THE FIBONACCI SEQUENCE AND THE GOLDEN RATIO || MATHEMATICS IN THE MODERN WORLD
Summary
TLDRThis video script explores the Fibonacci sequence, a series of numbers where each term is the sum of the two preceding ones, starting from one. Leonardo of Pisa, known as Fibonacci, introduced the sequence through a rabbit reproduction problem. The script delves into the sequence's prevalence in nature, such as in the spiral patterns of sunflowers and pinecones, and its relation to the golden ratio, approximately 1.618. It also demonstrates how to calculate terms in the sequence, concluding with an example to find the 15th term, which is 610.
Takeaways
- 📚 Lesson One focused on nature and patterns, while Lesson Two introduced the Fibonacci sequence.
- 🔢 Leonardo Pisano, known as Fibonacci, was a prominent European mathematician of the Middle Ages, credited with introducing the Arabic numeral system to Europe.
- 🐰 The Fibonacci sequence was discovered through an investigation into the reproduction of rabbits, where each new pair of rabbits produces another pair.
- 🌱 The sequence starts with two ones and each subsequent number is the sum of the two preceding ones, forming a pattern like 1, 1, 2, 3, 5, 8, 13, and so on.
- 📈 The Fibonacci sequence can be used to predict the number of rabbit pairs after a given number of months, with 144 pairs after one year in the example provided.
- 🌻 The Fibonacci sequence is observed in nature, such as the spiral structure of sunflowers and the arrangement of pinecones.
- 🌺 The number of petals in many flowers is often a Fibonacci number, and this can also be seen in the structure of fruits like pineapples.
- 🎨 The golden ratio, approximately 1.618, is closely related to the Fibonacci sequence and is often found in art and architecture, including the Mona Lisa.
- 🔍 The golden ratio can be expressed using the formula \( a \over b = (a + b) \over a \), where \( a \) and \( b \) are consecutive Fibonacci numbers.
- 📝 To find a specific term in the Fibonacci sequence, you start with the first two terms as one and continue by adding the last two terms to get the next one.
- 👋 The video concludes by encouraging viewers to like, subscribe, and hit the bell for more educational content.
Q & A
Who is Leonardo Pisano, also known as Fibonacci?
-Leonardo Pisano, nicknamed Fibonacci, was the greatest European mathematician of the Middle Ages, born in 1170 and died in 1240. He is known for introducing the Arabic number system to Europe.
What is the origin of the Fibonacci sequence?
-The Fibonacci sequence was discovered after an investigation on the reproduction of rabbits. It represents the number of rabbit pairs in a field over a period of time, assuming a constant rate of reproduction.
How is the Fibonacci sequence defined mathematically?
-The Fibonacci sequence is an infinite series of numbers where each number after the first two is the sum of the two preceding ones. It starts with 1, 1, and then follows with 2, 3, 5, 8, 13, and so on.
What is the significance of the Fibonacci sequence in nature?
-The Fibonacci sequence is observed in nature in various forms, such as the spiral structure of sunflowers and pine cones, the arrangement of petals in flowers, and the pattern of pineapple crowns.
How many rabbit pairs would there be at the end of one year according to the Fibonacci sequence?
-According to the script's illustration, there would be 144 pairs of rabbits at the end of one year if the sequence follows the Fibonacci pattern.
What is the golden ratio and how is it related to the Fibonacci sequence?
-The golden ratio, often denoted by the Greek letter phi (φ), is approximately equal to 1.618. It is related to the Fibonacci sequence as the ratio between successive Fibonacci numbers converges to the golden ratio.
Can the golden ratio be found in art and architecture?
-Yes, the golden ratio is often found in art and architecture, believed to provide aesthetically pleasing proportions, such as in the structure of the Mona Lisa.
How can one find the 9th term of the Fibonacci sequence?
-To find the 9th term of the Fibonacci sequence, start with the first two terms as 1 and 1, then continue adding the last two terms to get the next term, resulting in 34 as the 9th term.
What is the 15th term of the Fibonacci sequence?
-Following the pattern of the Fibonacci sequence, the 15th term is 610.
Do plants perform mathematical calculations?
-Plants do not perform mathematical calculations, but their growth patterns often follow the Fibonacci sequence, which is a result of natural processes.
Outlines
📚 Introduction to Fibonacci Sequence and Nature
The script begins with an introduction to the Fibonacci sequence, named after Leonardo Pisano Bigollo, known as Fibonacci. He was a prominent European mathematician from the Middle Ages who introduced the Arabic numeral system to Europe. The sequence, which is an infinite series of numbers starting with 1 and 1, follows a pattern where each number is the sum of the two preceding ones. The script also delves into the origin of the sequence, which was inspired by the hypothetical scenario of rabbit reproduction. It further explains how the Fibonacci sequence appears in nature, particularly in the spiral patterns found in sunflowers and pine cones, illustrating the prevalence of this mathematical concept in natural phenomena.
🌼 Fibonacci Sequence in Nature and the Golden Ratio
This paragraph explores the presence of the Fibonacci sequence in various aspects of nature, such as the arrangement of petals in flowers and the structure of fruits like pineapples. It emphasizes that while plants do not perform mathematics, their growth patterns often follow the Fibonacci sequence. The script introduces the concept of the golden ratio, denoted by the Greek letter phi (φ), which is approximately 1.618. This ratio is derived from the Fibonacci sequence and is found in various natural structures. The golden ratio is also related to the Fibonacci sequence through a specific formula. The script concludes with an example of how to calculate the ninth and fifteenth terms of the Fibonacci sequence, demonstrating the application of the sequence in mathematical problems.
