** Integers **are groups of numbers that are defined as the union of positive numbers, and negative numbers, and zero is called an Integer. ‘Integer’ comes from the Latin language word that means ‘whole’ or ‘intact’.

**do not include fractions or decimals.**

**Integers****What are Integers?**

**What are Integers?**

So, ** What are Integers?** If a set is constructed using all-natural numbers, whole numbers, and negative numbers, then that set is referred to as an Integer set.

Integers include positive numbers, negative numbers, and zero. Integers are represented by the symbol Z such that,

Z = {… -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, …}

Numbers greater than zero are called positive numbers. Example: 1, 2, 3 4 . . .**Positive Numbers:**Numbers less than zero are called negative numbers. Example: -1, -2, -3 -4..**Negative Numbers:**is neither positive nor negative.**Zero (0)**

## Types of Integers

Integers are classified into three categories:

**Zero (0)****Positive Integers (i.e. Natural numbers)****Negative Integers (i.e. Additive inverses of Natural Numbers)**

### 1. Zero

Zero is a unique number that does not belong to the category of positive or negative integers. It is considered a neutral number and is represented as “0” without any plus or minus sign.

### 2. Positive Integers

Positive integers, also known as natural numbers or counting numbers, are often represented as Z+. Positioned to the right of zero on the number line, these integers encompass the realm of numbers greater than zero.

Z+ → 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,….

### 3. Negative Integers

Negative integers mirror the values of natural numbers but with opposing signs. They are symbolized as Z–. Positioned to the left of zero on the number line, these integers form a collection of numbers less than zero.

Z– → -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, -13, -14, -15, -16, -17, -18, -19, -20, -21, -22, -23, -24, -25, -26, -27, -28, -29, -30,…..

## How to Represent Integers on Number Line?

As we have discussed previously, it is possible to visually represent the three categories of integers – positive, negative, and zero – on a number line.

Zero serves as the midpoint for** integers on the number line**. Positive integers occupy the right side of zero, while negative integers populate the left side. Refer to the diagram below for a visual representation.

**Rules of Integers**

**Rules of Integers**

When two positive integers are added together, the result is always an integer. This principle underscores the fundamental nature of integers as encompassing positive values beyond the realm of natural numbers.**Addition of Positive Integers:**Similarly, the sum of two negative integers results in an integer. This mirrors the concept of integers extending into the realm of negative values, broadening the spectrum of numerical representation.**Addition of Negative Integers:**The product of two positive integers yields an integer. This rule reaffirms the idea that the multiplication of positive values remains rooted within the integer domain, offering a bridge between arithmetic operations and this distinct number set.**Multiplication of Positive Integers:**Similarly, when two negative integers are multiplied, the outcome is an integer. This rule showcases the symmetrical nature of integers, where even multiplicative interactions among negative values lead back to the realm of integers.**Multiplication of Negative Integers:**The intriguing aspect of integers reveals itself when considering the sum of an integer and its inverse. Remarkably, this sum always equals zero, illuminating the unique property that integers inherently possess. The interplay between positive and negative values results in a harmonious equilibrium at the integer zero.**Sum of an Integer and Its Inverse:**Delving deeper into the properties of integers, the product of any integer and its reciprocal, which is defined as one divided by the integer, is consistently equal to 1. This principle showcases the unity that exists within the intricate realm of integers.**Product of an Integer and Its Reciprocal:**

## Arithmetic Operations on Integers

The four basic Maths operations performed on integers are:

- Addition
- Subtraction
- Multiplication
- Division

### Addition of Integers

Addition of integers is similar to finding the sum of two integers. Read the rules discussed below to find the sum of integers.

**Example: Add the given integers:**

**3 + (-9)****(-5) + (-11)**

**Solution:**

- 3 + (-9) = -6
- (-5) + (-11) = -16

### Subtraction of Integers

Subtraction of integers is similar to finding the difference between two integers. Read the rules discussed below to find the difference between integers.

**Example: Add the given integers:**

**3 – (-9)****(-5) – (-11)**

**Solution:**

- 3 – (-9) = 3 + 9 = 12
- (-5) – (-11) = -5 + 11 = 6

### Multiplication of Integers

Multiplication of integers is achieved by following the rule:

- When both integers have same sign, the product is positive.
- When both integers have different signs, the product is negative.

Product of Sign | Resultant Sign | Example |
---|---|---|

(+) × (+) | + | 9 × 3 = 27 |

(+) × (–) | – | 9 × (-3) = -27 |

(–) × (+) | – | (-9) × 3 = -27 |

(–) × (–) | + | (-9) × (-3) = 27 |

### Division of Integers

Division of integers is achieved by following the rule:

- When both integers have the same sign, the division is positive.
- When both integers have different signs, the division is negative.

