Integers

Integers are a type of number that includes whole numbers and negative numbers. This means they do not include fractions or decimal numbers. For example,  -6, -5, -3, 0, 1, 2, and 9. Integers are represented using the letter Z.
 

• The positive numbers are the numbers greater than zero, i.e., 1, 2, 3, 4, …
   
• The numbers that are less than zero are negative numbers. For example, -1, -2, -3, -4, …
   
• Zero is a whole number and is neither positive nor negative.

Based on the type of numbers, integers can be classified into three categories 
 

• Zero
• Positive Integers
• Negative Integers

Zero: Zero is a neutral number, as it is neither positive nor negative; it is represented as 0. 

Positive Integers: Positive integers are numbers that are greater than 0 and are represented as Z+. In a number line, it is represented to the right of zero. For example, 1, 2, 3, 4, 5, … 

Negative Integers: Negative integers are natural numbers with opposite signs, so they are represented as Z-. In a number line, negative integers are listed to the left of zero. For example, -1, -2, -3, -4, -5, …  

A number line visually represents integers as a straight line. Integers include all the positive numbers, negative numbers, and zero. In a number line, when representing the integers, zero is marked in the center, representing a neutral point. The numbers on the right are the positive integers, as the values keep increasing. So, the numbers on the left side are negative integers, and the values of the numbers decrease toward the left.   

There are four basic arithmetic operations on integers:

• Addition of Integers
• Subtraction of Integers
• Multiplication of Integers
• Division of Integers

Now, learn them in detail. 

Addition of Integers

Adding two or more integers is called integer addition, and the result is called the sum. The sum may increase or decrease depending on the signs of the numbers involved. When adding the integers, make sure to follow these rules. 

1. When adding integers with the same sign, add their absolute values and give the sum the same sign.
   
   For example, 5 + 5= 10, - 5 + (-5) = -10.
    
2. When adding the integers of different signs, that is, one positive and the other negative, find the difference of the numbers and give the sign of the larger number to the sum.
   
   For example, 5 + (-3) = 2, 9 + (-16) = -7.

Subtraction of Integers

We subtract numbers to find the difference between them. Here, the result depends on the integers being positive or negative. Follow these rules when subtracting two integers,
 

1. Convert the subtraction to addition by taking the opposite of the second number.
    
2. Apply the rules of addition of integers. For example, 5 – 3 = 25 – (3) = 5 + 3 = 8

Multiplication of Integers

Multiplication of integers is the process of finding the product of two or more numbers. To multiply integers, follow these steps;
 

1. (+) × (+) = + ⇒ example 5 × 6 = 30
    
2. (–) × (–) = + ⇒ example (-5) × (-6) = 30
    
3. (–) × (+) = – ⇒ example (-5) × (6) = -30
    
4. (+) × (–) = –  ⇒ example (5) × (-6) = -30

Division of Integers

The division of an integer is splitting a number into groups. There are different rules to be followed, such as;
 

1. (+) ÷ (+) = + ⇒ example 30 ÷ 6 = 5
    
2. (–) ÷ (–) = + ⇒ example (-30) ÷ (-6) = 5
    
3. (–) ÷ (+) = – ⇒ example (-30) ÷ (6) = -5
    
4. (+) ÷ (–) = – ⇒ example (30) ÷ (-6) = -5

Integers follow several important properties, such as : 

• Closure Property
• Associative Property
• Commutative Property
• Distributive Property
• Additive Inverse Property
• Multiplicative Inverse Property
• Identity Property

Closure Property: The set is closed under an operation if performing that operation on any two elements of the set results in another element within the set.
 

For any two integers, p and q
 

• p + q ∈ Z 
• p – q ∈ Z
• p × q ∈ Z

Associative Property: The associative property states that grouping of integers in both addition and multiplication does not affect the result.

That is, p + (q + r) = (p + q) + r and p × (q × r) = (p × q) × r. 

Commutative Property: The Commutative property states that the order of the integers does not alter both addition and multiplication.

For example, p + q = q + p and p × q = q × p. 

Distributive Property: The product of a number with two addends will be the same as the sum of the products of the number with the addends is the distributive property.

That is: p × (q + r) = (p × q) + (p × r)
 

Additive Inverse Property: The sum of any positive integer and its negative integer is zero, which means p + (-p) = 0.

Multiplicative Inverse Property: Except for 1 and -1, most integers do not have multiplicative inverses within the set of integers. 

Identity Property: Identity property states that the sum of any number with 0 is equal to the number itself. That is, a + 0 = a. Same as the product of an integer with 1 is equal to the integer itself.

That is, a × 1 = a
 

In the real world, we use integers in different fields, from balancing budgets to measuring temperature. Now, let’s learn some applications of integers. 

• Integers are used to represent gains and losses in temperature, elevation, navigation, and so on.

• In population studies, we use integers to represent the number of individuals in a region.

• In computer science and programming, we use integers for efficient computation and logical control of algorithms, software applications, and digital systems.

• Integers are used to record credits and debits in financial transactions.