Pythagoras's Constant

 Pythagoras's constant

Introduction:

Pythagoras's constant:

 

   The square root of 2 , i.e., √2 is known as the Pythagoras's constant.

Geometrical interpretation:

   Pythagoras's constant geometrically represents the length of  diagonal of a square with unit lengh.

  Pythagoras's constant is also the length of the  hypotenuse of a right isosceles triangle of unit base and unit perpendicular.

Properties:

 Pythagoras's constant √2 is an irrational number. More interesting fact is that, it is the first irrational number.

Decimal expansion:

The decimal expansion of pythagoras's constant is: 

   √2= 1.414213562373...

Simple continued fraction of Pythagoras's constant:

The simple continued fraction of Pythagoras's constant is:

If you find any incorrect information or know more about this topic, please mention in the comment section!!!

Imaginary unit

Imaginary unit

 Imaginary unit: brief history

    Does every quadratic equation have a real solution?

   This is the first question,  where a fresh concept or a new branch of mathematics has started. Yes, it's the story behind complex analysis or complex numbers or imaginary numbers.

  The equation behind this brand new concept is:

                   

  

The problem was to find the roots of this equation. As we all know that, one of the main features of a real number is it's square is always positive. Here, the problem started when mathematicians tried to find the root of a negative real number. i.e. x=√(-1) =???

   This is the scene when the concept of complex numbers and imaginary unit begins. 

    The square root of (-1) becomes an icon and mathematicians represented it uniquely by the symbol "i". i.e.  i=√-1.

          

 Again, since every number has two possible square roots: one positive and one negative; (-1) demands  a negative square root also. This idea introduces "-i". i.e. , -i=-√-1.

  This is the brief  history of imaginary unit.

Properties of imaginary unit 'i':

  (1) i=√-1.

  (2) i×i=(√-1)×(√-1)=-1.

  (3) i×i×i=(i×i)×i=(-1)×i=-i.

  (4) i×i×i×i= (i×i)×(i×i)=(-1)×(-1)=1.

Importance of imaginary unit:

   Imaginary unit i is the key to complex numbers or complex analysis. It is one of the main pilar for this branch of mathematics. To represent a complex number 'z' we need  i. If the real part and imaginary part of a complex number 'z' is 'x' and 'y' respectively ; then , z=x+iy. 

   Clearly, to represent a complex number we need i. 

    Again to perform operations with complex numbers (like, sum, multiplication, subtraction, division) we use the properties of imaginary unit.

 To sum up, the study of complex numbers is impossible without imaginary unit.

So, we may say, "imaginary unit is the hero of complex analysis".

Euler number

Euler number

There are several famous and widely used  irrational numbers. We  might be familiar with the most popular  number pi;   but what about  the number e?

So, Let's start with...

e = 2.7182818284590452353602874713527...

Why Euler number?

The number e is called Euler's Number because it was first used by Leonhard Euler in the 1700s.. However, another mathematician named John Napier used the number back in the 1600s with logarithms. Napier just didn't call it e yet.

It is equal to the base of the natural logarithm.

ln x = log e(x)

Let's approximate Euler number 🤜🤛

Euler's number is irrational, which means that the decimal never terminates  or ends, and it does not repeat. The digits after the decimal continue indefinitely. That means that it is impossible to write an exact value to represent e, but there are some expressions that are approximate values of e.

One possible way to approximate the value of e is:

There are another ways to do this.

One of them is to use the expression (1 + 1/n) ^n ; as the value of n increases, the expression becomes closer and closer to the value of e .

What Euler number gives us?✍️

The number , becomes helpful in many different mathematical situations, like determining the compounded interest on continuously compounded bank accounts. In fact, this very use of the value of e is how Euler came up with the number.

Once you get deeper into your maths journey, you will find that  e turns up everywhere! Euler's number is especially helpful in engineering, probability and trigonometry applications. For example, it is used in Newton's heating and cooling, it is used to relate trigonometric functions to hyperbolic functions, it is used in probability to represent the normal distribution and it is even used in calculations with electric circuits!

The 100 decimal digits of Euler number e 🤔

182845904523536028747135266249775724709369995957

9676277240766303535475945713821785251664274...

Let's end up e  with the  most amazing equation:

.

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Golden number

Golden number

Can  you imagine that mathematics  makes things beautiful?

There are certain geometric shapes and ratios that crop up again and again in art and nature. Artists and architects use these mathematical elements to make their work pleasing to the eye, while nature presumably has her own reasons for including it. One idea that has endured since ancient times is the Golden Ratio, also known as Divine Proportion.

It's really golden thinking...isn't it?

