Binomial theorem

Binomial theorem

 We all know that a polynomial with two terms is called a Binomial.

 Now, Between these two terms, both may be variable or a combination of a variable and a constant number.

As for example, 3x-4y, 2x+6, ax+b, 2x² -3y², a+ x ,..etc.

Now,  the square or cube of the binomial (a+x) can be determined easily by the formulas or basic multiplication methods.

  But , it is very hard and labourious to find the n-th power(n is any positive integer) of the binomial (a+x) by basic multiplication. So, the multiplication method fails to find the value of (a+x)ⁿ.

 Then, how can we find the value of (a+x)ⁿ=?

  In this case algebra helps us. To solve these type of problems, algebra gives us a special formula. This formula is known as  Binomial theorem.

  Definition:

In algebra a general formula is used to expand a binomial with power n( any positive integer n) as a series of finite terms.This formula is called  binomial theorem.

 The binomial theorem was discovered by sir Issac Newton.

Statement of binomial theorem:

For any positive integer n and any real number a and x ; the expansion of the binomial (a+x) with power n, i.e, the expansion of (a+x)ⁿ is given by:

(a+x)ⁿ= aⁿ + C(n,1)×(aⁿ⁻¹)×x¹ + C(n,2)× (aⁿ⁻²)×x² +...+xⁿ= aⁿ+ n×aⁿ⁻¹×x + {n(n-1)/2! }×aⁿ⁻²×x² +...+xⁿ.

Here, C(n,1) , C(n,2),...C(n,n) are called the binomial coefficients.

 Note:

The number of terms in the expansion of (a+x)ⁿ are  always finite and equals to (n+1).

Pascal's triangle and binomial theorem:

In 1660 Pascal introduced an expansion of a binomial with power n.

He observed that,

  (a+x)⁰ = 1

  (a+x)¹= a¹+ x¹

  (a+x)² = a² + 2ax + x²

  (a+x)³ = a³ + 3a²x + 3ax² + x³

  (a+x)⁴ = a⁴ + 4a³x + 6a²x² +  4ax³ +x⁴

      and so on.

Pascal observed the special pattern or relation between the power(exponent) 'n' and binomial coefficients.

Power(exponent) of binomial     coefficients

            0                                                    1

            1                                                  1      1

            2                                              1      2     1

            3                                           1      3      3   1

     And so on.

Expansion of some important binomial expressions:

(a-x)ⁿ = aⁿ - C(n,1)×{aⁿ⁻¹ }x¹ + C(n,2)×{aⁿ⁻²} x² -...+(-1)ⁿ × xⁿ.

(1+x)ⁿ = 1+C(n,1)×x + C(n,2)×x² +...+xⁿ.

(1-x)ⁿ = 1 - C(n,1)×x + C(n,2)×x² -...+(-1)ⁿ ×xⁿ.

Note:

(1) The (r+1)-th term in the expansion of (a+x)ⁿ = t(r+1) ={C(n,r)×aⁿ⁻ʳ}×xʳ.

(2)The (r+1)-th term in the expansion of (a-x)ⁿ = t(r+1) = {C(n,r)×aⁿ⁻ʳ}×(-1)ʳ ×xʳ.

(3) If n is even the middle term of (a+x)ⁿ  will be the (n/2 +1)-th term of the expansion.

(4) If n is odd, there will be two middle terms and they will be {(n-1)/2 +1}-th term and {(n+1)/2 + 1}-th term.

(5) The sum of the all coefficients in the expansion of (a+x)ⁿ is equals to 2ⁿ.

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Harmonic progression

Harmonic progression

Let's observe the following two sequences of numbers  {1,3,5,7,...} and {1,1/3,1/5,1/7,...}.

   It is very clear that the first one is an Arithmetic progression (A.P.), and the second one is a sequence of reciprocals of the terms of the first one.

   The second sequence of numbers is known as Harmonic progression (H.P.).

Definition:

  A sequence of numbers {a,b,c,...} are in H.P if and only if the sequence of numbers {1/a, 1/b, 1/c,...} are in A.P.

 Thus, a sequence of numbers forms an H.P. if and only if the sequence of  their  reciprocals are in A.P. 

Therefore, the a sequence of numbers {a,b,c,...} forms a H.P. if the following condition is satisfied.

  Let, {a,b,c,...} are in H.P. Then, {1/a,1/b,1/c,...} are in A.P.

  So, 1/b - 1/a = 1/c - 1/b .

Which is the required condition that  a sequence of number {a,b,c,...} will form a H.P.

In particular, three numbers will form a H.P. , if the ratio of first and third number, is equals to the ratio of the differences between first , second and second, third respectively.

So, {p,q,r} will form a H.P. if , p/r = (p-q)/(q-r).

     But, we should remember that, there is no specific formulas to find the sum of a H.P.

So, to find the sum of a finite number of terms of a H.P. ; we first find the  corresponding A.P. and then using the sum formulas for an A.P. , the required sum is  obtained.

So, the formulas of the A.P.  are also useful for  a H.P.

The n-th term of an A.P.

 If "a" be the first term and "d" be the common difference of an A.P. having the n-th term t(n) , then , 

   t(n)=a+(n-1)d.

The sum of first n terms of an A.P.

   If  "a" , "d", t(n) are the first term, common difference and n-th term of an A.P. respectively, then , the sum of first n terms denoted by s(n) is given by:

  s(n)= (n/2) ×{a+t(n)}

 or, (n/2)×{2a+(n-1)d}, where , we use, t(n)= a+(n-1)d.

Harmonic Mean:

If three numbers are in H.P. then the middle number is called the Harmonic mean.

So, if {x,y,z} are in H.P., then y is called the Harmonic mean of x and z.

 Let, a and b are two numbers and H be their Harmonic mean. 

So, 1/H -  1/a = 1/b - 1/H 

or, 2/H = 1/a + 1/b

or, H= 2ab/(a+b).

Some intersting facts:

(1) If a, b, c are in H.P. then,  1/(bc) , 1/(ac), 1/(ab) are also in H.P.

(2)If a, b, c are in H.P. then, a/(b+c-a) , b/(c+a-b) , c/(a+b-c) are also in H.P.

(3) If a, b, c are in H.P. then, a(b+c) , b(c+a), c(a+b) are in A.P.

(4) If a², b², c² are in A.P. then, (b+c) , (c+a) and (a+b) are in H.P.

(5) If a, b, c are in H.P.  then, a/(b+c), b/(c+a) , c/(a+b) are also in H.P. (a+b+c ≠ 0).

(6) If a, b, c,d are in A.P. then, abc, bcd, abd, acd  are in H.P.

 Note:

Arithmetic- Geometric series:

 Each term of these type of series are expressed as the product of two terms ; one from A.P. and other from G.P.

  The general form of an Arithmetic- Geometric progression is, 

 a×1 + (a+d)×r + (a+2d)×r²+...

To find the sum of an Arithmetic - Geometric progression  a special method is used.

An example of an Arithmetic- Geometric progression is given by:

 1+2a+ 3a² + 4a³+..., a not equals to 1.

