# Integers

Have you ever seen temperature scale?
If you have seen it then you must be aware of the numbers represented on it. In the middle of the scale there is a red mark which is denoted by zero. Below the red mark there starts increasing negative numbers. Above the red mark, there is increasing positive numbers. There are many such examples. These type of representation is said to be integer representation.

## Negative Numbers:

-1, -2, -3, -4,…….. or negative signed numbers are called negative numbers. As we go higher negative numbers value decreases. As, -2 has less value than -1.

These negative numbers in numerical form (negative signed natural numbers) are said to be negative integers.

## Positive numbers:

1, 2, 3, 4……… are positive numbers and as we go higher the magnitude of numbers increases. 2 is greater than 1.

Natural numbers are called positive integers.

## Definition of integers:

“Negative numbers, 0 along with positive numbers collectively are integers”.

Or

“A bigger collection of numbers formed by whole numbers and their negatives”.

Example: …….. -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ……… are integers.

## Successor:

If we add 1 to the given number, then it is the successor of that number.

## Predecessor:

If we subtract 1 from given number than it is the predecessor of that number.

## Properties of integers:

1. Integers either added or subtracted give integers.
2. Addition of integers is commutative.
As, a + b = b + a
3. Addition of integers is associative.
As, a + (b + c) = (a + b) + c
4. 0 is an integer which is identity under addition.
As, a + 0 = 0 + a = a
5. When positive integers are added we get a positive integer.
As, [(+2) + (+4) = (+6)]
6. When negative integers are added it gives negative integer.
[(-2) + (-4) = (-6)]
7. When there is addition of positive and negative integers then we subtract the smaller from bigger and the integer we got after subtraction will contain the sign of bigger integer.
As, [(-5) + (+2) = (-3)] because in it 5 is bigger and 2 is smaller we subtracted (5 – 2) and sign of 5 will be on the answer.
8. Product of two positive integer is always positive. [(+) x (+) = (+)]
As, [(+4) x (+2) = (+8)]
9. Product of two negative integer is always positive. [(-) x (-) = (+)]
As, [(-4) x (-6) = (+24)]
10. Product of positive integer and negative integer is always negative. [(+) x (-) = (-)]
As, [(-5) x (+2) = (-10)]
11. When we consider the product of even number of negative integers it is always positive.
12. When we consider the product of odd number of negative integers it is always negative.

Properties of integers under multiplication:

1. Product of two integers is always an integer. That is, a × b is an integer for any two integers a and b.
2. Multiplication of integers is commutative. That is, a × b = b × a for any integers a and b.
3. The integer 1 is the multiplication identity, i.e., 1 × a = a × 1 = a
4. Multiplication of integers is associative, i.e., (a × b) × c = a × (b × c) for three integers a, b and c.
5. Distributive property is followed by integers under addition and multiplication.
That is, a × (b + c) = a × b + a × c for any integers a, b and c.

## Integer examples:

1. Find (-45) + (35).

Solution:
subtract smaller from bigger i.e. (45 – 35 = 10)

sign of bigger will be considered = -10

1. Find (6) x (-7).
Solution:
We know that [(+) x (-) = (-)] (6) x (-7) = – 42
2. Find [5 + {2 x (-2)}] Solution:
First {2 x (-2)} = -4
then, [5 – 4] = 1