What is an Integer – Definition & Properties

The word ‘integer’ is a Latin word that means ‘whole’ or ‘intact’. The term integers are used to denote a set consisting of all natural numbers and their corresponding negative numbers including $0$.

You use integers where you want to represent numbers in two opposite forms like describing temperature above/below freezing point, debit/credit of money, a geographical level above/below sea level, and elevator level when it is above/below the ground level. 

Let’s find out more about integers and their properties.

What is an Integer?

Integers are the collection of whole numbers and their corresponding negative numbers. Similar to whole numbers, integers also do not include fractional numbers. Thus, we can say that integers are numbers that can be positive, negative, or zero, but cannot be a fraction. 

The examples of integers are $1$, $2$, $7$, $-8$, $-12$, $-15$, etc. The set of integers is represented by $Z = \{…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …\}$.

Thus a set of integers consists of

• Positive Numbers: A number is positive if it is greater than zero. Example: $1$, $2$, $3$ . . .
• Negative Numbers: A number is negative if it is less than zero. Example: $-1$, $-2$, $-3$ . . .
• Zero ($0$) is defined as neither a negative number nor a positive number. It is a whole number.

Note: The notation $Z$ for the set of integers comes from the German word ‘Zahlen’, which means ‘numbers’.

Integers on a Number Line

Just like other numbers, the set of integers can also be represented on a number line. A number line of integers is used for the comparison of numbers that are placed at equal intervals on an infinite line that extends on both sides, horizontally. 

The number $0$ is placed at the centre of the line. 

The positive numbers starting from $1$ and moving on to $2$, $3$, and so on are placed on the right-hand side of $0$ at equal intervals. 

Similarly, the negative numbers starting from $-1$ and moving on to $-2$, $-3$, and so on are placed on the left-hand side of $0$ at equal intervals.

Properties of Integers

You can perform any of the following four basic operations on natural numbers. 

• Addition
• Subtraction
• Multiplication 
• Division

Each of these operations shows one or more of the following properties:

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

Let’s understand these properties of whole numbers in detail.

Closure Property of Integers

The closure property states that if any two numbers from a set are operated by an arithmetic operation then their result also lies in the same set.

The closure property is exhibited by the operations of addition, subtraction, and multiplication in the set of integers. The operation division does not show exhibit closure property in the set of integers.

Closure Property of Addition of Integers

It states that when two integers are added, then their sum is also an integer.

Mathematically, it is represented as if $a, b \in Z, \text {then  } a + b \in Z$.

For example, $-6$ and $13$ are integers, then their sum $-6 + 13 = 7$ is also an integer.

Similarly, $16$ and $-63$ are integers, then their sum $16 + \left(-63 \right) = -47$ is also an integer.

Closure Property of Subtraction of Integers

It states that when two integers are subtracted, then their difference is also an integer.

Mathematically, it is represented as if $a, b \in Z, \text {then  } a \- b \in Z$.

For example, $7$ and $12$ are integers, then their difference $7 – 12 = -5$ is also an integer.

Similarly, $-12$ and $18$ are integers, then their difference $-12 – 18 = -30$ is also an integer.

Closure Property of Multiplication of Integers

It states that when two integers are multiplied, then their product is also an integer.

Mathematically, it is represented as if $a, b \in Z, \text {then  } a \times b \in Z$.

For example, $7$ and $12$ are integers, then their product $7 \times 12 = 84$ is also an integer.

Similarly, $-14$ and $16$ are integers, then their product $-14 \times 16 = -224$ is also an integer.

Note: The operation division does not show closure property in the case of integers.

Commutative Property of Integers

The commutative property deals with the ordering of numbers in an operation. It states that the result remains the same even if the order of numbers in the operation is changed or swapped.

The commutative property is exhibited by the operations addition and multiplication in the set of integers. The operations subtraction and division do not show exhibit commutative property in the set of integers.

