Course in Mathematics in the modern world

1. The Nature of Mathematics
1.1 Mathematics in the World
1.2 Fibonacci Numbers
1.3 The Golden Ratio

2. Learning Objectives
At the end of the lesson the
student should be able to:
Identify patterns in nature and regularities
in the world.
Explain the importance of mathematics in
one’s life
Express appreciation for mathematics as a
human endeavor

3. As rational creatures we tend to
identify and follow patterns
whether consciously or
subconsciously.
Recognizing patterns feels
natural, like our brain is hardwired
to recognize them.

4. What are Patterns in Nature?
These are repeated designs or
behaviors found naturally. They
show consistency and structure
and often follow mathematical
principles.

5. EXAMPLES

6. Is a curved that emanates from
a central point, winding around
it with increasing distance.
Found in snail shells, galaxies,
Sunflowers, and hurricanes.
1. SPIRALS

7. Patterns and numbers in nature and
the world

8. 2. FRACTALS
Seen in fern leaves,
snowflakes, and
coastlines. They
look similar at
different scales.

9. 3. SYMMETRY
Present in
butterflies,
flowers, and
human faces.

10. 4. WAVES
In water, sound,
and light.

11. 5. TESSELLATIONS
Patterns of shapes
that fit perfectly
without gaps, like
honeycombs
made by bees.

12. EXAMPLES OF PATTERN

14. What is the next figure in the given pattern?

18. WHO IS FIBONACCI?

19. FIBONACCI
A GREAT EUROPEAN MATHEMATICIAN OF THE
MIDDLE AGES. HIS FULL NAME AN ITALIAN IS
LEONARDO PISANO, WHICH MEANS LEONARDO OF
PISA, BECAUSE HE WAS BORN IN PISA, ITALY IN
(1170-1240). FIBONACCI is the shortened word for
the Latin term. “filius Bonacci,” which stands for
“son of Bonacio” His father’s name was
GUGLIELMO BONACCIO.
FIBONACCI. Discovered the pattern of numbers
form the set {1,1,2,3,5,8,13,….} 70

20. FIBONACCI. Observed numbers in nature.
His most popular contribution perhaps is the
number that is seen in the petals of flowers.
A calla lily flower has only 1 petal, trillium
has 3, hibiscus has 5, cosmos flower has 8,
corn marigold has 13, some asters have 21,
and a daisy can have 34,55, or 89 petals,
surprisingly, these petal counts represent the
first eleven numbers of the FIBONACCI
sequence. Not all petal numbers of flowers,
however follow this pattern discovered by
Fibonacci.

21. Corn Marigold, Calla Lily,
Trillium,
Hibiscus,
Cosmos

25. The principle behind the Fibonacci
numbers is as follows:
o Letbe the nth integer in the Fibonacci sequence, the next (n+1)th
term
o Consider the first few terms below:
Let =1 be the second term, the third term Is found by +=
1+1 = 2
o The fourth term
o To find the new nth Fibonacci number, simply add the two numbers
immediately preceding this nth number.

26. n= 3: 1+1 = 2
n= 4: = 1+2 = 3
n= 5: = 2+3 = 5
n= 6: = 3+5 = 8
n= 7:
n= 8: = 8 + 13 = 21
N= 9: = 13 + 21 = 34

27. These numbers arranged in increasing order
can be written as the sequence
{1,1,2,3,5,8,13,21,34,55-89,…..)
Similarly when you count the clockwise and
counter clockwise spiral in the sun flower
seed head. It is interesting to note that the
numbers 34, and 55 occur which are
consecutive Fibonacci numbers. Pineapple
also have spiral formed by their hexagonal
nubs.

31. The Golden Ratio

32. The Golden Ratio
The ratio of two consecutive FIBONACCI NUMBERS
as n becomes large, approaches the golden ratio;
that is:
Lim
The golden Ratio denoted by φ is sometimes
called the Golden mean or golden section.

37. JOHANNES KEPLER
(Known for his laws of planetary
motion) he observed that dividing a
Fibonacci number by the number
immediately before it in the ordered
sequence yields a quotient
approximately equal to 1.618. this
amazing ratio is denoted by the
symbol φ called the Golden Ratio.

38. It is interesting to note that the ratio
of two adjacent Fibonacci
Numbers approaches the golden
ratio. That is:
As an n becomes large. This is
indeed a mystery. What does the
golden ratio have to do with a
rabbit population model?

39. The following table gives values of the
ratio
n n
3 = 2 = 1.617647.59
4 = 1.5 = 1.618181818
5 = 1.617977528
6 =1.6 = 1.61805556
7 =1.625 = 1.618025751
8 = 1.615384615 = 1.618037135
9 =1.619047619 = 1.618032787

40. KEPLER once claimed that GEOMETRY
has two great treasures;
1.Theorem of Pythagoras. This treasure
we may compare to measure of
gold. (
2.The division of a line into extreme
and mean ratio. This may we may
name a precious jewel.

44. The division of a line into EXTREME and
MEAN ratio.

50. Ratio & Proportion