Quadratic equation

 Quadratic equation 

•A polynomial equation of degree two is called a quadratic equation.

• General form of a quadratic equation (variable is x ) is: 

• A few examples of Quadratic equation are as follows: 

2x^2 +3x=0

x^2+1=0

3x^2=0.

• Solution of a quadratic equation : 

A quadratic equation can be solved in the following way: 

By factorising: 

Quadratic equation 

This formula is known as Shreedhar Acharya formula.

• Discriminant: 

The term : (b^2-4ac)  is known as discriminant of the quadratic equation. It determines the nature of roots of the Quadratic equation.

* If the discriminant of the quadratic equation is zero, the quadratic equation has real and equal roots. 

* If the discriminant is greater than zero , the quadratic equation has real and distinct roots.

* If the discriminant is less than zero, the quadratic equation has imaginary roots.

• Relation between roots and coefficients: 

If p and q are two roots of the Quadratic equation, then: 

p+q= (-b/a)       and    pq= (c/a).

That means: 

ap gp full form

 ap and gp full form:

ap gp full form

ap gp full form

In mathematics, more specifically in algebra ap and gp are very important topics.

 ap means Arithmetic progression or A.P.

And ,

gp means Geometric progression or G.P.

A.P. or ap or Arithmetic progression is a series of numbers: 1, 3, 5, ...  Or, 2,4,6,...

Whereas, 

G.P. or Geometric progression is a series of numbers: 1, 3,9,...   Or, 2,4,8,...

There are some differences between ap and gp.

ap , gp, ap gp full form

Arithmetic progression VS Geometric progression: ap vs gp

ap vs gp

Arithmetic progression VS Geometric progression:

1,2,3,...    VS   3,9,27,...

Let's Closely and carefully watch both series...🤫

 What did we see?

In the first series: 1,2,3...

  1, 1+1, 2+1,...and so on...

That means, adding 1 to the previous term of the series we get the next term.

Also, the series starts with 1 and two consecutive terms has a common difference of 1.

This type of series is known as Arithmetic progression (ap or, A.P.)

Now, let's go to the second series: 3,9,27,...

 3, 3×3, 9×3,...and so on...

Here, if we multiply the previous term with 3, we get the next term.

 The series starts with 3 and two consecutive terms has a common ratio 3.

This type of series is known as Geometric progression (gp or, G.P.)

 

The following image clearly and briefly reflects ap vs gp. Difference and relation between Arithmetic progression and Geometric progression: ap vs gp.

i.e., in a more general mathematical form:

 ap or A.P. or Arithmetic progression looks like:  

                      a, a+d, a+2d,...

a= first term, d= common difference.

And, gp or G.P. or Geometric progression looks like:

                       a, ar, ar×r,...

a= first term, r= common ratio.

 In brief, ap vs gp :

 ap or Arithmetic progression is related to addition of terms with a fixed number.

And, gp or Geometric progression is related to multiplication of terms with a fixed number.

Thanks.

Radical expression

Radical expression

Any mathematical expression containing a radical symbol(√) is called a radical expression. Generally we use the symbol '√' 

to determine the square root of a number. But, this symbol may be used to represent the 'cube' , 'fourth', or higher roots of a number.

Definition:

In mathematics, any expression containing the radical symbol (√) is known as radical expression. 

Let's explain the matter...

  Let, 'n' be any positive integer (n>1) and 'a' be any real number; then the expression, ⁿ√a or  (a)ˡ/ⁿ  is called a radical expression. Here, 'a' is known as radicand and the symbol '√' is known as radical. Here, the form  (a)ˡ/ⁿ  has a special name: "exponent form". 

  Examples:

   √16=4=(16)ˡ/² , √5,√7,...etc.

History:

The term "radical" is derived from a Latin word 'radix'. In Latin 'radix' means 'root'. 

In 1600s radical expressions were first used in England. Then the uses of radical expression spread worldwide.

Properties of radicals:

If a(>0) and b(>0) , then

(1) √a×√b = √(a×b)

(2)√(a/b)=(√a)/(√b)

(3) √(a+b) is not equals to (√a+√b).

(4) √(a-b) is not equals to (√a-√b).

Note:

  The expression (√a+√b) is called the conjugate of the expression (√a-√b). Therefore, the expressions (√a+√b) and (√a-√b) are conjugate to each other. so, we can use the conjugate to rationalize the denominator of a radical expression.

Simplified radical expression:

A radical expression is said to be in simplified radical form,  if each of the following are true:

(1) All exponents in the radicand must be less than the index.

(2) Any exponents in the radicand can have no factors in common with the index.

(3)No fractions appear under a radical.

(4) No radicals appear in the denominator of a fraction.

   As for example, simplyfy: ⁹√(a⁶).

