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Maths

Integers Rules Explained with Signs and Operations

Integers Rules Explained with Signs and Operations

Q: 1. What are the basic rules of integers?

The basic rules of integers explain how to add, subtract, multiply, and divide positive and negative numbers correctly.Addition: Same signs → add and keep the sign; different signs → subtract and keep the sign of the larger number.Subtraction: Change subtraction to addition of the opposite.Multiplication: Same signs → positive; different signs → negative.Division: Same signs → positive; different signs → negative.These integer rules are essential for solving arithmetic problems involving negative numbers.

Q: 6. What is the absolute value of an integer?

The absolute value of an integer is its distance from zero on the number line, always written as a non-negative number.Notation: |a||5| = 5|−5| = 5Absolute value is used when comparing magnitudes in integer operations.

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What Are the Rules for Adding Subtracting Multiplying and Dividing Integers

In Mathematics, every concept involves numbers. In this article, you will be learning about how to add integers and how to subtract two integers. Addition and Subtraction operations come under the Arithmetics part of Mathematics. Arithmetic operations other than addition and subtraction are multiplication and division. When you have to find the sum of two variables, you apply the addition operation between the two numbers or variables. When the difference between the two numbers has to be found, then the subtraction operation is applied between the two numbers or variables. These arithmetic operations are applicable to not just integers, but also real numbers, whole numbers, rational numbers, etc. The table below shows what property is applicable to addition and subtraction.

Property	Addition	Subtraction
Closure	j + k ∈ Z	j - k ∈ Z
Commutative	j + k = k + j	j - k ≠ j - k
Associative	( j + k ) + l = j + ( k + l )	( j - k ) - l ≠ j - ( k - l )
Distributive	j x ( k + l ) = j x k + j x l	j x ( k - l ) = j x k - j x l

Rules for Addition and Subtraction of Integers

Integers are a variety of numbers. They range from positive numbers to negative numbers and also includes the number zero. Integers are anything other than fractions. The rules applicable for addition and subtraction are same for both natural numbers as well as integers because natural numbers are nothing but integers itself. Before we move forwards and learn the rules for addition and subtraction, let’s understand the positive and negative rules that are applied during addition and subtraction operations.

Positive and Negative Rules

When two integers have the same sign or different signs, the product will have either a negative or positive sign. To understand it better, follow the table shown below.

Signs	Result
( + ) x ( + ) = ( + )	Positive x Positive = Positive
( + ) x ( - ) = ( - )	Positive x Negative = Negative
( - ) x ( + ) = ( - )	Negative x Positive = Negative
( - ) x ( - ) = ( + )	Negative x Negative = Positive

Rules

When a positive integer is multiplied by a positive integer the result is going to be positive.
When a negative integer is multiplied by a positive integer the result is going to be negative.
When a positive integer is multiplied by a negative integer the result is going to be positive.
When a negative integer is multiplied by a negative integer the result is going to be positive.
When there is no symbol present before the integer, it is considered to be positive.

Addition of Integers

There are three possibilities when the addition operation is applied between two integers:

The addition operation is applied between two positive integers.
The addition operation is applied between negative positive integers.
The addition operation is applied between one positive integer and one negative integer.

Addition Rules for Integers

Types of Numbers	Operation Applied	Final Result	Example
( + ) + ( + )	Add	Positive ( + )	1449 + 1854 = 3303
( - ) + ( - )	Add	Negative ( - )	(- 1449 ) + ( - 1854 ) = - 3303
( - ) + ( + )	Subtract	Positive ( + )	( - 1449 ) + ( 1854 ) = 405
( + ) + ( - )	Subtract	Negative ( - )	( 1449 ) + ( - 1854 ) = -405

Subtraction of Integers

There are three possibilities when the subtraction operation is applied between two integers:

The subtraction operation is applied between two positive integers.
The subtraction operation is applied between negative positive integers.
The subtraction operation is applied between one positive integer and one negative integer.

To Easily Solve Difference Problems, Follow the Steps Given Below:

Step 1: Convert the given negative sign to a positive sign.

Step 2: After converting the negative sign, find the inverse of the upcoming number.

For example: find the difference between ( - 7654 ) and ( - 9342 )

Step 1: Convert the given negative sign to a positive sign ( - 7654 ) - ( - 9342 )

⇒ ( - 7654 ) + ( 9342 )

Step 2: After converting the negative sign, find the inverse of the upcoming number.

