All you need to know about Natural numbers and Integers..

Natural number forms the base of our Math. It is an integral part of our calculation and without having a sound knowledge about the numbers, we won’t be able to go further in our calculation. Generally, in mathematics, the natural numbers are those used for counting or ordering something. suppose if we have to count different objects or have to do various operations like multiplication, addition etc in our day-to-day life and we know nothing about the numbers. Do you think that the task would be easier for us..? I think it won’t.

So to clear your Fundamentals about the natural numbers and the operations associated with it, our team has made a short note on natural numbers and their operations. After the summary, you will also find a set of questions to boost up your revision. Let’s quickly jump to the topic without wasting any time.

Natural numbers:

All the counting numbers (Cardinal number) are called natural numbers.
Eg: 1, 2, 3,4,5,6,121,254 etc.

Whole numbers:

All the natural numbers together with 0 are known as the whole Number. That is

                                         [ Whole number=0+natural numbers ]
Eg:
0, 1, 2,3,4,5,121,254 etc.

Integers:

All natural numbers, 0 and negatives of counting numbers are called Integers.
Eg: -1,-2, 0,1,10001 etc.

Positive Integers:  An integer whose value is greater than zero are known as Positive Integer. 1,2 ,3 ,121, 256,10001 etc all are Positive integers

Negative Integers:  An Integer whose value is less than zero are known as Negative Integer. -1,-2,-3,-10001,-67 etc all are negative integers

Zero(0):  It is also an integer but neither positive nor negative.

ADDITION Of Integer

The addition of integer is simple to understand if you know all the properties of addition of an integer. Let’s discuss various addition property of integer in brief.

Properties of Addition of Integers:

(A) Closure property of addition of Integer: The sum of two integers will always be an Integer.

Eg: 5+4=9 (Sum of two integers 4, 5 is also an integer 9)

(B) Commutative law of addition of integer:  if a and b are two integers then according to the Commutative law of addition.

                                                      ( a+b = b+a )
Eg:
 -6+10=4 and 10+(-6)=4 so  -6+10=10+(-6) according to Commutative law of addition.

(C) Associative law of addition of Integer:  if a, b and c are three integers then according to Associative law of addition,

 (a+b)+c=a+(b+c)

Eg: let the integers are 2,-4 and 6 then

{2+(-4)}+6=4 and  2+{(-4)+6}=4 so
{2+(-4)}+6=2+{(-4)+6}, according to Associative law of addition.

(D) Existence of additive identity of Integer: for any integer a we have

a+0=0+a=a ) is known as additive identity of an integer.
Eg: 5+0=0+5=5

(E) Existence of additive inverse of Integer: for any integer a we have

{ a+(-a)=(-a)+a=0 } ” -a’ is known as additive inverse of an integer a.
Eg: 6+(-6)=(-6)+6=0

SUBTRACTION of integer:

Subtraction of integer can be fun if you know the basic rule of subtraction of integer while subtracting integers.

Properties of subtraction of integers:

(A) Closure property of subtraction of Integer: if a and b are any two integers, then their subtract (a-b), will always be an integer.
Eg: 6-4=6+(-4)=2 [ Here 6, 4 are integers and their subtract 2 is also an integer]

(B) Subtraction of integers are not commutative: if a and b are two integers then

 (a-b) ≠ (b-a)
Eg
: if 6 and -4 are two integers then 6+(-4)=6-4=2 and (-4)-6=-4-6=-10 so 6+(-4) ≠(-4)-6

(C) Subtraction of integers are not associative: if a, b and c are three integers then

  (a-b)-c ≠ a-(b-c)
Eg:
if 2,3 and 4 are three integers then (2-3)-4=-5 and 2-(3-4)=3 so (2-3)-4 2-(3-4).

Multiplication of integer:

Multiplication of different integer always follows a rule. Let’s check the basic rule behind the multiplication of integer.

Properties of multiplication of integers:

(A) Closure property of multiplication of Integer: The product of two integers will always be an integer.