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Geometric progression

Geometric progression

{ 1,3,9,27,...}

 { 2,4,8,16,...}

 { 3,12,48,192,...}

What is the similarly among the above sequences of numbers?

  It is very clear. All the terms of the sequence (excluding the first term) can be obtained easily just by multiplying a constant number with the previous term.

This special type of sequence of numbers is known as Geometric progression (G.P.).

Definition:

A sequence of numbers {u(1), u(2), u(3),...}

 forms  a geometric progression if the value of the ratio, u(n+1)/u(n) is a constant for every positive integer n.

The ratio u(n+1)/u(n) is called the common ratio of the geometric progression.

General form of a G.P.

The most general form of a geometric progression is { a,ar,ar²,ar³,...}, where "a" and "r" are the first term and common ratio of the geometric progression.

The n-th term of a G.P.

If t(n) be the n-th term of a geometric progression whose first term is "a" and common ratio is "r" then,

  t(n)=arⁿ⁻¹.

example:
   find out the 10-th term of the geometric progression {2, 6, 18,...}.
    Here, the first term is 2 and the common ratio is 3.
  So, the 10-th term is= 2×(3)¹⁰⁻¹ =2×3⁹.
   

Sum of first n-terms of a G.P.

If "a"and "r" are  the first term and common ratio of a geometric progression, then the sum of the geometric progression upto first n-terms is:

   Assuming, "r" not equals to 1,

S(n)= a×{(1-rⁿ)/(1-r)}, |r|<1.

and, S(n)= a×{(rⁿ -1)/(r-1)}, r>1 or, r<-1.

    In particular if r=1, S(n) =a+a+...+(upto n terms)= na.

    example:
        find the sum of n terms of the geometric progression {1,3,9,27,...}.
        Here, the first term is 1 and the common ratio is 3(>1).
        So, the sum of n terms is
   = {1×(3ⁿ -1)}/(3-1)
   = (3ⁿ -1)/2.

Geometric mean:

 If x be the Geometric mean of a and b then,  x/a =b/x

      or, x²=ab

      or, x=√(ab) or, -√(ab).

Thus the Geometric mean of two numbers is the square root of their product.
      example:
        In the Geometric progression {2,4,8}, 4 is the geometric mean between 2 and 8.

     In general in a Geometric progression with finite  number of terms, all the terms between the first and last terms are the geometric means between the first and last term.

   example:
     In the geometric progression {1,3,9,27,81}, the terms 3,9 and 27 are the geometric means between 1 and 81.

Note:

(1) If the product of three terms of a geometric progression  is given then we consider the terms as, a/r, a,ar.

(2)If the product of  4 terms of a geometric progression is given , then we consider the terms as, a/r³, a/r, ar, ar³.

(3) If the first term and the common ratio of a geometric progression  is known, we can find any term of the geometric progression.

(4) If we know two terms (consecutive) of a Geometric progression, we can determine the whole geometric progression.

(5) The reciprocal terms of a  geometric progression also forms a geometric progression.

The infinite Geometric series:

A series of the form:

  a+ar+ar³+...+arⁿ+...infinity, 

is called an infinite geometric series.

   As for example, 1+1/2+1/(2²)+...+infinity.

 Sum of an infinite geometric series:

 If , -1<r<1 , then , sum is =a+ar+ar²+...+infinity= a/(1-r).

 In particular, if a=1, sum is= 1+r+r²+...+infinity=1/(1-r).

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Arithmetic progression

Arithmetic progression

Let's see the sequence of numbers below:

 {1, 3, 5,7,9,...}

{2,4,6,8,10,...}

{5, 10, 15, 20,...}

There is a common special property in each of the above sequences: the difference between any two consecutive terms of the sequence is the same.

 This special type of sequence of numbers is known as Arithmetic progression (A.P.).

Definition:

A sequence of numbers in which the difference between every pair of consecutive terms are the same (constant); is called an Arithmetic progression and the difference is called as common difference.

As for example,  the sequence of numbers 

{ 2,5,8,11,14,17,... } is an arithmetic progression , having a common difference 3.

The n-th term of an A.P.

If "a" be the first term and "d" be the common difference of an arithmetic progression  having the n-th term t(n); then,

   t(n)=a+(n-1)d.

example:
    find out the 7-th term of the arithmetic progression {5, 12, 19,...}.
     Here, the first term is 5.
      The common difference is 7.
       so, the 7-th term t(7) is= 5+{(7-1)×7}
                                                = 5+42
                                                = 47.

The sum of first n terms of an A.P.

If  "a" , "d", t(n) are the first term, common difference and n-th term of an arithmetic progression respectively ; then , the sum of first n terms denoted by s(n) is given by:

  s(n)= (n/2) ×{a+t(n)}

 or, (n/2)×{2a+(n-1)d}, where , we use, t(n)= a+(n-1)d.

example:
    calculate the sum of the arithmetic progression {2, 5, 8,...,152}.
       Here, at first we are to find out the number of terms in the givrn arithmetic progression.
          Here, t(n)= 152 and a=2, d=3.
                so, 2+(n-1)×3 =152
                 or, n-1= 50
                  or, n=51.
   So, the number of terms in the given  arithmetic progression is 51.
 Now, the sum of the serirs is
   =(51/2)×(2+152)
   =51×77
   =3927.
  Therefore, the sum of the given arithmetic progression is 3927.

Properties of an A.P.

(1) If we add or subtract a constant term with each term of an arithmetic progression ; the new sequence will form a new arithmetic progression.

(2) If we multiply or divide a constant term with each term of an arithmetic progression , the new sequence of numbers will form a new arithmetic progression.

(3) If the sum of three terms of an arithmetic progression is given then we can consider the terms as, a-d, a, a+d.

(4) If the sum of four terms of an arithmetic progression is given then we can consider the terms as, a-3d, a-d, a+d, a+3d.

(5) In an arithmetic progression the sum of equidistant terms from the begining and ending sides is equals to the sum of the first and last term of the arithmetic progression.

Arithmetic mean

 If three terms (consecutive)  are in arithmetic progression; the middle term of them is called the arithmetic mean.

i.e., if a, b  be the terms of an arithmetic progression , and x be their arithmetic mean (A.M.) then, a, b,x are in arithmetic progression.

i.e, x-a= b-x

or, x=(a+b)/2.

Thus, The arithmetic mean  of two terms in an arithmetic progression  is the half of their sum.

As for example, in the arithmetic progression  {3,6,9,12,...};

6 is the arithmetic mean of 3 and 9.


Note:
   (1)  If the number of tetms of an arithmetic progression is even , then there are two middle terms. The middle terms are the (n/2)-th term and (n/2 +1)-th term of the
arithmetic progression.
    (2) If the number of terms of an arithmetic progression is odd, then there is only one middle term. The middle term is the {(n+1)/2}-th term of the arithmetic progression.


     example:
       find out the middle term or terms of the arithmetic progression {3,7,11,...,95}.
        Here, the first term is 3 , common difference is 4 and the n-th term is 95.
      So, 3+(n-1)×4 = 95
       or, n-1=23
       or, n= 24.
 Thus,  the number of terms of the given arithmetic progression is even. So, there are two middle terms. The middle terms are the 12-th and 13-th terms of the arithmetic progression.