⇒ ( - 7654 ) + ( - 9342 ) ( opposite of 9 is - 9 )

⇒ ( - 7654 ) + ( - 9342 ) = -16,996 (Perform addition operation)

Solved Problems

1. Evaluate the following: a ) 10 + 14, b) ( - 6 ) - ( 12 ), c) ( + 31 ) + ( - 21 )

Solution:

a ) 10 + 14 = 24 ( apply addition operation )

b ) ( - 6 ) - ( 12 ) = - 18 ( addition is performed two negatives numbers )

c) ( + 3 ) + ( - 2 ) = 10

Step 1: Convert the given negative sign to a positive sign ( + 31 ) + ( - 21 )

⇒ ( + 31 ) - ( + 21 )

Step 2: After converting the negative sign, find the inverse of the upcoming number.

⇒ ( + 31 ) + ( - 21 ) ( opposite of 9 is - 9 )

⇒ ( + 31 ) + ( - 21 ) = 10 (Perform addition operation)

FAQs on Integers Rules Explained with Signs and Operations

1. What are the basic rules of integers?

The basic rules of integers explain how to add, subtract, multiply, and divide positive and negative numbers correctly.

Addition: Same signs → add and keep the sign; different signs → subtract and keep the sign of the larger number.
Subtraction: Change subtraction to addition of the opposite.
Multiplication: Same signs → positive; different signs → negative.
Division: Same signs → positive; different signs → negative.

These integer rules are essential for solving arithmetic problems involving negative numbers.

2. How do you add integers with different signs?

To add integers with different signs, subtract their absolute values and keep the sign of the number with the greater absolute value.

Example: 7 + (−10)
Step 1: Subtract 10 − 7 = 3
Step 2: Keep the sign of −10
Final answer: −3

This method is commonly used when working with positive and negative integers.

3. What is the rule for subtracting integers?

The rule for subtracting integers is to change subtraction into addition of the opposite number.

Formula: a − b = a + (−b)
Example: 5 − (−3)
Step 1: Change to 5 + 3
Final answer: 8

This is often called the “keep-change-change” rule in integer arithmetic.

4. What is the rule for multiplying integers?

The rule for multiplying integers is that the product of two integers with the same sign is positive, and with different signs is negative.

(+) × (+) = +
(−) × (−) = +
(+) × (−) = −
(−) × (+) = −

Example: −4 × 6 = −24.

5. What is the rule for dividing integers?

The rule for dividing integers is the same as multiplication: same signs give a positive result, different signs give a negative result.

(+) ÷ (+) = +
(−) ÷ (−) = +
(+) ÷ (−) = −
(−) ÷ (+) = −

Example: −20 ÷ 5 = −4.

6. What is the absolute value of an integer?

The absolute value of an integer is its distance from zero on the number line, always written as a non-negative number.

Notation: |a|
|5| = 5
|−5| = 5

Absolute value is used when comparing magnitudes in integer operations.

7. How do you compare integers?

To compare integers, use their positions on the number line: numbers to the right are greater, and numbers to the left are smaller.

Positive numbers are always greater than negative numbers.
Among negatives, the number closer to zero is greater.
Example: −3 > −7

Understanding integer comparison helps with inequalities and ordering numbers.

8. What are the properties of integer operations?

Integer operations follow properties such as closure, commutative, associative, and distributive properties.

Closure: Sum or product of integers is an integer.
Commutative: a + b = b + a; a × b = b × a.
Associative: (a + b) + c = a + (b + c).
Distributive: a(b + c) = ab + ac.

Note: Division is not commutative or associative.

9. Why does a negative times a negative equal a positive?

A negative times a negative equals a positive because of the distributive property of integers.

Using: 0 = (−2) × [3 + (−3)]
Expanding: (−2 × 3) + (−2 × −3) = −6 + (−2 × −3)
To make the sum 0, (−2 × −3) must be +6.

Therefore, (−) × (−) = + to keep arithmetic consistent.

10. What are common mistakes when solving integer problems?

Common mistakes in solving integer problems usually involve sign errors and forgetting integer rules.

Forgetting to change subtraction into addition of the opposite.
Mixing up multiplication and division sign rules.
Ignoring parentheses in expressions like −(−4).
Confusing absolute value with negative numbers.

Carefully applying the integer sign rules and checking each step helps avoid these errors.