Number line with positive and negative integers, along with examples of solved basic operations.

Operations with Integers: Rules, Differences, Examples, and Step-by-Step Solved Exercises

What Are Operations with Integers?

Operations with integers are one of the fundamental topics in arithmetic and algebra. After learning about natural numbers, the next step is to understand how to work with positive numbers, negative numbers, and zero. This knowledge is essential for solving mathematical problems, interpreting real-life situations, and progressing to more advanced topics such as equations, functions, and algebra.

Integers allow us to represent quantities above and below a reference point. For example, a temperature of 8 °C can be represented as +8, while a temperature of 5 degrees below zero is expressed as −5. Similarly, a positive bank balance indicates available money, while a negative balance represents a debt.

Mastering operations with integers means understanding how addition, subtraction, multiplication, and division work when positive and negative signs are involved. Although the rules may seem difficult at first, they follow a logical mathematical structure that becomes much easier to understand with practice.

What Are Integers?

The set of integers consists of:

Negative numbers.
Zero.
Positive numbers.

It is represented by the letter ℤ:

ℤ = {..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ...}

Unlike natural numbers, integers allow us to represent situations involving losses, decreases, or positions below a reference point.

Everyday Examples

Temperature: −8 °C
Floor of a building: −2
Bank balance: −$350
Altitude: −120 meters relative to sea level
Profit: +250 pesos
Age: 18 years

These examples demonstrate that integers are a useful tool for representing real-life situations in different contexts.

Rules for Operations with Integers

Before solving operations with integers, it is important to understand two key concepts: absolute value and sign rules.

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of whether the number is positive or negative.

Examples:

|7| = 7
|−7| = 7
|15| = 15
|−15| = 15

Absolute value is useful for comparing magnitudes and determining how to operate when numbers have different signs.

Rules of Signs

The rules of signs simplify multiplication and division involving integers.

Sign × Sign	Result
(+) × (+)	+
(+) × (−)	−
(−) × (+)	−
(−) × (−)	+

The same sign rules also apply to division.

A simple way to remember them is:

Same signs → positive result
Different signs → negative result

Basic Operations with Integers

Addition of Integers

Case 1: Same Signs

Add the absolute values and keep the common sign.

Example:

(+12) + (+8) = +20

Another example:

(−15) + (−6) = −21

In both cases, the absolute values are added and the common sign is maintained.

Case 2: Different Signs

Subtract the absolute values and keep the sign of the number with the greater absolute value.

Example:

(+18) + (−7)

First, subtract the absolute values:

18 − 7 = 11

The number with the greater absolute value is +18, so the result is positive.

Result:

+11

Another example:

(−20) + (+6)

Subtract the absolute values:

20 − 6 = 14

The number with the greater absolute value is −20, so the result is negative.

Result:

−14

Subtraction of Integers

Subtracting an integer is equivalent to adding its opposite.

Example:

8 − (−4)

Change the subtraction into addition and use the opposite of −4:

8 + 4 = 12

Another example:

−6 − (+9)

Change the subtraction into addition and use the opposite of +9:

−6 + (−9) = −15

This rule makes it easier to solve problems and avoid confusion when working with positive and negative signs.

Multiplication of Integers

First, multiply the absolute values. Then, apply the rules of signs.

Examples:

7 × 5 = 35

−7 × 5 = −35

7 × (−5) = −35

−7 × (−5) = 35

A simple way to remember the rule is:

Same signs → positive result
Different signs → negative result

Division of Integers

The procedure for dividing integers is similar to multiplication.

Examples:

20 ÷ 4 = 5

−20 ÷ 4 = −5

20 ÷ (−4) = −5

−20 ÷ (−4) = 5

It is important to remember that division by zero is not defined in mathematics. Therefore, a number cannot be divided by zero.

Differences Between Natural Numbers and Integers

Although both sets consist of whole numbers without decimal parts, they have important differences.

Characteristic	Natural Numbers	Integers
Include negative numbers	No	Yes
Include zero	Depends on the convention	Yes
Mainly used for counting	Yes	Not always
Represent losses or debts	No	Yes
Set	ℕ	ℤ

Natural numbers are ideal for representing quantities and counting, while integers allow us to describe situations involving both positive and negative values.

Applications of Integers in Everyday Life

Integers appear in many everyday situations, including:

Temperatures above and below 0 °C.
Bank account balances.
Underground and above-ground floors in buildings.
Profits and losses in businesses.
Altitudes above and below sea level.
Sports scores and point differences.
Computer programming and data analysis.

The signs are different, so the result is negative.

Answer:

−7

Common Mistakes When Working with Integers

Many students make mistakes because they memorize the rules without understanding the mathematical reasoning behind them. Some of the most common mistakes include:

Confusing the rules for addition with the rules for multiplication.
Forgetting to change subtraction into addition of the opposite.
Failing to compare absolute values when adding numbers with different signs.
Assuming that the result of an operation must always be positive.
Attempting to divide by zero.

The best way to avoid these mistakes is to practice with a variety of exercises and analyze each step carefully before reaching the final answer.

Frequently Asked Questions (FAQ)

What Are Operations with Integers?

They are mathematical procedures involving addition, subtraction, multiplication, and division with positive numbers, negative numbers, and zero.

What Is the Difference Between Natural Numbers and Integers?

Integers include negative values, while natural numbers are primarily used for counting quantities and do not include negative numbers.

How Do You Add Numbers with Different Signs?

Subtract the absolute values and keep the sign of the number with the greater absolute value.

Why Does a Negative Times a Negative Equal a Positive?

This rule follows from the properties of integer multiplication and maintains consistency with fundamental mathematical principles, including the distributive property and established numerical patterns.

Can You Divide by Zero?

No. Division by zero is not defined in mathematics.

Conclusion

Operations with integers represent a fundamental step in learning mathematics. Understanding how signs, absolute values, and the rules for each operation work allows students to solve problems with greater confidence and accuracy.

Beyond simply passing an examination, mastering this topic makes it easier to understand more advanced concepts such as algebra, equations, and functions. Consistent practice and understanding the reasoning behind each procedure are essential for developing strong mathematical thinking that can be applied both academically and in everyday life.

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Resource Academic Record

Collection

Mathematics

Knowledge Center

Arithmetic

Knowledge Area

Integers

Topic

Operations with Integers: Rules, Differences, Examples, and Step-by-Step Solved Exercises

Level

Basic–Intermediate

Estimated Reading Time

12 minutes

Resource Type

Article

Competencies Developed

Mathematical reasoning · Problem-solving · Numerical operations · Logical thinking

Library Path

Digital Library → Mathematics Collection → Knowledge Center: Arithmetic → Knowledge Area: Integers