Note:

      The sum of first n natural numbers is :

S(n)=1+2+...+n = (n/2)(n+1).
       The sum of squares of first n natural numbers is:
 S(n)=1²+2²+3²+...+n² =(n/6)(n+1)(2n+1).
        The sum of cubes of first n natural numbers is:
S(n)=1³+2³+3³+...+n³={(n/2)(n+1)}².

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Division by zero

Division by zero

Are you shocked!

Oh no! I am not taking about how to divide a number by zero. Actually, that is not possible. But, some of us asks why?

Why we can not divide a number by zero?

 So, let's discuss about the matter with a deep inner sight!

  Before we start our journey, let's highlight the matter with an real life experience.

   In family we always share many things between our family members. Let , in a family there are 7 members and there are 14 cakes , so each family member can get 14/7=2 cakes.  

Again think that, in a family there are 7 members but there are no cake or 0 cake.

Then each member get 0 cake. So, 0/7=0.

  But, can you think about a matter that, in a family there are no members (just suppose) and there are 7 cakes and you have to distribute the cakes among them!

  How is it possible!!!

  Yes, this is the fact in the case of 7/0 also.

     

Yes, this the hidden truth!

There is an another method to realise that, why division by zero is undefined.

We know that, If we divide a number (dividend) by another number (divisor), we have a result(quotient), (assuming that remainder is 0).

Now, if we multiply the quotient with the divisor we will get the dividend back.convere is also true.

As for example, 14/7=2 and 7×2=14 and conversely also.

That's right!

Now,  if we divide 7 by zero, i.e., 7/0 and let result is 7/0.

But if we multiply 7/0 with 0 what we get?

We get, (7/0)×0=7×(0/0)=7.(Assuming that we can divide 0 by 0)

But, we know that, 0 ×(any number)=0.

So, (7/0)×0= 0.

Thus, we get, two different results, i.e., a contradiction arises.

So, our assumtions are wrong.

So, 7/0 is undefined.

Here, we must remember that, 0/0 is also undefined, it is called the indeterminate form.

So, our, conclusion is division by zero is undefined.

Did you know:Hero of mathematics is zero.

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Logarithm

Logarithm

History:

The scottish mathematian John Napier discovered logarithm. His discovery in logarithm was published in 1614. But the logarithm he  had discovered was very different from the modern logarithm.

  After John Napier , another mathematian Henry Briggs introduced base-10 logarithm. Which was very  easier to use. After that, many other mathematian contributed their theories about logarithm. In 1730, Euler defines exponential function (eˣ) and natural logarithm. The developments of modern logarithm was extended in 18-th century. Till now mathematians are updating logarithm and  introducing new theories.

Introduction:

We know that, if "a" and "x" be real , a not equal to 0, then a  and x are called the base and power/exponent/index of a in aˣ.

  Now, we can get the value of M in aˣ=M, if the values of a and x are given. As for example, if a=2, x=3 then, M= 2³=8.

 Again we can find the value of a from  aˣ=M, if x and M are given. As for example, of x=2, M=4, a=+2 or -2.

   But if the values of a and M are given , we can't get the value of x  easily from aˣ=M.

  As for example, if a=3, M=9, we get, x=2 very easily but if a=2, M=5, we are unable to get the value of x , easily by algebraic methods. In this we will use a different method which is called Logarithm.

Definition of Logarithm:

If aˣ=M, (a>0,M>0, a not equal to 1) then x is called the Logarithm of M to the base a ,and expressed as: x=logₐ M.

Converse is also true.

    Note:

(1) If we do not specify/mention the base, Logarithm is meaningless.

(2) The values of a logarithm of a number with respect to different bases will be different.

(3) The value of Logarithm for a negative number is undefined. i.e, if aˣ=-M(a and M both are positive, then value of x will be imaginary.

(4) The logarithm of 1 with respect to any base a(not equal to 0) is always 0.

(5) If a and M both are same positive number, the the value of x or value of Logarithm will be 1(as, log a a=1).

 (6) If, x= logₐ M , then , aᴸᵒᵍₐᴹ=M.

(7) Logarithm of zero is undefined. 

Laws of Logarithm:

(1) log ₐ (MN) = log ₐ (M) + log ₐ (N).

(2) log ₐ (M/N)= log ₐ (M) - log ₐ (N).

(3) log ₐ (M) = log ₓ (M) × log ₐ (x).

(4) log ₐ (M^n)= n×log ₐ (M).

(5) log ₓ (a) = 1/{log a (x) }.

(6) log ₐ (x) × log ₓ (a)= 1.

Where, M, N, a,x>0, a and b not equals to 1, n be any real number.

Some problems:

   problem:1

    If log ₓ (243)=10, then find the value of x.

     
        we have, 243= 3⁵.
         now, log ₓ (243) = 10
          or, x¹⁰ =243 = 3⁵
          or, x² = 3
           or, x =√3.
   so, the value of x is √3.

   problem:2
    If log ₇¹/² (343) = x , then what is the value of x?

       we have, 343=7³.
        now, log ₇¹/² (343) = x
            or, 7ˣ/² = 343=7³
             or, x/2 =3
              or, x=6.
  so, the value of x is 6.

  problem:3
   Calculate, log ₂ log ₂ log ₂ (16) =?
     we have, log ₂ log ₂ log ₂ (2⁴)
                    = log ₂ log ₂ (4 log ₂ 2)
                    = log ₂ log ₂( 2²)
                    = log ₂ (2 log ₂ 2)
                    = log ₂ 2 = 1.       [log ₐ a =1]

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Laws of indices

Laws of indices

Before discussing about the laws of indices, we will discuss about base, index and root.

Base and index:

If m is an integer,

  aᵐ = a×a×a×...×a(m times).

  Here, ''a" is called base and "m" is called index of "a".
   That is, in aᵐ , the number "a" itself is called base and how many times we multiply it is called it's index or power.

   As for example,

    2⁵= 2×2×2×2×2

    (-3)⁵= (-3)(-3)(-3)(-3)(-3)

     x⁴= x×x×x×x

     Here, 2, -3, x are called base and 4,5 are index.

    Root:

If a and x be two real numbers and n be any positive integer  such that, aⁿ=x , then, a is called as n-th root of x and denoted by , ⁿ√x  or (x)ˡ/ⁿ.

 In particular, if n=2,3 then a is called as the square and cube roots of x respectively.
example, Let, a²= 64, a=?
 Here, we have, 64=8².
Now, a² = 8²
       or, a=+8 or -8.
Here 8 is the square root of 64.

Note:

For, square root of a number 25(say) , we have two results +5 and-5.

For, cube root of a number, one and only one is positive.

In general, for n-th root  of a number one and only one positive root.

