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Integers are a fundamental concept in mathematics. They include all positive whole numbers, negative whole numbers, and zero. The term “integer” comes from the Latin word integer, which means “whole” or “intact.” This implies that integers are complete numbers without any fractions or decimals. In this article, we will learn about integers, their definition, and their key properties.
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What Are Integers?
Integers are a set of numbers that consist of positive numbers (like 1, 2, 3), negative numbers (like -1, -2, -3), and zero. They form a crucial part of number theory and are used in various mathematical operations and real-life situations.
Definition of an Integer
An integer is a number that does not have any decimal or fractional part. It includes positive numbers, negative numbers, and zero. For example, -5, 0, 1, 5, 8, 97, and 3,043 are all integers.
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Properties of Integers
- Integers do not include fractions or decimals. They are whole numbers that can be positive, negative, or zero.
- Integers are closed under addition, subtraction, and multiplication. This means that adding, subtracting, or multiplying two integers will always result in another integer.
- Dividing two integers does not always result in an integer. For example, dividing 1 by 2 gives 0.5, which is not an integer.
- Every integer has an additive inverse. For instance, the additive inverse of 5 is -5, because 5 + (-5) equals zero.
- Addition and multiplication of integers are commutative, meaning the order in which you add or multiply integers does not affect the result.
- Integers is a combination of all whole numbers, both positive and negative, as well as zero. By combining positive numbers, negative numbers, and zero, we form the set of integers.
The Set of Integers
The set of integers is often denoted by the letter Z and it includes:
- Positive Numbers: These are numbers greater than zero. Examples include 1, 2, 3, and so on.
- Negative Numbers: These are numbers less than zero. Examples include -1, -2, -3, and so on.
- Zero: Zero is a special number in this set. It is neit
Integers on a Number Line
A number line is a visual tool that displays numbers along a straight, horizontal line. This line extends infinitely in both directions, with numbers placed at equal intervals.
- Positive integers are placed to the right of zero.
- Negative integers are placed to the left of zero.
- Zero is positioned at the center of the line.
By using a number line, you can easily compare integers and see their relative positions. Integers, both positive and negative, can be effectively visualised on a number line. This visual representation helps in understanding arithmetic operations and the relationships between numbers.
Below given are some key points to remember when placing integers on a number line.
- On a number line, numbers to the right are always greater than those to the left.
- Positive Numbers are positioned to the right of zero since they are greater than zero.
- Negative Numbers are placed to the left of zero because they are smaller than zero.
- Zero is neither positive nor negative and is typically placed at the center of the number line.
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Properties of Integers
Integers have several important properties that help in performing various mathematical operations. Below mentioned are the main properties:
1. Closure Property
The closure property indicates that the set of integers is closed under specific operations. This means performing these operations on integers will always result in an integer:
Addition: a+b∈Z
Subtraction: a−b∈Z
Multiplication: a×b∈Z
Division: Note that division is not always closed for integers (e.g., 5÷2 is not an integer).
2. Associative Property
The associative property states that changing the grouping of integers does not change the result. This property applies to addition and multiplication:
Addition: a+(b+c)=(a+b)+c
Multiplication: a×(b×c)=(a×b)×c
3. Commutative Property
The commutative property means that changing the order of integers does not affect the result. This applies to addition and multiplication:
Addition: a+b=b+a
Multiplication: a×b=b×a
4. Distributive Property
The distributive property describes how multiplication distributes over addition: a×(b+c)=(a×b)+(a×c)
5. Additive Inverse Property
The additive inverse property states that adding an integer and its opposite results in zero:
For any integer a+(−a)=0
6. Multiplicative Inverse Property
The multiplicative inverse property says that multiplying an integer by its reciprocal results in one:
For any integer =1 (Note: This property holds only for non-zero integers.)
7. Identity Property
Integers follow identity properties for addition and multiplication:
- Additive Identity: Adding zero to an integer gives the integer itself: a+0=a
- Multiplicative Identity: Multiplying an integer by one gives the integer itself: a×1=a
These properties are essential for understanding and performing operations with integers.
Integer Operations
When working with integers, we perform four basic arithmetic operations:
- Addition of Integers
- Subtraction of Integers
- Multiplication of Integers
- Division of Integers
Addition of Integers
Adding integers involves finding the sum of two or more integers, where the outcome can either increase or decrease depending on whether the integers are positive or negative. Here are the key rules and examples for adding integers:
Rules for Adding Integers
- When both integers have the same sign, add their absolute values and keep the same sign for the result.
Example: 2 + (5)
Absolute values: ∣2∣=2 and ∣5∣=5
Difference: 5 + 2 = 7
- When one integer is positive and the other is negative, subtract the smaller absolute value from the larger absolute value, and assign the sign of the integer with the larger absolute value to the result.
Example: (−2)+5
Absolute values: ∣−2∣=2 and ∣5∣=5
Difference: 5−2=3
The larger absolute value is 5, which has a positive sign.
Therefore, (−2)+5=3
Subtraction of Integers
Subtracting integers involves finding the difference between two integers, where the result may increase or decrease depending on the integers’ signs. To simplify subtraction, follow these steps:
Rules for Subtracting Integers
- Change the subtraction problem into an addition problem by changing the sign of the subtrahend (the number being subtracted).
- Use the rules for adding integers to solve the resulting addition problem.
Example: Subtract the integers 7 and 10
Rewrite the Problem: 7−10 can be expressed as 7+(−10) by converting the subtraction into an addition problem and changing the sign of 10.
Absolute values: ∣7∣=7 and ∣−10∣=10
Difference: 10−7=3
The larger absolute value is 10, which has a negative sign.
Therefore, 7−10=−3.
Multiplication of Integers
Multiplying integers involves following specific rules based on the signs of the integers. Here are the basic rules:
Rules for Multiplying Integers
| Product of Signs | Result | Example |
| (+) × (+) | + | 2×3=6 |
| (+) × (-) | – | 2×(−3)=−6 |
| (-) × (+) | – | (−2)×3=−6 |
| (-) × (-) | + | (−2)×(−3)=6 |
Example: Multiply (−6)×3
Using the rules for multiplication, the product of a negative and a positive integer is negative.
Therefore: (−6)×3=−18
Division of Integers
Dividing integers involves distributing or grouping one integer into a specified number of groups. Below given are the rules for dividing integers:
Rules for Dividing Integers
| Division of Signs | Result | Example |
| (+) ÷ (+) | + | 12÷3=4 |
| (+) ÷ (-) | – | 12÷(−3)=−4 |
| (-) ÷ (+) | – | (−12)÷3=−4 |
| (-) ÷ (-) | + | (−12)÷(−3)=4 |
Example: Divide (−15)÷3
Using the division rules, when dividing a negative integer by a positive integer, the result is negative.
Therefore: (−15)÷3=−5
Integers: FAQs
What are integers?
Integers are whole numbers that include positive numbers, negative numbers, and zero. They do not include fractions or decimals. Examples of integers are −5, 0, and 8.
What is the difference between positive and negative integers?
Positive integers are numbers greater than zero (for example, 1, 2, 3), while negative integers are numbers less than zero (for example, −1, −2, −3). Zero is neither positive nor negative.
What are the basic operations of integers?
The basic operations on integers include addition, subtraction, multiplication, and division. Each operation follows specific rules based on the signs of the integers involved.
