Expansions

$\begin{array}{c}1.{(a+b)}^2=a^2+\mathrm{2ab}+b^2\\ 2.{(a-b)}^2=a^2-\mathrm{2ab}+b^2\\ 3.(a+b)(a-b)=a^2-b^2\\ 4.{(a+\frac{1}{a})}^2=a^2+\frac{1}{a^2}+2\\ 5.{(a-\frac{1}{a})}^2=a^2+\frac{1}{a^2}-2\\ 6.(a+\frac{1}{a})(a-\frac{1}{a})=a^2-\frac{1}{a^2}\\ 7.{(a+b+c)}^2=a^2+b^2+c^2+2(\mathit{ab}+\mathit{bc}+\mathit{ca})\\ 8.{(a+b)}^3=a^3+b^3+\mathrm{3ab}(a+b)=a^3+{\mathrm{3a}}^{2}b+{\mathrm{3ab}}^2+b^3\\ 9.{(a-b)}^3=a^3-b^3-\mathrm{3ab}(a-b)=a^3-{\mathrm{3a}}^{2}b+{\mathrm{3ab}}^2-b^3\\ 10.(a+b)(a^2-\mathit{ab}+b^2)=a^3+b^3\\ 11.(a-b)(a^2+\mathit{ab}+b^2)=a^3-b^3\\ 12.(a+b+c)(a^2+b^2+c^2-\mathit{ab}-\mathit{bc}-\mathit{ca})=a^3+b^3+c^3-\mathrm{3abc}\\ 13.(x+a)(x+b)(x+c)=x^3+(a+b+c)x^2+(\mathit{ab}+\mathit{bc}+\mathit{ca})x+\mathit{abc}\end{array}$

See the Video below for problem solving on Expansions

Understanding Irrational Numbers

Madam :   Archana ,Can You Pick the Integers from the Circle ?

Archana : It's very easy mam. 2 and 5

Madam : Whatever you picked are integers that's fine but you   

                missed a couple more.

Archana : Is it so mam but I can't find anymore integers here.

Madam :  Why don't you consider zero and negative integers as integer too. -1 and 0 both are integers.

Archana : What are the other numbers are they called as non-integers ?

Madam :  Definitely the other numbers are non integers but they don't  fall in the same category.   

Archana : It's very easy .Let me do it.

   

                7/2 = 3.5 

                -3/5 = -0.6

                1/3   = 0.3333........ It's a non terminating value.

                 Root 2 = 1.41421356.....It's a non terminating as well as non-recurring value.

Madam : Now you understand why I told you not to put the other numbers in the same category.

                The terminating ones are called as Terminating Rational number. e.g 7/2 , -3/5

                 The non-terminating and repeating or recurring  one is called as

                 Non-Terminating Rational number .e.g 1/3

                 How to express a non terminating decimal ?

                1/3 = 0.3333...   = 0.3 (put a dot over 3)

                9/110 = 0.0818181...   = 0.81(put a dot over 8 and 1) or (draw a line over 8 and 1)

                   Put a dot(.) above first and last repeating digit or draw a line above the repeating digits.

                                                  

                                                                                                              

           I know what are you going to ask me next .

           What is a rational number ?

            

          Any number that can be expressed in the form of m/n where m ,n are integers 

          and n not equals to 0 is called as Rational Number.

        

        Is 5 a rational number ?

        Yes because 5 can be expressed as 5/1 so here m=5 and n=1

        Here we can see m and n are integers and n not equals to zero so 5 is a 

        Rational Number .

      

          

     The number which can not  be expressed as Rational Number are called as

      Irrational Number 

      

     

•     Few theorems on number systems

         

        1.  If (I square)  is divisible by  P then I is divisible by P too.where I is a non zero integer
              and P is a prime number.

         2. If ( I1 x I2) is divisible by P then either ( I1 or I2 is divisible by P) or both (I1 and I2 are 
               divisible by P) where I1,I2 are non-zero integers and P is a prime number.

         3  If I to the power n is divisible by P then I is divisible by P too where I is a non-zero 
             integer , n is a positive integer and P is a prime number .
       
           Is root 2 a rational number ? 

           The answer is let us assume root 2 as a rational number and check whether it satisfies

           the properties of a rational number or not.

           Check the proof in the video tutorial below.

    

Surds

We have already discussed about irrational numbers like root 2, root 3 but still we missed some of the 

irrational numbers like cube root of 2 , cube root of 7 ,etc - -- these numbers can't be expresses as cubes of any rational number.Such numbers are called as surds.

So    n th root is a surd if

  1.a is a positive rational number.

  2. n > 1 and n is an integer.

  3. nth root of a is not a rational number.

We can represent 

•     nth root of a = a to the power of 1/n
•     (nth root of a) to the power of n = a
•     (nth root of a) (nth root of b) = nth root of (ab)
•     (nth root of a)/(nth root of b) = nth root of (a/b) 

                       where n>1 a,b are positive rational number.

 Rationalisation

 

Multiplying a surd with another surd to make it a rational number is called as rationalisation.

Example : 

              multiply  (5 + root3)  with (5 - root3) to make it arational number (5square -3)=23

               This is called rationalisation.

Representation real numbers in a Line 

Representation of real numbers in a line are given in the video tutorial below