In Maths, integers are the numbers which can be positive, negative or zero, but cannot be a fraction. These numbers are used to perform various arithmetic operations, like addition, subtraction, multiplication and division. The examples of integers are, 1, 2, 5,8, -9, -12, etc. The symbol of integers is “Z“.
In terms of sets, the set of integers includes zero, set of whole numbers, set of natural numbers (also called counting numbers) and their additive inverses (-1,-2,-3,-4,-5,..). Integers are the subset of real numbers.
Example of integers: -100,-12,-1, 0, 2, 1000, 989 etc…
As a set, it can be represented as follows:
Z= {……-8,-7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,……}
Table of contents: |
What are Integers?
We have different types of numbers in Mathematics, such as;
- Real Numbers
- Natural Numbers
- Whole Numbers
- Rational numbers
- Irrational numbers
- Even and Odd Numbers, etc.
What are Integers in Maths?
The word integer originated from the Latin word “Integer” which means whole. It is a special set of whole numbers comprised of zero, positive numbers and negative numbers and denoted by the letter Z.
Examples of Integers: – 1, 6, 15.
Fractions, decimals, and percents are out of this basket.
Symbol
The integers are represented by symbol ‘Z’.
Z = {…-3,-2,-1,0,1,2,3,..}
Also, read:
Rules of Integers
Rules defined for integers are:
- Sum of two positive integers is an integer
- Sum of two negative integers is an integer
- Product of two positive integers is an integer
- Product of two negative integers is an integer
- Sum of an integer and its inverse is equal to zero
- Product of an integer and its reciprocal is equal to 1
Now, let us discuss the addition, subtraction, multiplication, and division of signed integer numbers with examples.
Addition of Signed Integer Numbers
While adding the two integers with the same sign, add the absolute values, and write down the sum with the sign provided with the numbers.
For example,
(+4) + (+7) = +11
(-6) + (-4) = -10
While adding two integers with different signs, subtract the absolute values, and write down the difference with the sign of the number which has the largest absolute value.
For example,
(-4) + (+2) = -2
(+6) + (-4) = +2.
Subtraction of Signed Integer Numbers
While subtracting two integers, change the sign of the second number which is being subtracted, and follow the rules of addition.
For example,
(-7) – (+4) = (-7) + (-4) = -11
(+8) – (+3) = (+8) + (-3) = +5
Multiplication and Division of Signed Integer Numbers
While multiplying and dividing two integer numbers, the rule is simple.
If both the integers have the same sign, then the result is positive.
If the integers have different signs, then the result is negative.
For example,
(+2) x (+3) = +6
(+3) x (-4) = – 12
Similarly
(+6) ÷ (+2) = +3
(-16) ÷ (+4) = -4
Properties of Integers
The major Properties of Integers are:
- Closure Property
- Associative Property
- Commutative Property
- Distributive Property
- Additive Inverse Property
- Multiplicative Inverse Property
- Identity Property
Closure Property
According to the closure property of integers, when two integers are added or multiplied together, it results in an integer only. If a and b are integers, then:
- a + b = integer
- a x b = integer
Examples:
2 + 5 = 7 (is an integer)
2 x 5 = 10 (is an integer)
Commutative Property
According to the commutative property of integers, if a and b are two integers, then:
- a + b = b + a
- a x b = b x a
Examples:
3 + 8 = 8 + 3 = 11
3 x 8 = 8 x 3 = 24
But for subtraction and division, commutative property does not obey.
Associative Property
As per the associative property , if a, b and c are integers, then:
- a+(b+c) = (a+b)+c
- ax(bxc) = (axb)xc
Examples:
2+(3+4) = (2+3)+4 = 9
2x(3×4) = (2×3)x4 = 24
This property is applicable for addition and multiplication operations only.
Distributive property
According to the distributive property of integers, if a, b and c are integers, then:
a x (b + c) = a x b + a x c
Example: Prove that: 3 x (5 + 1) = 3 x 5 + 3 x 1
LHS = 3 x (5 + 1) = 3 x 6 = 18
RHS = 3 x 5 + 3 x 1 = 15 + 3 = 18
Since, LHS = RHS
Hence, proved.
Additive Inverse Property
If a is an integer, then as per additive inverse property of integers,
a + (-a) = 0
Hence, -a is the additive inverse of integer a.
Multiplicative inverse Property
If a is an integer, then as per multiplicative inverse property of integers,
a x (1/a) = 1
Hence, 1/a is the multiplicative inverse of integer a.
Identity Property of Integers
The identity elements of integers are:
a+0 = a
a x 1 = a
Example: -100,-12,-1, 0, 2, 1000, 989 etc…
As a set, it can be represented as follows:
Z= {……-8,-7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,……}
For more properties, click here.
Types of Integers
Integers come in three types:
- Zero (0)
- Positive Integers (Natural numbers)
- Negative Integers (Additive inverse of Natural Numbers)
Zero
Zero is neither a positive nor a negative integer. It is a neutral number i.e. zero has no sign (+ or -).
Positive Integers
The positive integers are the natural numbers or also called counting numbers. These integers are also sometimes denoted by Z+. The positive integers lie on the right side of 0 on a number line.
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,…. |
Negative Integers
The negative integers are the negative of natural numbers. They are denoted by Z–. The negative integers lie on the left side of 0 on a number line.
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,….. |
Applications
Integers are not just numbers on paper; they have many real-life applications. The effect of positive and negative numbers in the real world is different. They are mainly used to symbolize two contradicting situations.
For example, when the temperature is above zero, positive numbers are used to denote temperature, whereas negative numbers indicate the temperature below zero. They help one to compare and measure two things like how big or small or more or fewer things are and hence can quantify things.
Some real-life situations where integers come into play are player’s scores in golf, football and hockey tournaments, the rating of movies or songs, in banks credits and debits are represented as positive and negative amounts respectively.
Solved Example
Question 1: Plot the following integers on the number line:
-121, -97, -82, -67, -43, -10, 0, 10, 36, 55, 64, 77, 110, 126, 147.
Answer:
In the above number line, each number has been plotted with a red dot.
Q.2: Solve the following:
- 5 + 3 = ?
- 5 + (-3) = ?
- (-5) + (-3) = ?
- (-5) x (-3) = ?
Solution:
- 5 + 3 = 8
- 5 + (-3) = 5 – 3 = 2
- (-5) + (-3) = -5 – 3 = -8
- (-5) x (-3) = 15
Q.3: Solve the following:
- (+5) × (+10)
- (12) × (5)
- (- 5) × (7)
- 5 × (-4)
Solution:
- (+5) × (+10) = +50
- (12) × (5) = 60
- (- 5) × (7) = -35
- 5 × (-4) = -20
Q.4: Solve the following:
- (-9) ÷ (-3)
- (-18) ÷ (3)
- (4000) ÷ (- 100)
Solution:
- (-9) ÷ (-3) = 3
- (-18) ÷ (3) = -6
- (4000) ÷ (- 100) = -40
Practise Questions
1. Sum of two positive integers is a positive integer. True or False?
2. What is the sum of first five positive integers?
3. What is the product of first five positive odd integers?
4. Plot the integers from -10 to +10 on the number line.
Video Lesson
To learn about the properties of integers and to solve problems on the topic, download BYJU’S – The Learning App from Google Play Store and watch interactive videos.
Frequently Asked Questions – FAQs
What are integers?
What is an integer formula?
What are the examples of integers?
Can integers be negative?
What are the types of integers?
Zero, Positive integers and Negative integers
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The set of positive and negative integers together with 0 is called
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The set of positive natural numbers and negative numbers together with 0 is called Integers.
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