Division of Sign | Resultant Sign | Example |
---|---|---|

(+) ÷ (+) | + | 9 ÷ 3 = 3 |

(+) ÷ (–) | – | 9 ÷ (-3) = -3 |

(–) ÷ (+) | – | (-9) ÷ 3 = -3 |

(–) ÷ (–) | + | (-9) ÷ (-3) = 3 |

## Properties of Integers

Integers have various properties, the major properties of integers are:

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

### Closure Property

Closure property of integers states that if two integers are added or multiplied together their result is always an integer. For integers p and q

- p + q = integer
- p x q = integer

**Example:**

(-8) + 11 = 3 (An integer)

(-8) × 11 = -88 (An integer)

### Commutative Property

Commutative property of integers states that for two integers p and q

- p + q = q + p
- p x q = q x p

**Example:**

(-8) + 11 = 11 + (-8) = 3

(-8) × 11 = 11 × (-8) = -88

But the commutative property is not applicable to the subtraction and division of integers.

### Associative Property

Associative property of integers states that for integers p, q, and r

- p + (q + r) = (p + q) + r
- p × (q × r) = (p × q) × r

**Example:**

5 + (4 + 3) = (5 + 4) + 3 = 12

5 × (4 × 3) = (5 × 4) × 3 = 60

### Distributive Property

Distributive property of integers states that for integers p, q, and r

p × (q + r) = p × q + p × r

**For Example, Prove: 5 × (9 + 6) = 5 × 9 + 5 × 6**

** LHS **= 5 × (9 + 6)

= 5 × 15

= 75

** RHS** = 5 × 9 + 5 × 6

= 45 + 30

= 75

Thus, LHS = RHS Proved.

### Identity Property

Integers hold Identity elements both for addition and multiplication. Operation with the Identity element yields the same integers, such that

- p + 0 = p
- p × 1 = p

Here, 0 is Additive Identity, and 1 is Multiplicative Identity.

### Additive Inverse

Every integer has its additive inverse. An additive inverse is a number that in addition to the integer gives the additive identity. For integers, Additive Identity is 0. For example, take an integer p then its additive inverse is (-p) such that

- p + (-p) = 0.

### Multiplicative Inverse

Every integer has its multiplicative inverse. A multiplicative inverse is a number that when multiplied to the integer gives the multiplicative identity. For integers, Multiplicative Identity is 1. For example, take an integer p then its multiplicative inverse is (1/p) such that

p × (1/p) = 1.

## Applications of Integers

** Integers **extend beyond numbers, finding applications in real life. Positive and negative values represent opposing situations. For instance, they indicate temperatures above and below zero. They facilitate comparisons, measurements, and quantification.

**feature prominently in sports scores, ratings for movies and songs, and financial transactions like bank credits and debits.**

**Integers**### Also Check,

- Rational Number
- Irrational Number
- Real Numbers

## Examples on Integers

**Example 1: Can we say that 7 is both a whole number and a natural number?**

**Solution:**

Yes, 7 is both whole number and natural number.

**Example 2: Is 5 a whole number and a natural number?**

**Solution: **

Yes, 5 is both a natural number and whole number.

**Example 3: Is 0.7 a whole number?**

**Solution:**

No, it is a decimal.

**Example 4: Is -17 a whole number or a natural number?**

**Solution:**

No, -17 is neither natural number nor whole number.

**Example 5: Categorize the given numbers among Integers, whole numbers, and natural numbers,**

**-3, 77, 34.99, 1, 100**

**Solution:**

Numbers Integers Whole Numbers Natural Numbers -3Yes No No 77Yes Yes Yes 34.99No No No 1Yes Yes Yes 100Yes Yes Yes

## Practice Questions on Integers

- Write three consecutive integers.
- Which of the following numbers is the largest: -6, 2, -3, or 0?
- Calculate the product of -7 and 9.
- Find the sum of -15, 20, and -8.
- And some word problems.
- If the temperature drops by 10 degrees Celsius and then rises by 7℃, what is the net change in temperature?
- A submarine is at a depth of 120 meters below sea level. If it rises 80 meters, what will its new depth be?

## Integers – FAQs

**1. What are Integers?**

**1. What are Integers?**

The union of zero, natural numbers, and their additive inverse is called Integers. It is mathematically denoted by the symbol Z.

**2. What are Consecutive Integers?**

**2. What are Consecutive Integers?**

Consecutive Integers are integers that are adjacent to each other on a number line. The difference between the two consecutive integers is “1”.

**3. Write the Examples of Integers.**

**3. Write the Examples of Integers.**

Examples of integers are -1, -9, 0, 1, 87, etc.

**4. What are the different types of Integers?**

**4. What are the different types of Integers?**

There are three different types of integers:

- Zero
- Positive Integers
- Negative integers

**5. Can Integers be Negative?**

**5. Can Integers be Negative?**

Yes, integers can be negative. Negative Integers are -1, -4, and -55, etc.

**6. State whether the given statement is True or False: “All Natural Numbers are Whole Numbers”?**

**6. State whether the given statement is True or False: “All Natural Numbers are Whole Numbers”?**

True, all natural numbers are whole numbers but not vice versa.