Commutative Property of Addition of Integers

It states that for any two integers their sum remains the same even if the positions of the numbers are interchanged or swapped.

Mathematically, it is represented as if $a, b \in Z, \text {then  } a + b = b + a$.

For example, $-5$ and $13$ are two integers. $-5 + 13 = 8$ and also $13 + \left(-5 \right) = 13 – 5 = 8$.

Similarly, for two integers, $-56$ and $-48$, $-56 + \left(-48 \right) = -104 $ and $-48 + \left(-56 \right) = -48 – 56 = -104$.

Commutative Property of Multiplication of Integers

It states that for any two integers their product remains the same even if the positions of the numbers are interchanged or swapped.

Mathematically, it is represented as if $a, b \in Z, \text {then  } a \times b = b \times a$.

For example, $-9$ and $11$ are two integers. $-9 \times 11 = -99$ and also $11 \times \left(-9 \right) = -99$.

Similarly, for two integers, $-4$ and $-8$, $-4 \times \left(-8 \right) = 32 $ and $-8 \times \left(-4 \right) = 32$.

Associative Property of Integers

The associative property deals with the grouping of numbers in an operation. It states that the result remains the same even if the grouping of numbers is changed while performing the operation.

The associative property is exhibited by the operations addition and multiplication in the set of integers. The operations subtraction and division do not show exhibit associative property in the set of integers.

Associative Property of Addition of Integers

It states that the sum of any three integers remains the same even if the grouping of the numbers is changed. 

Mathematically, it is represented as if $a, b, c \in Z, \text {then  } \left (a + b \right) + c = a + \left(b + c \right)$. 

For example, for three integers $2$, $-8$ and $15$, $\left( 2 + \left(-8 \right) \right) + 15 = -6 + 15 = 9$ and $2 + \left(-8 + 15 \right) = 2 + 7 = 9$.

Similarly, for three integers $-10$, $12$ and $-4$, $\left(-10 + 12 \right) + \left(-4 \right) = 2 + \left(-4 \right) = -2 \text { and } -10 + \left(12 + \left(-4 \right) \right) = -10 + 8 = -2$. 

Associative Property of Multiplication of Integers

It states that the product of any three integers remains the same even if the grouping of the numbers is changed. 

Mathematically, it is represented as if $a, b, c \in Z, \text {then  } \left (a \times b \right) \times c = a \times \left(b \times c \right)$. 

For example, for three integers $4$, $3$ and $-9$, $\left( 4 \times 3 \right) \times \left(-9 \right) = 12 \times \left(-9 \right) = -108$ and $4 \times \left(3 \times \left(-9 \right) \right) = 4 \times \left(-27 \right) = -108$.

Similarly, for three integers $5$, $-10$ and $3$, $\left(5 \times \left(-10 \right) \right) \times 5 = -50 \times 5 = -250 \text { and } 5 \times \left(-10 \times 3 \right) = 5 \times \left(-30 \right) = -150$. 

Distributive Property of Integers

The distributive property of integers deals with the splitting of the distribution of integers through addition and subtraction while performing the multiplication operation.

There are two forms of distributive property of integers.

• Distributive property of multiplication over addition
• Distributive property of multiplication over subtraction

Distributive Property of Multiplication Over Addition of Integers

It states that for any three integers the expression of the form $\left(a + b \right) \times c$ can be solved as $a \times b + a \times c$.

For example, $\left(5 + 3 \right) \times 9$ can be solved as $5 \times 9 + 3 \times 9 = 45 + 27 = 72$.

This also $\left(5 + 3 \right) \times 9$ on solving gives $8 \times 9 = 72$. 

Similarly, $\left(150 + 6 \right) \times \left(-4 \right)$ can be solved as $150 \times \left(-4 \right) + 6 \times \left(-4\right) = -600 – 24 = -624$.

This also $\left(150 + 6 \right) \times \left(-4\right)$ on solving gives $156 \times \left(-4\right) = 624$. 