Laws of indices

(1) (aᵐ)×(aⁿ)=aᵐ⁺ⁿ
(2) (aᵐ)÷(aⁿ)=aᵐ⁻ⁿ
(3) (aᵐ)ⁿ= aᵐⁿ
(4) (ab)ᵐ= (aᵐ)×(bᵐ)
(5)(a÷b)ᵐ= (aᵐ)÷(bᵐ)
(6) If m is a positive number, a⁻ᵐ=1÷(aᵐ). Here, a⁻ᵐ is called reciprocal of aᵐ.
(7) If m,n are integer, aᵐ/ⁿ means (aᵐ)ˡ/ⁿ ; i.e., n-th root of aᵐ.
(8) If m=0, a⁰ is meaningless, a⁰=1.
(9) If a,m,n real and aᵐ=aⁿ then, m=n, where, a not equal to 0,1,-1.
(10) If a, b, m are real , and aᵐ= bᵐ then, either a=b or m=0.

   Some examples:

  (1) calculate, (2⁵)×(2⁻³) =?

    Ans:  we have, (2⁵)×(2⁻³) = 2⁵⁻³ = 2² = 4.

  (2) calculate, (8²)÷ (2³) =?

    Ans:  we have, 8²= (2³)² = 2⁶.

   Now,(8²)÷(2³) =(2⁶)÷(2³) = 2³=8.

  (3) simplify,( 2⁵)× (5⁵)=?

Ans: we have, (2⁵)×(5⁵) = (2×5)⁵ = 10⁵.

  (4) simplify, (9⁴)÷ (3⁴)=?

  Ans: We have, (9⁴)÷(3⁴)= (9÷3)⁴ =3⁴.

  (5) calculate, {(⁵√8)⁵/²} ×{(16)⁻³/⁸ }=?
 Ans: we have, (⁵√8)⁵/² =(8)¹/² =(2³)¹/²=2³/².
           Also, (16)⁻³/⁸ =(2⁴)⁻³/⁸ = 2⁻³/².
    so,{(⁵√8)⁵/²} ×{(16)⁻³/⁸ }=2³/² × 2⁻³/² =2⁰=1.

  (6) Arrange the following numbers in increasing order: 2⁶³ , 3⁴⁵ , 5²⁷ , 6¹⁸ .

  Here, 2⁶³ = (2⁷)⁹ = (128)⁹ ;
             3⁴⁵ =(3⁵)⁹ = (243)⁹ ;
              5²⁷ =(5³)⁹ = (125)⁹ ;
              6¹⁸ = (6²)⁹ = (36)⁹ ;
  Since, 36<125<128<243
   so, 6¹⁸<5²⁷<2⁶³<3⁴⁵.
 Therefore, the increasing order is :
                       6¹⁸, 5²⁷, 2⁶³ ,3⁴⁵.
                 
        

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seven

 seven

What is seven?

In Mathematics, the short answer is :

 Seven is a natural number ,also a prime number; which is denoted by "7".

   But, this is not all about seven!

  Let's chat with seven...

    We know that seven is considered as a lucky number. It is also called a happy number, safe number...

   But , what Mathematics say about 7???

  Let's see...

    7 is the 4-th prime number. It is a Mersenne prime (as, 2³ -1 =7); not only that,  it is also a double Mersenne prime(as, the exponent 3 itself be a Mersenne prime).

 7 is a factorial prime. It is a Harshad number also.
    Let us consider the random experiment of rolling two standard die simultaneously.
 The occurrence of getting 7 as result is 6 times(1-6,6-1,2-5,5-2,3-4,4-3) out of 36(6×6) times. Thus the the probability of getting 7 is=6/36=1/6.
    The last digit of Graham's number is seven.
   Seven  is the only dimension, besides the familiar three , in which a vector cross product can be defined .
  A seven-sided shape is called a heptagon .
  Seven  is the lowest dimension of a known exotic sphere . But, there may exist as yet unknown exotic smooth structures on the four-dimensional sphere.

   The "Millennium Prize" Problems are seven problems in mathematics which were stated by the "Clay Mathematics Institute" in 2000. Currently, six of the problems remains unsolved .

  Now, let's see what world say about 7:

(1) There are seven days in a week.

 (2) There are seven seas/oceans in the world ( North Atlantic, South Atlantic, Arctic, North Pacific, South Pacific, Indian, Southern).

(3) There are seven continents in the world (Asia, Europe, North America, South America, Africa, Australia, Antarctica).

(4) seven classical planets (i.e., the seven moving objects in the sky visible in the naked eye) ( Mars, Jupiter, Venus, Saturn, Mercury, moon and the sun itself!).

(5) Seven colours in rainbow (VIBGYOR).

(6) There are seven basic  musical notes(Indian version: sa, re, ga , ma, pa , dha, ni. Western version: do, re, me, fa, so, la, te.).

(7) There are seven logic gates: 

  NOT, AND, OR, NOR, NAND,XOR, XNOR.

(8) There are seven rows in the periodic table.

(9) There are seven heveans.
(10)In China, the entire seventh month of the lunar calendar is considered the Ghost month.
      We will close this topic with an interesting story. Yes, there is an interesting story about seven.
     Seven men were accused of Christianity around the  250 AD , when the  Roman emperor Decius ruled. They took refuge in a cave and fell asleep.The emperor saw his chance to get rid of them once and for all and ordered the cave to be sealed.
Many decades later a farmer opened the cave and found the Seven Sleepers.
They woke up believing they had only slept a day.
In 1927 the “Gotto” near Ephesus was excavated.
The ruins of a church was found and on the walls inscriptions dedicated to the Seven Sleepers.

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Polynomial

Polynomial

When we think about polynomials , one question  arises in our mind.

  What is a polynomial???

  Yes, to know about polynomials we shall start with its definition. So, let's start:

 Definition:

 In mathematics, an expression consisting of variables (x,y,z...etc) and  coefficients ( known/unknown) , and that involves addition, subtraction, multiplication and non negative integral powers/exponents of variables ; is called a polynomial.

    A polynomial in a single variable  x is of the form: 

  a₀xⁿ+ a₁xⁿ⁻¹ +...+aₙ .

   Types of polynomial:

Polynomial Degree:

Zero polynomial: a polynomial of degree zero is called a zero polynomial or a constant.
   As for example 0, 1,2,... are zero polynomials.

 Linear polynomial: A polynomial of degree 1 is called a linear polynomial.

 As for example  2x, x+3, x/5,... are linear polynomials.

 Similarly, polynomials with degree 2,3,4,5 are called quadratic, cubic, quartic, quantic polynomials respectively.

  Also polynomials are named differently according to the number of terms.

  A polynomial with  single, dual and triple terms  are called monomial, binomial, trinomial respectively.

 A polynomial with real coefficients is called a real polynomial and a polynomial with complex coefficients is called a complex polynomial.

A polynomial with integer coefficients is called an integer polynomial.

 A polynomial with one variable is called a univariate polynomial.

 As for example ( x+2) is an univariate polynomial.

A polynomial with two variables is called a bivariate polynomial.

 As for example  (x+3y+60) is a bivariate polynomial.

A polynomial with more than one variables is called a multivariate polynomial.

 As for example  (4x+y+7z-50) is a multivariate polynomial.

