 
Hello friends, today
we will learn about Fibonacci Sequence and The golden ratio
and also the connection between them
let's learn about Fibonacci Sequence first
this sequence can be formed from two numbers
0 and 1
let's form the sequence
0 and 1 are first two terms
in order to get the third term
we have to add 0 and 1
so get 1 as the third term
similarly to get the fourth term
we have to add the last two i.e. 1 and 1 so we get 2
similarly to get the fifth we have to add the last two
so get get 3
just like that 5 8 13 21 34
 
till infinity
so this is Fibonacci sequence
0 1 1 2 3
but what's special in this
to get that we should understand the golden ratio first
 
we can  understand it with a line
this is a line and i cut it from this point
larger length is a and smaller is b
 
now we can calculate the value of a/b
and a+b/a from here
 
if we know the value of a and b
there exist some value of a and b
such that  a/b=a+b/a
 
 
there are
so that equal value will be the Golden ratio
 
 
let's find the value of golden ratio
we have a line of length 100
and we cut it into two parts
70 and 30
we get the value a/b as 2.33
and a+b/a as 3.33
 
 
we will make it equal by adjusting the line
 
 
we get 61.8 and 38.2
 
so we get equal value of a/b and a+b/a
that value is 1.618
this value is known as golden ratio
 
looks simple
but this number appears in nature showing beauty of art and architecture
 
how it shows? beauty we will learn it this videop
it's time to learn the connection between golden ratio and Fibonacci sequence
 
this is our Fibonacci sequence
the special about this series is that
if we divide any two numbers of this series we get approximately equal to golden ratio!
 
 
 
 
 
the bigger numbers we take the more accurate answer we get
actually golden ratio is an irrational number
 
this is only upto 3 decimal places
 
 
the bigger numbers we take the more accurate answer we get
 
now it's time
to learn how this series and golden ration describe the beuty of nature
 
we take these
to form squares
like I've taken 13 and formed the square of side 13
 
before 13 it's 8
so I've formed the square of length 8
similarly square of length 5
side of square 3 and side of square 2
 
now I'm making arcs from the corner of the square
 
like that
so we get a spiral shape
 
and this spiral is found in every beautiful things of nature
 
like
In flowers we see a spiral like that
 
it's that spiral only!
It shows the beauty of nature
 
we get this spiral in every beautiful things of nature
like you have watched various paintings y
 
you will somewhere the value of golden ratio
like for e.g. the frame of the painting can be equal to golden ratio
or the length of painting can be equal to that
something will always be equal to that
This means the painters
would have known this that taking this length will make our painting beutifull
but we can't verify it that they intentionally have taken this length
or they just paint the picture and the length become equal to golden ratio
and becomes a legendary art work
 
we denote golden ratio with a symbol phi
 
now I've told you that when
 
a/b is equal to a+b/a
then that value is golden ratio
now golden ratio has a symbol phi
so we can replace a/b with phi
 
now we separate a
so a/a + b/a
a cancel out and becomes 1
now b/a is inverse of a/b
and a/b is equal to phi
so b/a becomes 1/phi
so this is the formula for golden ratio
when you solve it you will get a quadratic equation
 
when you find the roots of it
the roots comes out to be equal to golden ratio
Fibonacci sequence was discovered by a mathematician naming Fibonacci
 
but actually it was Indians who first discovered it as suggested by some mathematicians
 
but like other theories
this theory also have no written record
so it is known by the name of Fibonacci sequence
do you know when we celebrate Fibonacci day
 
Fibonacci day is celebrated  on 23rd November
this contains a mystery too
 
because we write 23 November as
23 / 11
since 11th month is November
now if we write 11 before 23 then
we get 1123
which itself a part of Fibonacci series
so 23 November is celebrated as Fibonacci day
 
This is all about Fibonacci Sequence and The Golden Ratio
Thanks for watching!