Mindmap
Keywords
💡Fibonacci Sequence
💡Leonardo Pisano Bigollo
💡Nature
💡Golden Ratio
💡Rabbit Reproduction
💡Sunflower
💡Pine Cone
💡Flower Petals
💡Pineapple
💡Arithmetic Progression
💡Pattern Recognition
Highlights
Lesson one focused on nature and patterns, while lesson two introduced the Fibonacci sequence.
Leonardo Pisano, known as Fibonacci, was a prominent European mathematician of the Middle Ages.
Fibonacci's contributions include introducing the Arabic numeral system to Europe.
The Fibonacci sequence was discovered through an investigation of rabbit reproduction.
The sequence starts with two ones and each subsequent term is the sum of the two preceding ones.
An example calculation shows that 144 pairs of rabbits would be present after one year.
The Fibonacci sequence is observed in nature, such as in the spiral structure of sunflowers.
Pine cones also exhibit the Fibonacci spiral, demonstrating its prevalence in nature.
The number of petals in flowers and the arrangement of seeds in pineapples often relate to Fibonacci numbers.
Plants do not 'do math' but their growth patterns follow the Fibonacci sequence.
The golden ratio, approximately 1.618, is closely related to the Fibonacci sequence.
The golden ratio can be expressed as a ratio between two numbers, following a specific formula.
The golden ratio is observed in various natural structures, such as the Mona Lisa's composition.
A method to find a specific term in the Fibonacci sequence is demonstrated using the ninth term as an example.
The 15th term of the Fibonacci sequence is calculated as 610 in the lesson.
The video encourages viewers to like, subscribe, and hit the bell for more educational content.
Transcripts
last discussion our lesson one
all about nature and pattern in lesson
two we will discuss
the fibonacci sequence
who is leonardo obisa
fibonacci okay by the way
ansaritang fibonacci is a nickname
leonardo so fibonacci is the greatest
european mathematician of the middle
ages
he was born in 1170 and died in 1240.
he introduced the arabic number system
in europe
okay let's discuss the origin
of fibonacci sequence
okay s
it was a pair of rabbit fibonacci
sequence was discovered
after an investigation on the
reproduction of
rabbits
let's consider this illustration
suppose a newly born pair of rabbits
one male and one female are put in a
field
rabbits are able to mate at the age of
one month
so that at the end of the second month a
female can produce
another pair of rabbits consider this as
a one male and one female of rabbit
okay so nagma matured sila
in a month after that they can produce
another pair
of rabbits so get another one scenario
sequence okay
fibonacci sequence is an integer in the
infinite sequence
1 1 2 3
five eight thirteen so ethiopian
patterns are number of players no
uh rabbit reproduction of which the
first two terms
are one and one and
its succeeding term is the sum of the
two
immediately preceding in short
we add the last two terms to get the
next term
okay how many pairs will be there in one
year
so gamma kyung illustration at in kanina
yamato illustration
paris after 12 months
so we have this pattern using this
pattern we have one
one two three five eight thirteen
so my kitten attend now
next term young last two terms
eight plus thirteen get twenty one so
parama young next at 21
21 plus 13 get 34. so 55
89 144 so ebig sabin
144 pairs will be there at the end of
one year so it in first second third
fourth
pip six seven eight nine ten eleven
twelve month
okay fibonacci sequence in nature paul
supa anubinati malay relates the
fibonacci sequence in nature so
or how we can relate fibonacci sequence
in nature
so pedi banati marinate
a fibonacci sequence in nature yes for
example
the sunflower okay fibonacci spiral
in sunflower no mai kita
you can try to observe the spiral
structure of the sunflower
so this is an example of fibonacci
sequence
another is the pine cone so
another is the pine cones so
you clearly show the fibonacci spiral
okay so nothing young arrangement no
but no pine cones okay so
also we can relate or we can uh
uh see fibonacci sequence in
different plants or like for example
flowers no so the number of
petals a number of petals of a flower
are often fibonacci numbers so
number of petals now pretty nothing
considers
and one two three four five six seven
eight so either number nine
petals and also
we can also relate fibonacci sequence in
fruits like the pineapple and its crown
so
and also the pineapple fruitless so
tinder not indita no it is an example of
pattern of fibonacci sequence
and here are another example of
fibonacci sequence
tanong can plants do math
no but their growth is based on this
sequence okay we can relate
uh another one class is the golden ratio
this is related on the
uh fibonacci sequence so anubian gold in
ration am
the golden ratio is often denoted by the
greek letter
pi this is approximately equal to 1.618
okay so the golden ratio can be
expressed
as the two ratio between two numbers so
you can use this
formula or this equation a over b
is equal to a plus b over a so this
illustration can be applied using this
formula
so san bernardino golden ratio in nature
so
subpainting knee mona lisa
structuring
golden ratio okay let's try to
find the indicated term of the fibonacci
sequence for example
let's find the ninth term of fibonacci
sequence so alumni
now first and second term is one okay
so that will be the c pattern
to find fibonacci sequence so therefore
i'm
nine term nut and i 34 so a nine
terminal fibonacci sequence is 34
next 15 term okay
so another input 15 term so in the
second
anatom nine terms 55 span
ten pound eleven twelve term third
thirteenth term fourteenth term at punk
ito
terms so ebxa bn and punk 15 term i 610
thank you for watching this video i hope
you learned something
don't forget to like subscribe and hit
the bell button
but updated ko for more video tutorial
this is your guide in learning your mod
lesson your walmart
channel
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