  Polynomial terms:

 Homogeneous polynomial:

A polynomial having more than one variable and each term of the polynomial having same degree n, is called a homogeneous polynomial of degree n.

 example:

   (x²+ 5xy+y²)  and (x³+3x²y+3xy²+y³) are homogeneous polynomials of degree 2 and 3 respectively.

Complete polynomial:

    A polynomial without any zero coefficient is said to be complete polynomial ; otherwise it is incomplete polynomial.

Vanishing polynomial:

    A polynomial all of whose coefficients are zero is called a vanishing polynomial.

Monic polynomial:

    A monic polynomial is an univariate polynomial in which the leading coefficient (the non zero coefficient of highest degree) is equal to one.

      So, a monic polynomial is of the form:     xⁿ+ aₙ_₁xⁿ⁻¹ +...+ a₁x + a₀ .

  Polynomial formula:

 (1)  addition,subtraction, multiplication of two or more polynomials are also polynomial.

(2)  division of two polynomials may not be a polynomial.

This is the basic idea about a polynomials.

(3) Derivatives and integration of a polynomial are also polynomials. 

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Factorial

Factorial

Factorial of a number

Definition: 

   The product of all consecutive positive integers from 1 to n, is called Factorial of n.

The factorial of a non negative integer is denoted by  n!.

  i.e., n!= 1.2.3...(n-2)(n-1)×n.

              = n(n-1)...3×2×1. , for all integers n greater than or equal to 1.

Factorial of 0

The factorial of zero is defined to be 1.

 0!=1.

Significance of n! 

We know that, the no. of permutations in choosing r elements from n elements is= n(n-1)(n-2)...(n-r+1).

Now, choosing  r=n, i.e., no. of permutations in taking n elements out of n elements is= n(n-1)(n-2)...3×2×1 = n!.

Significance of 0!

Simply, we can say that, the no. of ways to choose  0 element from the empty set is: 

0!/(0!×0!)=1.

 More generally, we can say that, the no. of ways to choose all the n elements among the set of n elements is:

   n!/(n!×0!)= 1.

Important property of factorial of n is: n!= n×(n-1)!.

 Note:

 ( a±b)! ≠ (a! ± b!).

  (a×b)! ≠ (a! × b!).

Some factorials:

   0! = 1.

    1!=1.

    2!=2.

    3!=6.

    4!=24.

    5!=120.

    6!=720.

    7!=50400.

     8!=40320.

     9!=362880.

     10!=3628800.

     11!=39916800.

      12!=479001600.

 Factorial of a non integer:

The factorial of a non integer can be defined using the gamma function such that, n!=π(n+1).

  Here one question may arise that, why the factorial of a negative integer does not exist?

   The simple answer is that, the factorial of a negative integer is not defind.

How does our computers calculate the value of factorial of a number?

     Mathematically  calculating  the factorial  of a number is easy. We  just multiply a bunch of numbers together. However, simple as it may seem, most computers don’t find the answer by just multiplying.  There are certain limitations.
     Factorials are always integer numbers because it is  the result of multiplying integers together.
     Modern computers covert our everyday numbers into binary, before they do any calculation. Older computers are limited to working with 32 binary digits, or bits. which translate to a maximum of 2,147,483,647 in decimal. For the newer 64-bit computers, it can store an integer in    decimal value upto 2⁶³ -1.
Which is a big number. But if you look at the table, you will see that a 32-bit computer can only calculate up to 12! and   a 64-bit computer gives accurate value upto 20!. Beyond these boundaries, most common computers  provides an approximate answer.

     

  Alternating factorial:
 The alternating factorial of a positive integer "n" is the absolute value of the  alternating sum of the factorials of the positive integers 1,2,3,...,n.
  Mathematically, we denote alternating factorial of 'n' as , af(n) and define as,
  af(n)= n! - af(n-1), [using recurrence relation]. Here, af(1)=1.
 As for example, af(4)= 4! -3! +2! - 1! = 19.

Exponential factorial:

The exponential factorial of a positive integer n is defined as the raising powers of the integers n-1, n-2, n-3,... exponentially (i.e., ((nⁿ⁻¹)ⁿ⁻²)ⁿ⁻³...  ).

 Using recurrence relation exponential factorial is defined as follows:

 aₙ = nᵃₙ_₁, where aₒ=1.

 As for example, 9 is an exponential factorial(as, 9=(3²)¹ ). 

    

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Armstrong number

Armstrong number

Let's see a magic!!!

The sum of  cubes of digits of 153 is 153 itself!

 

 Yes! It's very interesting.
These types of numbers are known as Armstrong number.

Definition:

    A number with n digits is called an Armstrong number if the sum of n-th powers of its all digits be the same number.

  i.e., for a three digit number the sum of cubes of its all digits must be equal with the original number. for a number with four digits the original number  must be equals to the sum of  fourth power of all its digits. And so on.

some examples are follows:

0,1,2,3,4,5,6,7,8,9,153,370,371,407,1634,...

  Interesting fact about an Armstrong  number with n digits is as mentioned in the definition is that , the sum of the  n-th power of the digits (for all  the  digits )is equals to the original number.
  How to check a number is Armstrong number or not:
   (1) First, find the number of digits of the given number (say, n).
   (2) Then, calculate the n-th powers  of  the all digits.
    (3) sum all the results.
     (4) Now, if the sum is equals to the original number ,then the original number is a armstrong number; otherwise it is not an armstrong number.

 Check 153 is an armstrong number or not:
  (1) The number of digits of 153 is 3.
  (2) The cubes of 1,5and 3 are 1, 125 and 27 respectively.
   (3) now sum of 1,125 and 27 is 153.
  (4) so, the sum 153 is equals to the original number 153. Thus 153 is an armstrong number.
 
   Check 121 is  an armstrong numbers or not:
  (1) The number of digits of 121 is 3.
  (2) The cubes of 1,2and1 are 1,4 and 1 respectively.
   (3) now the sum of 1, 4 and 1 is 6.
   (4) so, the sum 6 is not equals to the original number 121. Thus, 121 is not a armstrong number.

It is to be noted that, an Armstrong number is also known as narcissistic number or a plus perfect number.

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Palindrome

Palindrome

Definition: a word, number, phrase or other sequence of characters which reads the same from  the both forward (beginning) and backward (ending) positions ; is called a Palindrome.

 e.g. madam, 121, noon ,...etc.

Types of Palindrome:

There are several types of Palindrome.

 (1) Characters, word and line palindromes:
  Characters, word(s) and line(s) which reads same from the both forward and backward positions, are these types of palindromes.

 example: noon, madam, refer, level,...

Note: A sequence of characters (string) palindrome is called a string palindrome. e.g. madam.

(2) Sentence or Phrase palindromes:
 A sentence or a phrase which reads the same from the both forward and backward positions are these types of palindromes.

 example:

   "Rats live on no evil stars"

    " Step on no pets"

[Please remember that: spaces are included and capitization and spaces are to be ignored in sentence palindrome.]

 number:

 The number palindrome are  called  as "Palindromic number" or "numeral palindrome".

   e.g. 121, 11, 22, 8, ...etc.

  Now, we will discuss about the Palindromic number or numeral palindrome.
   

Palindromic number

Actually, a palindromic number is a number which remains unchanged when its digits are reversed.

   As we can see that if the digits of "121" are reversed we get "121", which remains unchanged. So, 121 is a palindromic number. But if we reverse the digits of "123" we get "321", which is a different number. So, 123 is not a palindromic number.
   How to check a number is palindromic or not:
 (1) Take the given number.
 (2) write the number from the ending position.
   (3) If the new number is equal to the original number, then the original number is palindromic; otherwise it is not a palindromic number.

      The first few decimal palindromic numbers are: 0,1,2,3,4,5,6,7,8,9,11,22,33,44,55,66,77,88, 99, 101,111,121,131,141,151,...202,212,...

      The palindromic prime numbers or palprimes( a prime number which is also a Palindromic number) are: 2,3,5,7,11,101,111,131,151,...

   The palindromic square numbers are: 1,4,9,121,484,676,...
   The palindromic cube numbers are:
   0,1,8,343,1331,...

    The binary palindromic numbers are: 0,1,11,101,111,1001,1111,...

   So, it is clear that, there are many palindromic numbers in different bases.
   Palindromic and anti palindromic polynomial:
   Let us consider a polynomial of degree n of the form:
    P= a(0)+a(1)x+a(2)x²+...+a(n)xⁿ.
  Now, P is called palindromic polynomial if,
 a(i)=a(n-i), for i=0,1,2,..,n ; and called anti - palindromic polynomial if, a(i)=-a(n-i), for, i=0,1,..,n.
Example:
  The polynomials, P(x)= (x+1)ⁿ is palindromic polynomial for all n. But the polynomials, R(x)=(x-1)ⁿ  is palindromic polynomial for even n and anti - palindromic polynomial for odd n.
  

  

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Prime number

Prime number

The first question of all :

What is a prime number?

Yes!

       A positive integer or a natural number which is greater than 1 and has exactly two factors 1 and itself, is called a Prime number.

     A Prime  number may be defined in another interesting way:

   A prime number is a positive integer greater than 1 which can not be expressed as the product of two smaller positive integers both of which are smaller than that positive integer.

   The family of Prime number starts with 2.

  

These are the prime numbers between 1 and 100.

Here one interesting thing to remember is that, 1 is not a prime number.

Yes, 1 is neither a prime number nor a composite number.

Test of primality:

To check a given number m is prime or not , we have the following steps.

(1) Find the square root of the given number, i.e. √m. Let, n=√m.

(2) now check that , m is divisible by the numbers (2 to n )or not. 

(3) If m is  completely divisible by any one number, then m is a  composite number; otherwise m is a prime number.

As for example,  let , we are to check 17 is prime or not.

Now, √17=4.123(approximately). Let, n=4.

Now, 17 is not divisible by any one of the numbers 2 to 4. So, 17 is a prime number. 

Again, let we are to check the number 16.we see that √16=4, and 16 is divisible by 2,4 .so, it is clear that 16 is a composite number.

There are various prime numbers like Fermat's prime and Mersenne prime.

Fermat's prime: a prime number of the form, 2ᵐ +1, where m= 2ⁿ and 'n' is a positive integer; is called a Fermat's prime. Some known Fermat's prime are: 3, 5, 17,...etc.

Mersenne(Marsenne) prime: It is a specific type of prime number which must be reducible in the form: 2ⁿ -1, where n is a prime number. Some of the known value of n for Mersenne prime are: 2,3,7,..

An interesting fact about prime number is that they are endless. We don't know which is the biggest one member of this family. Till now the biggest prime number :

 the Great Internet Mersenne Prime Search announced that a computer owned by Jonathan Pace in Germantown, Tennessee, discovered a new prime number. At 23,249,425 digits, the number, known as M77232917, is now the largest known prime.

  

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Pascal's triangle

Pascal's triangle

 One of the most interesting triangular array or pattern of numbers (binomial coefficients) is called the Pascal's triangle.

  It is globally named after  Blaise Pascal, a French mathematian.

     

                                      1 

                                 1         1

                             1        2          1

                         1       3        3         1

                      1      4       6           4     1

                  1      5      10      10        5      1

   

  There is a rule for the element/entry in m- th row and n- th column of Pascal's triangle.

  The element/entry is: m!/[n! ×(m-n)! ].

   There are many interesting facts about the Pascal's triangle:

    (1): The horizontal sums of each row is a power of 2.

         

    (2): each horizontal line/row of Pascal's triangle is a power of 11.

         

Note: but for 6 th line the digits overlaps.

i.e. 15101051 = 1(5+1)(0+1)051=161051.

(3): The sum of diagonal elements of  Pascal's triangle represents the Fibonacci sequence.

(4): The interesting fact is Pascal's triangle gives the combinations of heads and tails in a toss of with a  coin. Not only that, it also gives us the probability of getting any no. of heads exactly.

As for example, if we toss a coin three times; the combinations of heads and tails are: HHH, HHT, HTH, THH, TTH, THT, HTT, TTT. Which is in the pattern: 1, 3,3,1.

Also we can obtain the probability of getting exactly two heads as follows:

There are total (1+3+3+1=8) outcomes or event points.(also, 2³=8). And no. of event points with exactly two heads is 3.

So, the probability of getting exactly 3 heads is: 3/8, which is also obvious result by the theory of Probability.

(5): If we observe the diagonals of Pascal's triangle, we can see that:

The first diagonal is a sequence of unity(1), The second is a sequence of Natural numbers(1,2,3,..), The third diagonal is a sequence of triangular numbers (1,3,6,10,...) and four is a sequence of tetrahedral numbers.

(6):The Pascal's triangle is symmetrical on both sides (left and right) like a mirror image.

  

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History of pi

History of pi

What is pi???

 If it is the question, then there are different answers.

  firstly, pi is the ratio of the circumference and radius of a circle of any radius.

   It is also obvious that pi is an irrational number.

    Actually pi a mysterious number of mathematics; a special constant.

  There are many stories about the discovery of pi.

  History of pi

  Pi was  seen for the first time  200 years before the birth of Christ. Pi was used in the old Babylon. Then the  value of pi was 3.125. The Rhind Papyrus informs that in around 1600 BC Egyptians used 3.1605 as the value of pi.

  The person who had found  the value of pi for the first time mathematically was the great greek mathematician Archimedes.

    It was in around 250 BC. The value of pi given by him was greater than 3.1408 and less than 3.1429.

   After 400 years later  another greek philosopher Tolemy  had shown that pi=3.14166.( He told that a circle has 360 sides).

  In the third AD  Liu-Hiu from china, calculated the value of pi upto five decimal places.  400 years after his work another Chinese mathematician Zu Chong determines the value of pi upto six decimal places.His pi lies between  3.1415926 and 3.1415927.

 The Indian mathematicians had thought about pi differently. Around 1400 AD Indian mathematician Madhava had calculated pi correct upto 11 significant figures, using a series( now known as Gregory series).

The value of his pi was given by:

   π/4= 1-1/3 + 1/5 - 1/7 + 1/9 -...

  (This series is obtained from Gregory series and better known as Leibniz series.)