The Golden ratio (symbol: Φ , the Greek letter "phi" )
is a special number  equal to 1.618(approximately).
 It is an irrational number that is a solution of  the quadratic equation:
x²-x-1=0; 

with a value of: φ =(1+√5)/2=1.618033989...
The golden ratio is also called the golden mean or golden section. Other names include extreme and mean ratio , medial section , divine proportion ,divine section , golden proportion , golden cut  and Golden number.


The idea behind the Golden ratio:


To find the golden ratio ,  we divide a line into two parts such  that:
the long part divided by the short part ; which is also equal to
the whole length divided by the long part.

Here  we take a straight line of length a+b; where a is the long part (red shaded portion) and b is the short part (blue shaded portion).Then the Golden ratio is represented algebrically as:
  a+b/b=a/b=φ
These numbers can be applied to the proportions of a rectangle, called the Golden rectangle. This is known as one of the most visually satisfying of all geometric forms ; hence, the appearance of the Golden ratio in art. The Golden rectangle is also related to the Golden spiral, which is created by making adjacent squares of Fibonacci dimensions.


The Golden ratio also appears in all forms of nature and science. Some intersting  places include:


Face beauty:

It appears around us in our daily lives, even in our aesthetic views. Studies have shown that when test subjects view random faces, the ones they deem most attractive are those with solid parallels to the Golden ratio. Faces judged as the most attractive show Golden ratio proportions between the width of the face and the width of the eyes, nose, and eyebrows. The test subjects weren't mathematicians or physicists familiar with " phi" they were just average people, and the Golden ratio elicited an instinctual reaction.

Flower petals: The number of petals on some flowers follows the Fibonacci sequence. It is believed that in the Darwinian processes, each petal is placed to allow for the best possible exposure to sunlight and other factors.


Seed heads: The seeds of a flower are often produced at the center and migrate outward to fill the space.
As  for example : sunflowers follow this pattern.


Pinecones: The spiral pattern of the seed pods spiral upward in opposite directions. The number of steps the spirals take tend to match Fibonacci numbers.
Tree branches: The way tree branches form or split is an example of the Fibonacci sequence. Root systems and algae exhibit this formation pattern.


Shells: Many shells, including snail shells and nautilus shells, are perfect examples of the Golden spiral.

Spiral galaxies: The Milky Way has a number of spiral arms, each of which has a logarithmic spiral of approximately  12 degrees. The shape of the spiral is identical to the Golden spiral, and the Golden rectangle can be drawn over any spiral galaxy.


Hurricanes: Much like shells, hurricanes often display the Golden spiral.
Fingers: The length of our fingers, each section from the tip of the base to the wrist is larger than the preceding one by roughly the ratio of phi.

Animal bodies: The measurement of the human navel to the floor and the top of the head to the navel is the Golden ratio. But we are not the only examples of the Golden ratio in the animal kingdom; dolphins, starfish, sand dollars, sea urchins, ants and honeybees also exhibit the proportion.


DNA molecules: A DNA molecule measures 34 angstroms by 21 angstroms at each full cycle of the double helix spiral. In the Fibonacci series, 34 and 21 are successive numbers.

Golden ratio myths & facs:
There are many misconceptions and misrepresentations about the Golden ratio. Some look too strongly for patterns and say it exists where it really doesn’t. Some whose goal is to spread  golden ratio myth say it doesn’t exist where it really does, missing the obvious and often not stating what proportions appear instead. People on both sides often just repeat what they’ve heard rather than personally performing the analysis required to support their conclusions. Intelligence and education are not always factors in getting to the truth, as even Ph.D.’s in mathematics sometimes get it wrong. . Let’s look at some of the common points of confusion and debate, covering beauty, the Parthenon, the UN Secretariat Building, the Great Pyramid, Nautilus shell, use by famous artists (Da Vinci, Botticelli, Seurat, etc.) and other topics.

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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!

    

  

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,...

   

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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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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! 

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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Decimal Number System

Decimal Number System

Hello!!!

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

Number

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

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

Number system

The way of representing numbers is called a number system.

Different types of number systems

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

Hexadecimal number system;

Octal number system;

Binary number system.

 Now we will discuss about the Decimal number system.

       

Decimal number system

This system is mostly used in calculations and measurements worldwide.

 Brief history:

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

Warring States period in China.

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

   Some non-mathematical ancient books

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

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

Descripton:

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

  0, 1, 2,3,4,5,6,7,8,9.

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

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

   Thousands   Hundreds  Tens  Ones/units

    10³                   10²          10¹       10⁰

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

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

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

How  do we build or devlop decimal  numbers?

