Integers Class 6 NCERT NOTES

INTEGERS

Whole Numbers
Whole numbers include zero and all-natural numbers i.e. 0, 1, 2, 3, 4, 5,6,7,8, and so on. Whole numbers are always positive.

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Negative Numbers
The numbers which lie to the left of zero on the number line are called negative numbers i.e. -1, -2, -3, -4, -5, and so on. They always have a negative sign. 

• 

 Zero
The number zero means an absence of value.

What are integers? 
Integers are nothing but collections of all positive and negative numbers including zero. For example - 0, -1, -3, 9, 45, -867, 948 are integers.

Steps to represent integers on the number line.

 1. Draw a straight line and mark a point as 0 on it
2. Points that are marked to the left should be replaced with negative integers i.e. -1, -2, -3, -4, -5, and so on. 
3. Points marked to the right should be replaced with positive integers i.e. 1, 2, 3, 4, 5, 6, and so on.

NOTE➡
"0" is neither positive nor negative but sometimes we take "0" as positive.

 What is the absolute value of an integer?
The absolute value of an integer is the numerical value of the integer without considering its sign. For example, the absolute value of -9 is 9 and of +9 is 9.

The symbol of absolute value.
The absolute value of an integer can be asked by written its symbol. The symbol to ask for absolute value is "|...|". For example➡ |-98| is 98 and |98| is 98.

Ordering of integers

• On a number line, the number increases as we move towards the right and decreases as we move towards the left.
• Hence, the order of integers is written as..., –5, –4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5...
• Therefore, – 3 < – 2, – 2 < – 1, – 1 < 0, 0 < 1, 1 < 2 and 2 < 3.
  

  Addition of Integers

  First,  Positive integer + Positive integer           
  Add the 2 integers and add the positive sign.
  Example ➡8 + 6 = 14
  
  Negative integer + Negative integer           
  Add the two integers and add the negative sign.
  Example ➡(-3) + (-18) = (-21)

          Positive integer + Negative integer                     
           Subtract and use the sign of the larger integer.
            Example: (+5) + (-2)
                            Subtract: 5 - 2 = 3
                            Sign of bigger integer (5): +
                            Answer: +3
            Example: (-5) + (2)
                 Subtract: 5-2 = 3
                 Sign of the bigger integer (-5): -
                 Answer: -3

Properties of Addition and Subtraction of Integers

Closure under Addition

• a + b and a – b are integers, where a and b are any integers.

Commutativity Property

• a + b = b + a for all integers a and b.              

Associativity of Addition

• (a + b) + c = a + (b + c) for all integers a, b and c.

Additive Identity

• Additive Identity is 0, because adding 0 to a number leaves it unchanged.
• a + 0 = 0 + a = a for every integer a.
  Multiplication of Integers
  • The product of a negative integer and a positive integer is always a negative integer.
    $10\times -2=-20$
  • The product of two negative integers is a positive integer.
    $-10\times -2=20$

Properties of Multiplication of Integers

Closure under Multiplication

• Integer x Integer = Integer

Commutativity of Multiplication

• For any two integers a and b, a $\times$ b = b $\times$ a.

Associativity of Multiplication

• For any three integers a, b and c, (a $\times$ b) $\times$ c = a $\times$ (b $\times$ c).

Distributive Property of Integers

• Under addition and multiplication, integers show distributive property.
• For any integers a, b and c, a $\times$ (b + c) = a $\times$ b + a $\times$ c.

Multiplication by Zero

• For any integer a, a $\times$ 0 = 0 $\times$ a = 0.

Multiplicative Identity

• 1 is the multiplicative identity for integers.
• a $\times$ 1 = 1 $\times$ a = a
  

  Division of Integers

$\left(\frac{positiveinteger}{negativeinteger}\right)or\left(\frac{negativeinteger}{positiveinteger}\right)\to$ The quotient obtained is a negative integer.

$\left(\frac{positiveinteger}{positiveinteger}\right)or\left(\frac{negativeinteger}{negativeinteger}\right)\to$ The quotient obtained is a positive integer.

Properties of Division of Integers

For any integer a,

• $\frac{a}{0}$ is not defined
• $\frac{a}{1}=a$

Integers are not closed under division.
Example: $(–9)÷(–3)=3$ result is an integer but $(-3)÷(-9)$$=\frac{-3}{-9}$ $=\frac{1}{3}=0.33$ which is not an integer.