  Before Madhava another great Indian Brahmagupta in 640 AD had obtained pi=√10. It is assumed that a Chinese Zhang Heng  in 130 AD had obtained the value of pi √10 400 years before Brahmagupta.

  In the Christian 9th century Muhammad - Al - Khwarizmi had calculated pi upto 4 significant figures. After many years Jamshid - al- Kashi had found the value of 2π correct upto 16 significant figures.

  In 1701 the value of pi correct upto 100 decimal places was given by  John Machin.

  In 1781 Swedish mathematian Johann Lambert proved that pi is an irrational number.

 In 1873 William Shanks had highlighted the value of pi upto 527 decimal places.

 Point to be noted that French mathematian Faancosis Viete first expressed pi as a product of infinite terms as:

 2/π=√1/2×√1/2+1/2×√1/2...

  John Wallis also expressed pi as a product of infinite terms.

  There were many people in the entire world who had viewed pi differently.

 Till now pi is not known comptely. 

   Actually,  pi is an endless mysterious number!!!
     

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Decimal Number System

Decimal Number System

Hello!!!

Before we discuss about the decimal number system ; we should introduce ourselves with the 'numbers' and 'number systems'.

Number

Generally , a mathematical object which is used to count , label and measure ; is called a number.

There are mainly two different types of numbers: Imaginary numbers  and   Real numbers.

Number system

The way of representing numbers is called a number system.

Different types of number systems

The number systems are mainly subcategories into :
Decimal or positional number system;

Hexadecimal number system;

Octal number system;

Binary number system.

 Now we will discuss about the Decimal number system.

       

Decimal number system

This system is mostly used in calculations and measurements worldwide.

 Brief history:

 World's first decimal multiplication table was made from bamboo slips, in the time period  305 BC; during the

Warring States period in China.

Many ancient cultures calculated with numerals based on ten, sometimes argued due to human hands (as, human hands has total ten fingers) typically having ten digits.

   Some non-mathematical ancient books

  like  "Vedas" dating back to 1900–1700 BCE make use of decimals.

  The Egyptian hieratic numerals, the Greek alphabet numerals, the Hebrew alphabet numerals, the Roman numerals, the Chinese numerals and early Indian Brahmi numerals are all non-positional decimal systems, and required large numbers of symbols.

Descripton:

 The base-10 or decimal number system contains ten single digits:

  0, 1, 2,3,4,5,6,7,8,9.

 But we can't use only the single digits for any requirement. So, their permutations and combinations are made for fullfill our requirements.

 To write 10 or more, we use 2 or more digits. Each of the digits of a higher value is associated with a place value. Each of these place values is associated with a power of ten.

   Thousands   Hundreds  Tens  Ones/units

    10³                   10²          10¹       10⁰

      A general expansion of a decimal number:  aₙ,...a₁,a₀ ,b₁ ,b₂..., bₙ is as follows:

 aₙ×10ⁿ + ...+a₁×10¹ +a₀×10⁰+ b₁×10⁻¹ +...+bₙ×10⁻ⁿ.

 As for example, 19= 1×10 + 9×10⁰.
                               112=1×10² + 1×10¹+2×10⁰.

How  do we build or devlop decimal  numbers?

   It is very easy to write the single digit decimal numbers(0-9). But mathematics can not think in single digits. We need multi-digit numbers for daily  calculations.
 So, we are devloping the number system.
   Let's take an example.
 when we write the  number next to 9 ; we just  make it a number with two digits by adding 1 to the left side and make the right side 0. So, the next number to 9 becomes 10.
The next numbers are devloped by just replacing the right most digit with 1,2,..  upto 9. When we reach a number with right most digit 9; we add 1 with the left most digit and make the other digits 0. This gives the next number. As for example 20 is the next number to 19. In this way we are getting the next number. We are increasing! We are devloping!

It is needless to talk about the necessity of decimal numbers. We are using decimal numbers almost everywhere in our daily life. Actually we are living  in decimals!

  

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Complex numbers

Complex numbers

complex numbers

    History of complex numbers:

 What is the root of the equation: x²+9=0.

 This is not a tough question in the present time. But this is the main cause behind invention of complex numbers. 

       

complex numbers

   We all know that , one of the main feature of a real number is it's square is always positive. But the problem  started when the mathematicians tried to find the square root of a negative real number.

  Yes, It was Heron of Alexandria who thought about this topic for the first time, probably in 1st century, 50 AD.

He was trying to find out the value of √(81-114) .But he gave up. After this for a long time nobody showed interest about this.

  But in 1500's when solutions of 3rd and 4th degree polynomial equations were discovered, mathematicians realised the necessity of square root of a negative number.  

  Finally in 1545 , Girlamo cardano, a famous mathematician wrote a book (title: Ars Magna) on the imaginary numbers. He solved the equation: x(10-x)=40. His solution was: (5 +√-15) and (5-√-15).

But he personally did not like to work with the imaginary numbers. So he did not work more on the complex numbers.

  Later in 1637, Rane Descartes came up with the  standard form(a+ib) of complex numbers. 

complex number:  Definition:

The square root of a negative real number is called a complex number. 

  In other words, a complex/ imaginary number is a number of the form: p+iq, where p and q are real and 'i' is considered as the imaginary unit or  "iota". Here , i=√-1, be the root of : x²+1=0.

    examples: 3+5i, -3+5i, -3- 5i, 3-5i,...etc.

   In a complex number, z=x+iy, x is called the real part of z and y is called the imaginary part of  z. 

   The order pair (x,y) of z=x+iy,  represents the complex number z. If x=0, then the number (0,y) is purely imaginary and if y=0, then the number (x,0) is purely real.

  complex numbers: Geometrical representation:

       

Complex numbers

Geometrically, a complex number (x+iy)  represents a point (x,y) in the complex plane or Argand plane. Here, we take (0,0) as origin and x- axis as the real axis and Y axis as the imaginary axis.

complex numbers formulas:

Modulus of a complex number:

Let,  (x+iy) be a complex number; where x,y are real numbers and i=√-1.Then the  positive square root of (x² + y²) is called the modulus of (x+iy) and denoted by mod(z) or |z|.

As for example, modulus of (3+4i) is √(3² + 4²) = √(9+16) = 5.

   Geometrically, modulus of a complex number is the distance of the  complex number  from the origin in the complex plane.

Amplitude or argument of a complex number:

Let, z=x+iy is a complex number and |z| not equals to zero. Then the value of  θ for which both the equations , x=|z|cosθ and y= |z|sinθ are satisfied; is called the amplitude or argument of z and denoted by arg(z) or amp(z).

 So it is clear that more than one value of θ can satisfy the equations. So, more than one value of argument may exist. But, the value of θ which also satisfy -π< θ(< or=)π , is called the principal value of argument. The value of argument of a complex number z is obtained from, y/x =tan(θ).
   There is a rule to find out the amplitude of a given complex number correctly. If the complex numbers be such that,
        (1) z=x+iy  then, arg(z) = tan⁻'(y/x) ;
        (2) z=-x+iy  then, arg(z)= π- tan⁻'(y/x) ;
        (3) z=-x-iy  then, arg(z) = -π+ tan⁻'(y/x) ;
        (4) z =x-iy  then, arg(z) = -tan⁻'(y/x) .

complex numbers calculator

  

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Real Numbers

Real numbers

  We all are familiar with the number system . Here we discuss about the Real Numbers. Real numbers are those values which represents a point in a straight line, known as the real line.