Eg: 5×7=35, 4×8=32

(B) Commutative law of multiplication of Integer: for any two integers a and b we have

  (a×b)=(b×a)
Eg:
let 2 and 3 are two integers then 2×3=3×2=6, so we can say that (a x b)= (b x a), according to the Commutative law of multiplication of integer.


(C) Associative law of multiplication of Integer:
for any integers a, b and c we have

  (a×b)×c=a×(b×c)
Eg: let 2, 3 and 4 are three integers then (2×3) ×4=2×(3×4)=24, so we can say that (a×b)×c=a×(b×c), according to the Associative law of multiplication of integer.

(D) Distributive law of multiplication over Addition: for any integers a, b and c we have

a×(b+c)=a×b+a×c=ab+ac

Eg: let 2, 3 and 4 are the three integers then, 2×(3+4)=2×3+2×4=14.

(E) Existence of multiplicative identity of integer: for any integer a we have

(a×1)=(1×a)=a

Here 1 is known as the multiplicative identity for integers

(F) Existence of multiplicative inverse of integer: for any integer, a, there exist a non-zero multiplicative inverse 1/a as

a×1/a=1/a×a=1

Here 1/a is defined as the multiplicative inverse of an integer.

(G) property of zero: for any integer a, we have

a×0=0×a=0

Eg:  3×0=0×3=0. Any number divided by zero results in zero.

DIVISION of integer:

division of integer is simple as compared to multiplication. what you have to do is to keep in mind the following properties of the division of an integer.

Properties of the division of integers:

(A) If a and b are integers then (a÷b) is not necessarily an integer.
Eg: let if 3 and 4 are the two integers then (3÷4=3/4) is not an integer.

(B) If a is an integer and a≠0 then  a÷a=1
Eg: let 3 is the integer so 3÷3=1 where (3≠0)

(C) If a is an integer then (a÷1=a).
Eg:
let 5 is an integer then (5÷1)=5

(D) If a is an integer and a≠0 then (0÷a)=0 but (a÷0) is not defined.
Eg: let 10 be the integer so (0÷10)=0 but (10÷0) is not defined

(E) If a, b and c are the integers, then (a ÷ b) ÷ c ≠a ÷ (b ÷ c)
Eg: let 4, 2 and 1 are the integers so (4÷2) ÷1=2 but 4 ÷ (2÷1) =2

Modulus of an integer

Modulus of any integer is a function which results in the positive value of the integer. A modulus always gives a positive value of the integer. Modulus of any integer a is denoted by |a| and given as

 |a| = a, if a is positive or zero and  -a, if a is negative.
Thus |10|= 10 and |-10|= -(-10)= 10

Above following notes on natural number and integer is designed by a list of highly qualified teachers of our family. After reading the note, now you will be able to answer questions based on natural numbers. Let’s quickly revise the topic with the help of below sample paper on the natural number.

Multiple choice questions:

The sum of two integers is 10. If one of the integers is 4 then other will b

(a) 10 (b)  4     (c)  6     (d) 2

2. What must be subtracted from 19 to get 13

(a) 10 (b)  49    (c)  6     (d) 11

3. How much less than -6 is from 10

(a)12     (b) 14     (c)  16     (d) 12

4. (-10)-(-6) equals to

(a  ) -3   (b)- 4     (c)  -6     (d) -2

5. (-0) divides by 16 will give

(a)10     (b)  4     (c)  6     (d) 0

6. How much the number 12 is exceeded by -6

(a)18     (b)  14     (c)  16     (d) 12

7. Which of the following is true

(a)10<-4     (b)  4>11     (c)  6 >-4  (d) 2+0=0

8. (-2)×(5) equals to

(a) 5    (b)  0   (c)  -10      (d)  10

9. The multiplicative inverse of 6 would be

(a)1/6   (b) 6    (c)  1/4  (d) 12

10. (-1)× (-1)× (-1)× (-1)………..up to 51 terms

(a)1   (b) 51    (c)  -11  (d)NOT

Hopefully, this concise note about natural numbers and integers will help you to clear your basic concept about integers and natural numbers. The Note on rules of natural number and integers will help the students as well as the aspirants, who are preparing for various Competitive examinations.

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