- An integer is a whole number that can be positive, negative, or zero, and does not include fractions or decimals. - Integers can be represented on a number line, with positive integers to the right of zero and negative integers to the left. - The four basic arithmetic operations for integers are addition, subtraction, multiplication, and division, and they follow specific rules depending on the signs of the integers.

3. •A perfect square is a positive integer that is obtained
by multiplying an integer by itself. In simple words,
we can say that perfect squares are numbers that are
the products of integers by themselves. Generally,
we can express a perfect square as x2, where x is an
integer and the value of x2 is a perfect square.
•Integers include positive numbers, negative
numbers, and zero. 'Integer' is a Latin word which
means 'whole' or 'intact'. This means integers do not
include fractions or decimals.

5. •What are Integers?
•Integers include all whole numbers and negative numbers.
This means if we include negative numbers along with whole
numbers, we form a set of integers.
•Integers Definition
•An integer is a number with no decimal or fractional part and
it includes negative and positive numbers, including zero. A
few examples of integers are: -5, 0, 1, 5, 8, 97, and 3,043. A
set of integers, which is represented as Z, includes:

6. • 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 is defined as neither a negative number nor a positive number. It
is a whole number.
• Set of Integers
• The set of integers is represented by the letter Z and it is written as
shown below:
• Z = {... -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, ...}
• Observe the figure given below to understand the definition of
integers.

8. Integers on a Number Line
•A number line is a visual representation of
numbers on a straight line. This line is used
for the comparison of numbers that are placed
at equal intervals on an infinite line that
extends on both sides, horizontally. Just like
other numbers, the set of integers can also be
represented on a number line.

10. Graphing Integers on a Number Line
• Positive and negative integers can be visually represented on a
number line. Integers on a number line help in performing arithmetic
operations. The basic points to keep in mind while placing integers
on a number line are as follows:
• The number on the right side is always greater than the number on
the left side.
• Positive numbers are placed on the right side of 0, because they are
greater than 0.
• Negative numbers are placed on the left side of 0, because they are
smaller than 0.
• Zero, which is not positive or negative, is usually kept in the middle.

11. Integer Operations
The four basic arithmetic operations associated with integers are:
• Addition of Integers
• Subtraction of Integers
• Multiplication of Integers
• Division of Integers
• There are some rules for performing these operations of integers.
Before we start learning these methods of integer operations, we need
to remember a few things.
• If there is no sign in front of a number, it means that the number is
positive. For example, 5 means +5.
• The absolute value of an integer is a positive number, i.e., |−2| = 2 and
|2| = 2.

12. Addition of Integers
•Adding integers is the process of finding the
sum of two or more integers where the value
might increase or decrease depending on the
integer being positive or negative. The
different rules and the possible cases for the
addition of integers are given in the following
section.

13. Rules of Integers in Addition
•While adding two integers, we use the following rules:
•When both integers have the same signs: Add the
absolute values of integers, and give the same sign as
that of the given integers to the result.
•When one integer is positive and the other is negative:
Find the difference of the absolute values of the numbers
and then give the sign of the larger of these numbers to
the result.
•Example: Add the given integers: 2 + (-5)

14. •Solution:
•Here, the absolute values of 2 and (-5) are 2 and 5 respectively.
•Their difference (larger number - smaller number) is 5 - 2 = 3
•Now, among 2 and 5, 5 is the larger number and its original
sign “-”.
•Hence, the result gets a negative sign, "-”.
•Therefore, 2 + (-5) = -3

15. Example: Add the given integers: (-2) + 5
• Solution:
• Here, the absolute values of (-2) and 5 are 2 and 5 respectively. Their difference
(larger number - smaller number) is 5 - 2 = 3. Now, among 2 and 5, 5 is the larger
number and its original sign “+”. Hence, the result will be a positive value.
Therefore,(-2) + 5 = 3
• We can also solve the above problem using a number line. The rules for the
addition of integers on the number line are as follows.
• Always start from 0.
• Move to the right side, if the second number is positive.
• Move to the left side, if the second number is negative.

16. •Example: Find the value of 5 + (-10) using a number line.
•Solution:
•In the given problem, the first number is 5 which is
positive. So, we start from 0 and move 5 units to the right
side.

17. Finally, we reach at -5. Therefore, the value of 5 + (-10) = -5