Mathematics

                                           1. Integers

Introduction:- To be able to subtract a larger number from a smaller number, and to be able to indicate the idea of oppositeness of things, man indicated each number with a positive and negative signs for them.

Definition :- Integers are, a bigger collection of numbers, formed by whole numbers and their negatives.

Properties of integers :- Closure property

                                          (1) Integers are closed under addition.

                  In general, for any two integers a and b, a+b is an integer.

                              For example :- 5+4=9;                                                                                                                       -8+6=-2                                                            

                                        (2) Integers are closed under subtraction. 

                In general, if a and b are two integers, a-c is also an integer.

                              For example:- 7-2=5;                                                                                                                        -22-(-9)=-13

                                        (3) Integers are closed under multiplication.

                 In general, for any two integers a and b, ab is an integer.

                               For example:- 3×4=12;                                                                                                                     -7×-5=35

                                        (4) Integers are NOT closed under division.   

                  In general, if a and b are two integers, a/b may not be integer.

                                     Commutative Property

                                        (1) Addition is commutative for integers.

                  In general, for any two integers a and b, a+b=b+a

                          

                              For example:- 2+(-3)=2-3=-1; but

                                                      (-3)+2=-3+2=-1

                                        (2) Subtraction is NOT commutative for integers.

                  In general, for any two integers a and b, a-b≠b-a

                     

                              For example:- 7-(-4)=7+4=11; but (-4)-7=-4-7=-11

                                                      5-(-2)=5+2=7; but (-2)-5=-2-5=-7

                                       (3) Multiplication is commutative for integers.

                  In general, for any two integers a and b, ab=ba

                                

                             For example:- 9×(-4)=-(9×4)=-36; but

                                                      (-4)×9=-4×9=-36

                                        (4) Division is NOT commutative for integers.

                   In general, for any two integers a and b, a/b≠b/a

                                 

                           For example:-  2/8=1/4; but

                                                    8/2=4/1

                                  Associative property

                                          (1) Addition is associative for integers.

                  In general, for any three integers a, b and c, a+(b+c)=(a+b)+c

              

                         For understanding this statement, let us take an example: 

                                             -4, -3 and -6

        (-6)+[(-4)+(-3)]                                          [(-6)+(-4)]+(-3)

          (-6)+(-7)                                                      (-10)+(-3)

              = -13                                                           =-13

                                            (2) Subtraction is NOT associative for integers.

                  In general, for any three integers a, b and c, a-(b-c)≠(a-b)-c

                     

                         For example:- 5-(6-4)= 5-2=3; but

                                                (5-6)-4= 1-4=-5

                                             (3) Multiplication is associative for integers.

                  In general, for any two integers a, b and c, a×(b×c)=(a×b)×c

               

                         For example:- [(-3)×(-2)×4]=6×4=24;

                                                 [-3×(-2×4)]=(-3)×(-8)=24

                                               (4) Division is NOT associative for integers.

                                              Distributive property

                          (1) Distributive property of multiplication over addition.

      In general, for any three integers a, b and c, a×(b+c)=a×b+a×c

                      For example:-      -2(4+3)=

  -2(7)=                                                                        (-2×4)+(-2×3)=

        -14                                                                               (-8)+(-6)=                 

                                                                                              -14

                        (2) Distributive property of multiplication over subtraction.

       In general, for any three integers a, b and c, a×(b-c)=a×b-a×c

                          

                    For example:-        -2(4-3)=

   -2(1)=                                                           (-2×4)-(-2×3)=

         -2                                                          (-8)-(-6)=  

                                                                                              -2

                                     Identity under addition

              '0' is the identity integer under addition. In general, for an integer a, a+0=a=0+a or in other words, a+0=0+a=a

 For example:- 9+0=9=0+9

Operations on integers (formula only):-  Addition and Subtraction:

 Positive(+)   {+/-}    Positive(+)   =  Positive(+) sign

Negative(-)   {+/-}    Negative(-)   =  Negative(-) sign

Negative(-)   {+/-}    Positive(+)    =   The sign of the greater number(+/-)

                                                                 Multiplication and division:

 Positive(+)  {×or/}  Positive(+)  =  Positive(+) sign

Negative(-)  {×or/}  Negative(-)  = Positive(+) sign

Negative(-)  {×or/}  Positive(+)  =  Negative(-) sign 

     

   

                                    2. Fractions   

Introduction:- In our daily life, we use fractions in different situations. We use them when a whole thing is divided into parts.

Definition :- A fraction means a part from the whole thing, or more generally, equal parts from the whole thing.

Example :- The fraction 2/4 means: 2 out of 4 parts.                                                                                                                                            

Components of fractions :- In a fraction, we call the top number, the Numerator and it indicates the part that we choose from the whole.

And the number at the bottom, we call it the Denominator, and it indicates the part in which the whole thing is divided.

Example :- The ratio of 20 boys and 30 girls in a class can be represented by the fraction 2/₃ (2 upon 3).

               In this fraction-2/₃, 2 is the numerator and 3 is the denominator.

 What is Quarter? 

        Suppose you cut a cake in the middle, horizontally and vertically. We will get four equal parts. That's one-fourth of the whole cake. So the fraction is: 1/4 of the whole cake. It is also called quarter.

Fraction on the number line(only an idea) :- On a number line, we can divide the length between 0 and 1 into four equal parts. And if you take a fraction, suppose 1/₄, it lies between 0 and 1 and one part can be one-fourth. 

Classification of fractions:- 

1. Decimal fractions:- Fractions in which the denominator is 10 or a higher power of 10 are called decimal fractions.

              Examples :-  2/10; 3/100; 4/1000, etc.

2. Vulgar fractions:- Fractions in which the denominator is other than 10 or any powers of 10 are called vulgar fractions.

              Examples :-  3/6; 8/16; 14/26; 17/36, etc.

3. Proper fractions:- Fractions in which the numerator is less than the denominator are called proper fractions.