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

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

  

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Complex numbers

Complex numbers

complex numbers

    History of complex numbers:

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

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

       

complex numbers

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

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

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

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

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

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

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

complex number:  Definition:

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

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

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

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

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

  complex numbers: Geometrical representation:

       

Complex numbers

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

complex numbers formulas:

Modulus of a complex number:

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

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

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

Amplitude or argument of a complex number:

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

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

complex numbers calculator

  

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Real Numbers

Real numbers

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

History:

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

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

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

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

  This is a brief history of real numbers.

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

         

                                                                              

  Now we will discuss about these numbers.

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

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

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

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

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

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

This is the family of Real numbers.

  

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The Lucky Number

The Lucky Number

It's a massive hit! what a short!

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

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

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

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

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

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

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

     (3) six is a harmonic divisors number.

     (4) six is a congurent number.

     (5) A cube has 6 faces.

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

      (7) Insects has 6 legs.

      (8) A standard die has 6 faces.

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

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

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

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


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

  

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Is 1 a prime number?

Is 1 a prime number?

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

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

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

Now let's prove our answer with mathematics.

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

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

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

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

  

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Magic of 9

Magic of 9

Hello!!!
   We are talking about nine(9), the biggest one digit decimal member of the natural number family (N).

      It is the second non unitary square prime number of the form:P² (P square).
    Some interesting facts about 9 are:
   (1) 9 is an exponential factor[9={(3)²}¹].
    (2) The sum of digits of any multiple of 9 is 9.
            2×9=18 and 1+8=9
            3×9=27 and 2+7=9
            4×9=36 and 3+6=9
             .................................
             9×9=81 and 8+1=9
             ..................................
  121×9=1089 and 1+0+8+9=18 and 1+8=9
           and so on...
    (3)
           12345679×9=111111111
           12345679×18=222222222
           12345679×81=999999999
            ..............................................
This is true for all multipliers of 9.
     (4) The difference between a 10 base positive integer and sum of its digit is a integral  multiple of 9.
       e.g. sum of digits of 41=5
                and 41-5=36 and 36=9×4, which is an integral multiple of 9.
      (5)
            73, 7+3=10
            793, 7+9+3 =19 and 1+9=10.
            7993, 7+9+9+3=28 and 2+8=10.
      (6) The sum of digits of a number added with 9 is always equals to the sum of the digits of the result.
           56+9=65 and sum of the digits of 65=11 , sum of the digits of 65=11.
      (7) 9 can be expressed in some interesting ways:
       9=1!+2!+3!
       9=25-16= square of 5- square of 4.
       9=cube of 1+ cube of 2.
     (8) The digits of any number which is divisible  by 9 , can be rearranged in any manner and the resulting number will be also divisible by 9.
         As for example, 567 is divisible by 9.
Now, if we rearrange the digits 5,6,7 in any manner we get 576,657,675,765,756. These all are divisible by 9.
       (9) The expression 'nine times out of ten' describes a likely probability.
       (10) If we arrange the digits 0-9 in a vertical column and the digits 9-0 in another vertical column parallel to the previous column and then add  the elements in each row; we get 9 as result in each row.
       0  + 9 =9
       1  + 8 =9
       2  + 7 =9
       3  + 6 =9
       4  + 5 =9
       5  + 4 =9
       6  + 3 =9
       7  + 2 =9
       8  + 1 =9
       9  + 0 =9.
        There are many surprising facts of 9 of such type.
      There were 9 planets in the solar system, before 2006(when Pluto was rejected as a planet).
      The atomic number of fluorine is 9.
       In Hindu philosophy , there are 9       universal elements named   water,earth,Air,fire,ether,time, space,soul and mind.
       The Nine(9) of Diamonds is called “The Curse of Scotland”. Many different stories are told about this card.
      The ninth sign of the Zodiac is Sagittarius; identified by the Greeks as a centaur which is a combination of a half human and a half horse. Centaurs are magical creatures known for their skills as archers, philosophers, and predictors of the future. Being the ninth sign of the zodiac, Sagittarius has been associated with the astrological ninth house.
     Navaratri is a 9 day festival dedicated to 9 forms of Durga.
   The Double Ninth festival is an old Chinese tradition celebrated on the ninth day of the ninth lunar month. In Taiwan this day is also dedicated to the senior citizens.
The festival is associated with chrysanthemums. Poems and paintings of chrysanthemums are made.Chrysanthemum tea and wine are enjoyed.
It is also traditional to hike in the mountains on this day. In the Gregorian calendar the Double Ninth festival falls in the month of October.
     In the English calendar the 9-th month is September. Coincidentally, "September" has 9 letters. So, the 9-th month has 9-th letter.
     According to yoga the human body has 9 doors.
    A human baby takes 9 months to develop in his/her mother's womb.
     So 9 is a symbol of life and creation!!!

  

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