + b can be understood by breaking it down:

30. How can you explain why subtracting a negative number makes a number larger?

Subtracting a negative number makes a number larger because it's equivalent to adding its positive counterpart. Think of it as removing a debt – if you remove a $5 debt, you're $5 richer. Mathematically, a - (-b) = a + b. This concept is key to understanding how negative numbers behave in subtraction.

31. How can you use the concept of temperature to explain integer addition and subtraction?

Temperature is an excellent real-world application for integer operations:

32. Why is it that adding two negative numbers always results in a negative number?

Adding two negative numbers always results in a negative number because:

33. How does the concept of "net change" relate to integer addition and subtraction?

The concept of "net change" in integer operations refers to the overall effect of multiple additions and subtractions:

34. Why is it important to understand the concept of "opposite operations" in integer arithmetic?

Understanding "opposite operations" in integer arithmetic is crucial because:

35. How does the concept of "taking away" change when dealing with negative numbers?

The concept of "taking away" with negative numbers can be counterintuitive:

36. Why is it that subtracting a smaller number from a larger negative number results in a number closer to zero?

When subtracting a smaller number from a larger negative number, the result is closer to zero because:

37. What's the difference between -(-5) and -5?

-(-5) is positive 5, while -5 is negative 5. The double negative in -(-5) cancels out, resulting in a positive number. This is similar to how in language, a double negative often creates a positive meaning. Understanding this difference is crucial for correctly interpreting and solving equations with nested negative signs.

38. How can you use the concept of opposite directions to explain integer subtraction?

Integer subtraction can be explained using opposite directions:

39. What's the importance of understanding that addition and subtraction can be done in any order when there are multiple operations?

Understanding that addition and subtraction can be done in any order (when properly accounted for) is important because:

40. What are integers and why do we need them?

Integers are whole numbers that can be positive, negative, or zero. We need them to represent quantities that can go below zero, like temperatures, elevations below sea level, or financial losses. They allow us to express a wider range of real-world situations mathematically.

41. What's the difference between absolute value and the actual value of an integer?

The absolute value of an integer is its distance from zero on the number line, regardless of its sign. For example, both 5 and -5 have an absolute value of 5. The actual value includes the sign, so -5 is less than 5. Understanding this difference is crucial for comparing integers and solving equations.

42. How does the concept of absolute value relate to addition and subtraction of integers?

Absolute value is the distance of a number from zero on the number line, regardless of its sign. In integer operations:

43. How can you explain why (-5) + (-3) = -8 using the number line?

Using the number line to explain (-5) + (-3) = -8:

44. How does the distributive property work with integer addition and subtraction?

The distributive property with integers works as follows: