Integers: In Mathematics, the term “integer” was derived from Latin. An integer denotes complete or whole. Integers are the set of positive and negative numbers along with zero. These numbers are an extension of natural numbers and whole numbers. Fraction and decimal numbers are not considered to be integers. 

In this article, let us study more about integers definition and examples with some solved questions.

What are Integers?

Integers can be defined as the set of natural numbers and their additive inverse including zero. The set of integers is \(\left\{ { \ldots .. – 3,\; – 2,\; – 1,\;0,\;1,\;2,\;3 \ldots .} \right\}.\) Integers are numbers which can not be fraction.

Integers are an extension of whole numbers and natural numbers. Natural numbers along with zero and negative natural numbers make integers. Whole numbers along with negative natural numbers make integers.

That is,

Whole Numbers + Negative Natural numbers = Integers

Natural Numbers + Zero + Negative Natural Numbers = Integers

Symbol

The set of integers are denoted by \(I\) or \(Z\).

\(I = Z = \left\{ { \ldots .. – 3,\; – 2,\; – 1,\;0,\;1,\;2,\;3 \ldots .} \right\}\)

Representing Integers on a Number Line

A number line can be defined as a straight line with numbers placed at equal intervals or segments along its length. A number line can be extended infinitely in any direction and is usually represented horizontally.

The integers can be represented on the number line as

The positive integers are represented to the right side of zero on the number line and negative integers are represented to the left side of zero.

The farther the numbers move to the right from zero on the number line, the value of the numbers increases and the farther the numbers move to the left on the number line from zero, the value of the number decreases.

When we compare any two numbers on a number line, the number on the right side of the number line will be greater.

Types of Integers

Integer numbers are basically of three types:

1. Positive integers, which are also called natural numbers.

2. Negative integers, which are the additive inverse of natural numbers.

3. Zero, which is neither negative nor positive.

Operations and Rules

Like in natural numbers and whole numbers, all the basic mathematical operations can be done on integers as well.

Addition of Integers

Unlike natural numbers or whole numbers, the major difference in integers is that integers include negative numbers as well.

So, the addition of integers has to be understood in detail when positive and negative numbers are involved.

Addition of Two Positive Integers

Similar to the way we add in natural numbers and whole numbers, when two positive integers are added, then we get a positive integer as the result.

That is, \({\text{Positive + Positive=Positive}}\)
For example, \(2 + 3 = 5\)
\(6 + 4 = 10\)

Addition of a Positive and Negative Integer

Before learning the addition of a positive and negative integer, we should first understand the concept of absolute value. The absolute value of a number is the numerical value of the number regardless of its sign. That is absolute value is neither negative nor positive, it is just the numerical value of the number.

For example,
The absolute value of \(5 = 5\)
The absolute value of \(-5 = 5\)
The absolute value of \(0 = 0\)

When a positive and negative integer has to be added, we should first take the absolute value of the two integers and take the difference between them. The answer will take the sign of the integer which have the bigger absolute value.

For example,
\(-2 + 3 = 1\)
Here, the absolute value of \(3 = 3\) and the absolute value of \(-2 = 2\)
So after finding the difference of absolute values, the answer will take the sign of the integer with the greater absolute value. That is of \(3\).
\(- 5 + 1 = – 4\)
Here, the absolute value of \(- 5 = 5\) and the absolute value of \(1 = 1\)
So, the difference of absolute values \(= 5 – 1 = 4\)
The answer will take the sign of the integer with the greater absolute value, that is of \(- 5\).

Addition of Two Negative Integers

The addition of two negative integers is done in a similar way to how we add two positive integers and the only change is in the sign of the answer. When two negative integers are added, the result will be a negative integer. So, when adding two negative integers, add the absolute value of both the integers and add a negative sign to the answer.

For example,
\(- 3 + – 2 = – 5\)

We can observe that however, we add two integers the sum is again an integer. That is, integers are closed under addition.

Subtraction of two integers

Subtraction of integers is done much like the addition of integers. Take the absolute value of integers. Then find their difference and make the sign of the integer with a greater absolute value.

Subtraction of Integers

Substraction of Two Positive Integers

When a smaller integer has to be subtracted from a bigger integer, perform the normal subtraction.

For example, \(5 – 3 = 2\)

When a greater integer has to be subtracted from the smaller integer, take the difference of their absolute values and put a negative sign for the answer.

For example, \(3 – 5 = – 2\)

Subtraction of Two Negative Integers

When two negative integers are subtracted, then it becomes the addition of a positive and negative integer.

For example, \( – 4 – \left( { – 2} \right) = – 4 + 2 = – 2\)

Subtraction of a Positive and Negative Integer

When a positive and negative integer is subtracted, check whether a negative number is to be subtracted from a positive number or a positive number is to be subtracted from a negative number.

If a positive integer is subtracted from a negative integer, then add both the integers and give a negative sign for the answer.

For example, \(- 3 + – 2 = – 3 – 2 = – 5\)

If a negative integer is subtracted from a positive integer, then add both the integers and give a positive sign for the answer.

For example, \(3 — 2 = 3 + 2 = 5\)

We can observe that however, we subtract two integers the difference is again an integer. That is, integers are closed under subtraction unlike whole numbers and natural numbers.

Multiplication of Integers

Multiplication of Two Positive Integers

When two positive integers are multiplied, the product will also be a positive integer.

For example, \(3 \times 2 = 6\)

Multiplication of Two Negative Integers

When two negative integers are multiplied, the product will be a positive integer.

For example, \(- 3\times – 2 = 6\)

Multiplication of a Positive and Negative Integers

When a negative and positive integer is multiplied, the product will be a negative integer.

For example, \(- 3\times 2 = – 6\)

We can observe that however, we multiply two integers the product is again an integer. That is, integers are closed under multiplication.

Division of Integers

The integers do not include fractions and decimals. So, the division of integers can be performed only when the quotient is an integer. In all other cases division of integers are undefined. Also, division by zero is not defined. Hence, the division is not closed under integers.

The sign of the quotient of the division of two integers is similar to the product.

Division of Two Positive Integers

When two positive integers are divided, the quotient will be a positive integer.

Division of Two Negative Integers

When two negative integers are divided, the quotient will be a positive integer.

Division of a Positive and Negative Integer

When a positive and negative integer is divided, the quotient will be a negative integer.

Properties of Integers

Integers hold all the properties of whole numbers and natural numbers. Integers hold some more properties.

The \(5\) main properties of integers are

1. Closure Property
2. Commutative Property
3. Associative Property
4. Distributive Property
5. Existence of Identity

Let us see the properties that hold for addition, subtraction, multiplication and division.

Closure Property

Integers are closed under addition, subtraction and multiplication. But, the division is not closed under integers as the integers do not include decimals or fractions and the division by zero is not defined.

We can easily represent this as

\({\text{Integer + Integer=Integer}}\)

\({\text{Integer-Integer=Integer}}\)

\({\text{Integer}} \times {\text{Integer=Integer}}\)

Commutative Property

The commutative property states that when the order or numbers are changed on interchanged, the answer will remain the same.

If \(p\) and \(r\) are two integers, then if \(p + r = r + p\), the addition is commutative in integers.

For example, \(1 + 2 = 2 + 1 = 3\)

So, addition is commutative in integers.

Considering the subtraction,

\(1 – 2 =  – 1\) and \(2 – 1 = 1\)

When numbers have interchanged the difference obtained in the subtraction is different. Hence, subtraction is not commutative in integers.

If \(p\) and \(r\) are two integers, then if \(pr = rp\), the multiplication is commutative in integers.

For example, \(1 \times 7 = 7\) and \(7 \times 1 = 7\)

So, multiplication is commutative in integers.

Considering the division,

\(2 \div 1 = 2\) and \(1 \div 2 = \frac{1}{2}\) which is not an integer.

When numbers are interchanged the quotient obtained in the division is different. Hence, the division is not commutative in integers.

Associative Property

The associative property states that when three or more numbers are operated together, the order of numbers did not affect the result.

If \(p,\,q\) and \(r\) are three integers, then if \(\left( {p + q} \right) + r = p + \left( {q + r} \right),\) the addition is associative in integers.

For example, \(1 + \left( {2 + 3} \right) = 6 = \left( {1 + 2} \right) + 3\)

If \(p,\,q\) and \(r\) are three integers, then if \(\left( {p \times q} \right) \times r = p \times \left( {q \times r} \right),\) the addition is associative in integers.

For example, \(1 \times \left( {2 \times 3} \right) = 6 = \left( {1 \times 2} \right) \times 3\)

When we consider subtraction, \(1 – \left( {2 – 3} \right) = 1 – \left( { – 1} \right) = 1 + 1 = 2\) and \(\left( {1 – 2} \right) – 3 = \left( { – 1} \right) – 3 = – 4\)

As the results are not the same, subtraction is not associative in integers.

Similarly, the division is also not associative.

For example, \(\left( {8 \div 4} \right) \div 2 = 2 \div 2 = 1\) and \(8 \div \left( {4 \div 2} \right) = 8 \div 2 = 4.\)

As the results are not the same, the division is not associative in integers.

Distributive Property of Multiplication over Addition

Distributive property states that when a number is multiplied by the sum of two numbers, then it is the same as taking the product of the numbers and adding them.

That is if \(p,\,q\) and r are three integers, then \(\left( {p + q} \right)r = p \times r + q \times r\)

For example, \(\left( {2 + 3} \right)4 = 5 \times 4 = 20\)

\(2 \times 4 + 3 \times 4 = 8 + 12 = 20\)

Existence of Identity

Identity property exists for addition and multiplication in integers.

Additive identity states that when an integer is added to the additive identity, the sum is the integer itself. The additive identity is \(0\).

For any integer \(a,~\,a + 0 = 0 + a = a\).

Multiplicative identity states that when an integer is multiplied to the multiplicative identity, the product is the integer itself. The multiplicative identity is \(1\).

For any integer \(a,\,a \times 1 = 1 \times a = a\).

Existence of Inverse

Inverse property exists for addition and multiplication in integers.

Additive inverse states that when an integer is added to the additive inverse, the sum yields zero.

For any integer \(a,~\,a + ( – a) = 0\)

So, the additive inverse of a is \( – a\) and the additive inverse of \( – a\) is \(a\).

Multiplicative inverse states that when an integer is multiplied to the multiplicative inverse, the product yields \(1\).

For any integer \(a,\,a \times \frac{1}{a} = 1\)

So, the multiplicative inverse of \(a\) is \(\frac{1}{a}\) and the additive inverse of \(\frac{1}{a}\) is \(a\).

Applications of Integers in Real Life

Integers represent many real-life situations.

In numbering the floors in high-storeyed buildings or malls, the basement is usually numbered as \( – 1\) or \( – 2\).

The temperature below zero degrees is represented by negative numbers.

The amount debited or borrowed from a bank is denoted by a negative sign and the credit or deposited is denoted by a positive sign.

When the level of the land is considered, below sea level is considered negative and above sea level is considered positive.

When a person gains weight, it is represented as positive and when loses weight, it is represented as negative.

In a battery cell, the cathode is represented by a positive sign and the anode is represented by a negative sign.

The acceleration of a car is represented as positive and deceleration is represented as negative.

Integers play a role even in history. AD is represented as positive and BC is represented as negative.

In this article we learned about the integers, how integers are defined, the representation of integers, the basic mathematical operations on integers, different properties on integers, the applications of integers in real life. Also, some interesting problems on integers and the frequently asked questions about integers are provided below.

Solved Examples

Q.1. What temperature change will a customer experience in a grocery store when they walk from the vegetable section at  \({20^{\text{o}}}\,{\text{C}}\) to the frozen meat section which is set to \({20^{\text{o}}}\,{\text{C}}\) ?

Ans: Temperature at the vegetable section \({20^{\text{o}}}\,{\text{C}}\)

Temperature at the frozen meat section \({-20^{\text{o}}}\,{\text{C}}\)

Hence, the difference in temperature
\( = {20^{\text{o}}}\,{\text{C-}}{20^{\text{o}}}\,{\text{C}} = {20^{\text{o}}}\,{\text{C}} + {40^{\text{o}}}\,{\text{C}}\)

Ans: Year in which a Mauryan emperor was born \( = 322{\text{ }}BC\)

Year in which a Mauryan emperor died \( = 298{\text{ }}BC\)

Hence, the number of years the emperor lived \( = 322 – 298 = 24\) years

Q.3. There are \(15\) villages in a district. If there are \(150\) houses in each village, find the total number of houses in the district.

Ans: Number of villages in the district \( = 15\)

Number of houses in each village \( = 150\)

Hence, the total number of houses in the district \( = 15 \times 150 = 2250\) houses

Q.4. A plane is flying at a height of \(35000\,{\text{feet}}\). It comes down to \({\text{10000 feet}}\) after an hour. What is the difference in height of the plane?

Ans: The height of the plane initially \(35000\,{\text{feet}}\)

The height of the plane after an hour \({\text{10000 feet}}\)

Hence, the difference in height of the plane \( = 35000\,{\text{feet-10000 feet}} = 25000\,{\text{feet}}\)

Q.5. A submarine was situated \({\text{500 feet}}\) below sea level. If it descends \({\text{200 feet}}\), what is its new position?

Ans: Height at which submarine is situated \({\text{500 feet}}\) below sea level

Height to which submarine descends \({\text{200 feet}}\) below sea level

Hence, the new position of the submarine below sea level \({\text{=500 feet-}}200\,{\text{feet}}\) \({\text{=3}}00\,{\text{feet}}\) below sea level

Q.6. The temperature of Thar desert is \({50^ \circ }\,{\text{C}}\) in the summer and \({5^ \circ }\,{\text{C}}\) in the winter. What is the difference in temperature?

Ans: Temperature of Thar desert in summer \({50^ \circ }\,{\text{C}}\)

Temperature of Thar desert in winter \({5^ \circ }\,{\text{C}}\)

Hence, the difference in temperature \({50^ \circ }\,{\text{C-}}{5^ \circ } = {45^ \circ }\)

Q.7. Find the product of \( – 2,\, – 4,\, – 1\) and \(3\)?

Ans: The product of two negative integers is positive and the product of a negative and positive integer is negative.

So, \( – 2 \times  – 4 = 8\)
\( – 1 \times 3 =  – 3\)
\(8 \times  – 3 =  – 24\)
Hence, \( – 2 \times  – 4 \times  – 1 \times 3 =  – 24\)

Q.8. Which is the largest negative integer?

Ans: The largest negative integer is \( – 1\).

Q.9. What is the quotient obtained when \( – 100\) is divided by \(50\)?

Ans: When a negative integer is divided by a positive integer, then the quotient will be positive.

So, \( – \frac{{100}}{{50}} = – 2\)

Q.10. Among \( – 7\) and \( – 2\), which is the greater integer?

Ans: In the number line, \( – 2\) lies on the right of \( – 7\) and so \( – 2\) is greater than \( – 7\).

Summary

In this article we learned about the integers, how integers are defined, the representation of integers, the basic mathematical operations on integers, different properties on integers, the applications of integers in real life, some interesting problems on integers and the frequently asked questions about integers.

FAQs on Integers

Q.1. What are integers in math?
Ans: An integer number is a number that can be positive, zero or negative but can not be fraction like \( \frac{5}{2}.\)

Q.2. What is the difference between a whole number and an integer?
Ans: The difference between a whole number and an integer is that an integer can also be a negative number but a whole number can never be negative.

Q.3. Is \(0\) a real number?
Ans: Yes! Zero is a real number.

Q.4. What are \(5\) examples of integers?
Ans: Integers include positive numbers and negative numbers along with zero. Five examples of integers can be \( – 5,\, – 1,\,0,\,19,\,100\).

Q.5. How is a set of integers represented?
Ans: The set of integers are represented by either \(I\) or \(Z\).

Q.6. What are the 3 types of integers?
Ans: The three types of integers are positive integers, negative integers and zero.

Q.7. What are the integers from \(1\) to \(10\)?
Ans: The integers from \(1\) to \(10\) are \(1,\,2,\,3,\,4,\,5,\,6,\,7,\,8,\,9\) and \(10\).

Q.8. What are the \(4\) operations on integers?
Ans: The \(4\) integer operations are addition, subtraction, multiplication and division.

Q.9. Is \(3.14\) an integer number?
Ans: No, \(3.14\) is not an integer number. Integers do not include decimals or fractions.

Q.10. What are the \(5\) properties of integers?
Ans: The main \(5\) properties of integers are closure property, commutative property, associative property, distributive property and existence of identity and inverse.

Other Related Articles

Odd Numbers	Even Numbers	Whole Numbers
Composite Numbers	Real Numbers	Natural Numbers
Co Prime Numbers	Rational Numbers	Prime Numbers

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