Distributive Property of Multiplication Over Subtraction of Integers

It states that for any three integers the expression of the form $\left(a – b \right) \times c$ can be solved as $a \times b – a \times c$.

For example, $\left(14 – 6 \right) \times 8$ can be solved as $14 \times 8 – 6 \times 8 = 112 – 48 = 64$.

This also $\left(14 – 6 \right) \times 8$ on solving gives $8 \times 8 = 64$. 

Similarly, $\left(100 – 9 \right) \times 5$ can be solved as $100 \times 5 – 9 \times 5 = 500 – 45 = 455$.

This also $\left(100 – 9 \right) \times 5$ on solving gives $91 \times 5 = 455$. 

Note: The distributive property does not hold for division in the case of integers.

Additive Identity Property of Integers

The additive identity property of integers is also known as the identity property of integers of addition, which states that adding $0$ to any integer, results in the number itself. This is due to the fact that when we add $0$ to any integer, it does not change the number and keeps its identity.

Mathematically, it is expressed as for any integer, $a$ there exists an integer $0$ such that $a + 0 = 0 + a = a$.

For example $7 + 0 = 0 + 7 = 7$, or $-9 + 0 = 0 + \left(-9 \right) = -9$.

Note: $0$ is called the additive identity of integers.

Multiplicative Identity Property of Integers

The multiplicative identity property of integers is also known as the identity property of integers of multiplication, which states that multiplying $1$ to any integer, results in the number itself. This is due to the fact that when we multiply $1$ to any integer, it does not change the number and keeps its identity.

Mathematically, it is expressed as for any integer, $a$ there exists an integer $1$ such that $a \times 1 = 1 \times a = a$.

For example $4 \times 1 = 1 \times 4 = 4$, or $-8 \times 1 = 1 \times \left(-8 \right) = -8$.

Note: $1$ is called the multiplicative identity of integers.

Additive Inverse Property of Integers

The additive inverse of an integer is its opposite number. If an integer is added to its additive inverse, the sum of both the numbers becomes zero ($0). 

Mathematically, it is expressed as for every integer $a$, there exists an integer $-a$, such that $a + \left(-a \right) = -a + a = 0$. $-a$ is called the additive inverse of integer $a$.

The simple rule is to change the positive number to a negative number and vice versa. 

For example, additive inverse of $11$ is $-11$, since, $11 + \left(-11 \right) = -11 + 11 = 0$.

Or, additive inverse of $-6$ is $6$, since, $-6 + 6 = 6 + \left(-6 \right) = 0$.

Conclusion

The set of integers is the set of numbers consisting of positive numbers (natural numbers), their corresponding negative numbers, and $0$. The integers exhibit the properties – closure property, commutative property, associative property, distributive property, additive identity property, additive inverse property, and multiplicative identity property.

Practice Problems

State True or False

• $0$ belongs to the set of integers
• $0$ does not belong to the set of integers
• $1$ is an additive identity of integers
• $0$ is an additive identity of integers
• $1$ is a multiplicative identity of integers
• $0$ is a multiplicative identity of integers
• The set of integers is represented by the letter $I$.
• The set of integers is represented by the letter $Z$.
• $-4$ is an additive inverse of $4$
• $-4$ is a multiplicative inverse of $4$
• $\frac {1}{4}$ is an additive inverse of $4$
• $\frac {1}{4}$ is a multiplicative inverse of $4$
• The multiplicative inverse of integers exists in the set of integers
• The multiplicative inverse of integers does not exist in the set of integers

Recommended Reading

• Associative Property – Meaning & Examples
• Distributive Property – Meaning & Examples
• Commutative Property – Definition & Examples
• Closure Property(Definition & Examples)
• Additive Identity of Decimal Numbers(Definition & Examples)
• Multiplicative Identity of Decimal Numbers(Definition & Examples)
• Natural Numbers – Definition & Properties
• Whole Numbers – Definition & Properties

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