Arithmetic progression VS Geometric progression: ap vs gp

ap vs gp

Arithmetic progression VS Geometric progression:

1,2,3,...    VS   3,9,27,...

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

 What did we see?

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

  1, 1+1, 2+1,...and so on...

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

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

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

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

 3, 3×3, 9×3,...and so on...

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

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

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

 

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

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

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

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

a= first term, d= common difference.

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

                       a, ar, ar×r,...

a= first term, r= common ratio.

 In brief, ap vs gp :

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

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

Thanks.

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".