History:

 Around 1000 BC the Egyptians used integers and simple fractions for basic  calculations. The use of irrational numbers was started in 600 BC. The concept of irrationality was implicitly accepted by early indian mathematicians.The Greek mathematicians realised the need for irrational numbers around 500 BC.

  The acceptance of zero, negative and fractional numbers was boosted  in the middle ages.  Firstly, the indian and chinese mathematicians took the whole responsibility. Then  the arabic mathematicians also joined them.

   In 16th century the use of decimal notation was started widely. The great mathematician Descartes introduced the term 'real' to describe the roots of a polynomial , distinguishing them from ''complex" ones in the 17th century.

  In 18th and 19th centuries the work on irrational and transcendental numbers had more  developed.

  This is a brief history of real numbers.

  Well, we know that, real numbers or the set of Real numbers (R), consists of the sets of Rational numbers, Irrational numbers , Integers, whole numbers.

         

                                                                              

  Now we will discuss about these numbers.

Rational numbers: These real number are such type that, they can be expressed in the form:p/q, where , p and q are both integers and q not equals to zero. 2/3, 3/5,0, 1...etc. point to be noted that: every integer is a rational number as it can be expressed as p/q  such that q not equals to 0. Here we also highlight that , p can be zero. So 0 is a rational number. 

Irrational numbers: These real numbers can not be expressed in the form: p/q, where p and  q are integers and q not equals to zero.                                          

                                Pi(π) is a well known irrational number. √2,√3 ...etc are also irrational numbers.

 The set of Rational numbers also consist of Integers and fractions.

Integers: The set of integers consists of positive integers{1,2,3,...} , negative integers {...-3,-2,-1} and zero. The number zero is also known as "zero integer". The set of positive integers are also called the natural numbers (N).  The set of numbers contains zero and the set of natural numbers is called whole numbers {0,1,2,3,...} .

                         At the end we are to think about the  fractions.which are also rational numbers. Actually the rational numbers excluding the integers are fractions. Generally a fractions is a part of a whole number. fractions are of three types: proper fractions (numerator<denominator) , improper fractions (numerator>denominator) and mixed fractions (combination of a whole number and a proper fraction).

This is the family of Real numbers.

  

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The Lucky Number

The Lucky Number

It's a massive hit! what a short!

 Yes, it's a short of luck for every batsman in cricket; it's the one and only six!!!

   In the 2011 world cup final the winng short for India was six!!! It was only six who had made a luck for the county!

  Not only in cricket , in the game of throwing a die; 6 is also the luck.

   Really! it's true that , 6 is a symbol of luck!
         

But in the world of mathematics 6 is not only a number of luck, but  it's very interesting also.

   (1) six is the product of two consecutive natural number (6=2×3).

    (2) six is the only even perfect number (a number which is equal to the sum of its all factors, excluding the number itself),(6=1+2+3). Again if we add all the factors of 6 including itself and divide the result by 2 ; the result is 6 itself.

     (3) six is a harmonic divisors number.

     (4) six is a congurent number.

     (5) A cube has 6 faces.

     (6) A standard guitar has 6 strings.A standard flute has 6 holes.

      (7) Insects has 6 legs.

      (8) A standard die has 6 faces.

      (9) A benzene molecule  has a ring of 6 carbons. Again 6 is the atomic number of carbon!

      (10) We have 6 senses!
      (11) There are six players on a volleyball team and an ice hockey team.
      (12)Every Braille cell (Braille is a reading and writing system for blind people) is made up of six dots; two columns consisting of three dots on each side. Various dots are raised to specify different letters.

    June is the sixth month in calender. June is named after Juno. Juno was the queen goddess in Roman mythology. She was married to Jupiter. Juno was the patron goddess of the Roman Empire. Her equivalent in Greek mythology was Hera.Juno was also the patroness of marriage.
  This may explain why so many  people consider the month of June to be a favorable time to get married.
 
    In the Bible, according to the Gospel of John, Jesus preformed his first miracle in Cana. At a wedding six stone jars were filled with water and Jesus turned the water into wine.

   In Buddhism ,  Samsara or The Wheel of Life is the six spheres of existence that all are trapped in.  These six are: beings in hell , hungry ghosts, animals, asuras , humans and devas. Everyone will upon death be reborn in a higher or lower state or class  depending on their own karma( good and bad deeds).
    The only way to break out of Samsara is by obtaining enlightenment or illumination.


    The American philosopher William James says that, "whenever two people meet , there are really 6 people present. There is each man as he sees himself, each man as the other person sees him and each man as he really is ".
      At last, We can recall the words of Saint Augustine.
“Six is a number perfect in itself, not because God created all things in six days; rather, the converse is true . God created all things in six days because the number is perfect.”

  

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Is 1 a prime number?

Is 1 a prime number?

 There are lots of confusion about the primality of the number one.

 So, naturally the question arises that is 1 a prime number?

The short answer is: no.
Yes it's simply  'no'!
   

Now let's prove our answer with mathematics.

Before we discuss the reason behind this , we shall recall the definition of a prime number.
    Definition: A prime number is a positive integer (natural number), which is greater than 1 and has exactly two factors 1 and the number itself. As for example, 2,3,5,7,11 etc.
    The definition can be stated in another way that: a prime number is a natural number (>1), which can not be expressed as the product of two smaller natural number both smaller than that number.
   

     Here, from the definition, the point to be noted that , 2 is the smallest prime number. So, every prime number is greater than 1. Thus  , we can't say 1 as a prime number.
 On the other hand, one can not be expressed as the product of two smaller natural numbers both smaller than 1 (as 1 is the least natural number).
 So, in this case also 1 failed to prove himself as a prime number.

  There are some other theories which  also prove that one is far more special than a prime number.
   (1) one is the unit of the positive integers.
   (2) one is the only multiple identity.
   (3) one is the smallest natural number.
   (4) one is the smallest positive integer which merits its own existence  by peano's axiom.
    (5) one is the only positive integer  which has only one factor or divisor 1.
    (6) The fundamental theorem of arithmetic states that,
"Every positive integer greater than one can be written uniquely as a product of primes, with the prime factors in the product written in order of nondecreasing size".
   Here we find the most important use of primes. They are the unique building blocks of the multiplicative group of integers. In discussion of warfare you often hear the phrase "divide and conquer." The same principle holds in mathematics. Many of the properties of an integer can be traced back to the properties of its prime divisors, allowing us to divide the problem into smaller problems. The number one is useless in this regard because a = 1 ×a = 1 ×1 ×a =1×1×1×a= ... That is, divisibility by one fails to provide us any information about a .

     Here, the interesting fact is , 1 is also not a composite number(a number which is not a prime number).
  So 1 is neither a prime number nor a composite number; it's unity!!!

  

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