# Integers : Types and Properties

What are Integers?
Integers are a type of rational numbers. Rational numbers of the form p/q where q= ±1 are called integers. They are denoted by “Z”.

Integers are of two types :
•   Negative integers
Negative integers are the set of negative numbers before 0. They do not have any fractional or decimal part. Eg. -1, -2, -3 and so on.

•   Positive integers / Whole numbers
Positive integers / whole numbers are the set of natural numbers including zero. They do not have any fractional or decimal part.

To distinguish between the positive and negative sides of 0, opposite signs are used, i.e. positive (+) and negative (-).

 Integers on Number Line

The opposite of an integer is called its negative or additive inverse.
5 + (-5) = 0. So 5 is the additive inverse of -5 and vice versa.
Each integer on the number line, except 0, consists of its mirror image of the opposite sign. Eg. Mirror image of 8 will be -8,that of -2 will be 2 etc.

Zero is lesser than every positive integer and greater than every negative integer.
It is not possible to find the greatest positive or least negative number.

### Absolute value of an integer:

The absolute value of an integer is the numerical value without taking into account its sign. The modulus sign represents the absolute value of the integer. Eg.|-1|=1, |2|=2. Thus, the absolute value of an integer is either greater than or equal to the integer.

Rules for operations on integers:
•   Addition of 2 positive or 2 negative integers:
Add their absolute values and prefix the sign of addends to the sum.
-2 + (-3) = -5
•   Addition of 1 positive and 1 negative integer:Find the difference of their absolute values and prefix the sign of the integer whose absolute value is greater.
2 + (-3) = -(3-2) = -1

Subtraction:
•   Subtracting 1 negative from another negative:
When there are 2 negative(-ve) signs placed side by side with no numeral in between, the 2 like signs  become a plus sign.
-2 – (-3) = -2 + 3 = 1
•   Subtraction of 1 negative from 1 positive:
Convert the like signs in case of negative signs and then add the integers.
3 - (-2) = 3 + 2 = 5
• Subtraction of 1 positive from 1 negative:
Convert the signs and then add the absolute values of integers add prefix minus sign.
-3 – (+2) = -3 -2 = -5

Multiplication:
•   Multiplication of 2 positive integers:
Multiply the values and prefix plus sign to the product.
2 x 3 = +6 or 6
•   Multiplication of a positive and a negative integer:
Multiply their absolute values and prefix minus sign to the product.
2 x (-3) = -6
•   Multiplication of 2 negative integers:
Multiply the absolute values and prefix plus sign to the product.
(-2) x (-3) = +6 or 6

Division:
•   Division of integers with like signs:
Divide their absolute values and prefix plus sign.
(-3) ÷ (-2) = +3/2 or 3/2
•  Division of integers with like signs:
Divide their absolute values and prefix minus sign.
(3) ÷ (-2) = -3/2