  We have , ⁹√(a⁶) = (a⁶)ˡ/⁹ = (a)⁶/⁹= (a)²/³= ³√a².

  So, simplified radical expression of ⁹√a⁶ is ³√a².

Method to simplyfy a radical expression:

There are many radical expressions , where the radicand is not a perfect square or cubes or higher powers of a number. In such cases to simplyfy the expression , we may use the following method.

  At first factories the radicand in all possible prime factors. Then collect the same prime factors and move them outside the radical sign. Then multiply the factors inside and outside the radical sign separately. The result is in the simplified form.

As for example, simplyfy: √480.

  we have, √ 480=√(2×2×2×2×3×2×5)

                             =2×2√(3×2×5)=4√30.

Some formulas to solve a radical expression:

(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.

If you find out any incorrect information or know anything more about this , please write it in the comment section!

Permutation and combination

Permutation

Let us arrange three types of fruits namely  Apple, banana and mango in all possible ways(each is different).
   Then we will get  six different varieties.
 Apple, banana, mango;
 Apple, mango, banana;
 Banana, mango, apple;
 Banana, apple, mango;
 Mango, apple, banana;
 Mango, banana, apple;

All are different!

The arrangement of things like this are known as Permutation.

 Definition: 

Permutation is arrangement of things in all possible ways.

 In permutation the order of things is  considered.

     As for example , let, we have to form a number  consisting of three digits using the digits 1,2,3,4 . To form this number the digits have to be arranged  in some order. Different numbers will get formed depending upon the order in which we arrange the digits. In this way , each arrangement of the digits is a permutation.

  Again , let , there are three prizes and nine participants in a competition.

 We are to distribute the prizes among the top three(first, second and third) participants. Then we are to choose three people out of nine. Now, the first winner can be chosen in 9 different ways. The second winner can be chosen in 8 different ways. And  the third winner  can be chosen in 7 different ways. Thus we have total 9×8×7 different ways to choose three winners from a set of 9 participants.

   We know that, 9!=9×8×7×...×2×1.

Now , 9×8×7=9!/6!. That is, 9!/(9-3)!.

   In general, there are n!/(n-k)!  different ways to arrange k elements out of n elements in some order. Generally, it is denoted by P(n,k).

  So, P(n,k) = n!/(n-k)!.

    

Combination

 Let we are to select 11 players out of 15 players to form a cricket team. We can select any 11 of the 15 players randomly. Here if we change the order of the players the team does not changes. So, in combination order is not considered.

Definition:

Combination is the selection of things. 

In combination  the order of things is not considered.

As for example, let we are to distribute 3  prizes(same) to 3 winners(first, second and third) out of 9 participants. Since prize is same for all, the order(first, second, third) does not matter. Now, we can select 3 winners from 9 participants in P(9,3) different ways. But here order is considered. So, if we does not consider the order we have total P(9,3)/3! ways.

  Thus , we can select 3 winners from 9 participants in P(9,3)/3! ways. The order is not considered here. This is a good example of combination. 

  Generally, combination of k elements out of n elements is denoted by C(n,k).

 C(n,k)= n!/{(n-k)!×k!} = P(n,k)/k!.

 This is the basic concept of permutation and combination.

 Some interesting problems on permutation and combination:

Problem 1:

 How many words can be formed using any four letters from the word "SPRITE" ?

    

   Here, the word "SPRITE" contains six different letters. We are to form words choosing four letters out of these six letters in all possible ways. So, we are to choose four letters from six letters in all possible different ways. Therefore, the number of ways is equals to P(6,4) = 6!/(6-4)! =6!/2!=360.

Finally, 360 words can be formed by choosing four letters from six letters of the word "SPRITE".

 Problem 2:

  Suppose, there are 30 students in a class. We are to form  quiz  teams of 5 students from these  30 students. How many different teams can be formed?

Here , we are to form teams of 5 students from 30 students in all possible different ways. So, the possible number of teams are equals to C(30,5) = 30!/{5!×(30-5)!} = 142506.

  

Note:

(1) P(n,r)= P(n-1 , r) +{r× P(n-1, r-1)}.

(2) C(n,r)+C(n,r-1)=C(n+1,r).

(3) C(n,r) ÷ C(n,r-1) =(n-r+1)/r.

(4) C(n,r) =C(n, n-r).

(5) If C(n,p) = C(n,q) then, p+q=n (p not equals to q).

(6) C(n,1)+C(n,2)+C(n,3)+...+C(n,n)=2ⁿ -1.

   

If you find out any incorrect information or know anything more about this , please write it in the comment section!

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.

If you find out any incorrect information or know anything more about this , please write it in the comment section!

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.

  If you find out any incorrect information or know anything more about this , please write it in the comment section!

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).

If you find out any incorrect information or know anything more about this , please write it in the comment section!

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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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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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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