Decimals

Decimals

Definition:

      Decimal is  a  way of representing numbers, where each digit or place value  is multiplied or divided or by different power of tens. There are two parts, mamey integral part and fractional part ; in a decimal number. These two parts are separated by a  'decimal point'(.). 

    Let us take a decimal number, say:

       417.83

Which can be expressed as:

      4×10²   + 1×10¹ +7×10⁰ +8×10⁻¹ + 3×10⁻².

(hundreds)(tens)(units)(tenths)(hundredths)

   So, each place value is divided by ten when we move forward (right side) to  the decimal point and each place value is multiplied by ten when we move backward(left side ) to the decimal point.

 It should be noted that every whole number is a decimal number.

 As for example, 7=7.0, 67=67.0, etc.

    

 Types of decimals:

 There are different types of decimal numbers. 

     (1) exact decimals,

     (2) recurring decimals;

     (3) infinite  decimals .

 Exact decimal:

    An exact decimal is a decimal number  which terminates after a few finite digits.

 It doesn't go forever. So, we can write down all its digits. 

   As for example, 83.4, 91.1428, 66.78 , are exact decimals.

 Recurring decimal:

     A recurring decimal is a decimal number which go on forever with some repeated digits at regular intervals.

   As for example, 0.333333... and 0.14285714285714... are recurring decimals.

   Infinite or non-repeating decimal:

   An infinite decimal is a decimal number which go on forever but don't contain  any repeated digit. It contains a large number of digits which are endless!

  As for example pi(π) is an infinite decimal.

Basic operations with decimals:

(1) The addition or subtraction of two decimals are performed as below.

  Let us take two decimals 83.4 and 91.142 .

   First, we add two zeroes on the right side of the decimal point of  83.4 after the digit  4 ; such that it's value remains uneffected. So, finally we get 83.4 as 83.400 . We have done this job to equate  the number of digits of the given numbers after the decimal point.

Now these two numbers are eligible for addition.

    Now let's add them.

 83.400+91.142 =174.542.

This is the process of addition of two decimals.

 The subtraction process is similar as the addition process.

(2)The multiplication with decimals are performed as below.

   Multiplying decimals is the same as multiplying two whole numbers. We just need to remember the following:

 (i) When we multiply a decimal number with a whole number;  if there is one digit after the decimal point in the question, there will be one digit after the decimal point in the answer.

  As for example, 83.4×7=583.8 .

 (ii) When we multiply two decimal numbers ;  if there are  'm'  digits and 'n'  digits   after the decimal point  in the first and second number respectively , there will be (m+n)  digits after the decimal point in the answer.

 As for example, 83.4×91.14=7601.076 .

(3) When  we divide  a decimal  number ('n' digits after the decimal point ) by a whole number; divide as usual manner but keep the decimal point after  ' n' digits  starting from the right side.

As for example, 22.2÷2=11.1 .

        If we are dividing a decimal  number by another decimal number, we  need to use the  equivalent fractions.

 As for example, 6.38 ÷ 0.07 means  6.38/0.07, which is the same as 638 / 7 (we have multiplied the numerator and denominator by 100).

 Always remember to  multiply the numerator and denominator by the same number. And make sure that the denominator is a whole number.

Importance of decimals:

 In our world whole numbers are not enough always.

The whole numbers  are generally used  to specify  discrete quantities. As for example, there are 60 students  in the classroom.  For counting students,  and other discrete quantities,  only the whole numbers are  required. But to measure the height or weight of students we need  continuous quantities. The need to describe continuous quantities often occurs in  our everyday life. In these cases decimal numbers always helps us.

    Decimals or decimal numbers are also converted to the  percentages easily. 

Decimals are perfectly compatible with the metric system of measurement.

Decimal numbers  fit on small calculator screens and are typed very easily.

 There are such infinite number of uses of decimals.

  In the whole world decimals are really very important. Actually we live and believe in decimals!

   

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

    

  

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.

   

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Even and odd numbers

Even and odd numbers

Let's enter in the world of numbers. There are several categories of numbers like integers(positive integers and negative integers), rational numbers, irrational numbers, whole numbers, fractions and much more. The integers (both positive and negative) can also be categorised as Even numbers and odd numbers. 

   So, Let's start with even and odd.

  Definition of Even and odd numbers:

An integer number (positive or negative) which is   divisible by two(2) is called an even number.

On, the other hand, an integer number (positive or negative) which is not divisible by 2 is called a odd numbers.

 So, it is very clear that, an even number is not a odd number and a odd number is not an even number.

  As for example,  8 is  divisible by 2 .So it  is an even number. But,  9  is not divisible by 2 . So, it is odd  a number.

The even and odd numbers can be defined in an another interesting way.

  An integer is called an even number if it has no remainder(or, remainder is 0) when it is divided by 2 . Similarly, an integer is called a odd number if it has remainder '1' when  it is divided by 2.

 As for example, 6 has no remainder when it is divided by 2. So, it is an even number.

But, 5 has remainder '1' when it is divided by 2. So, 5 is a odd number.

In  a  more formal way we can define even and odd numbers.

 A number  m  is called an even number if it can be expressed as , m= 2n such that n is an integer.

 Again, a number  k is called a odd number if it can be expressed as , k=2n+1 such that n is an integer.

 Here, 4 can be expressed as, 2×2 +0. So, it is an even number. And, 7 can be expressed as, 2×3 +1 . So, it is a odd number.

Basic operations with even and odd numbers:

(1) even number+even number=even number.

Example: 12+6=18.

(2) even number+ odd number =odd number.

Example:  24+5=29.

(3) odd number+odd number=even number.

Example:  5+7=12.

(4) even number - even number= even number.

Example: 8-4 =4.

(5) even number - odd number= odd number.

Example: 10-5=5.

(6) odd number - odd number = even number.

Example, 7-5=2.

(7) even number × even number= even number.

Example: 8×4=32.

(8) even number × odd number = even number.

Example: 4×7=28.

(9) odd number× odd number= odd number.

Example: 5×3=15.

Zero is an even number.

Yes , it's true. Zero is an even number.

Here is the proof.

 Zero is divisible by 2. Also, 0 can be expressed as, 0= 2×0.

So, it is clear that, 0 is an even number.

Here are some even and odd numbers.

 Even numbers: 0,2,4,6,8,10,12,14,16,18,20,...

 Odd numbers: 1,3,5,7,9,11,13,15,17,19,...

   

 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.

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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.

  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).

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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.

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

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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