MATHEMATICS IN THE MODERN WORLD
MR. LEONIL MENDOZA
LESSON 1: NATURE OF MATHEMATICS LESSON 1.1: PATTERNS AND NUMBERS IN NATURE AND THE WORLD
Patterns
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can be regular, repeated and recurring forms or design.
patterns identify relationship, find logical connections to form generalizations and predictions.
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Symmetry
❖ It indicates that you can draw an imaginary line across an object and the resulting parts are mirror of each other.
❖ Example: Butterfly- The butterfly is symmetry about the axis indicated by the dotted line. Note that the left and right
portions are the same. This type of symmetry is called bilateral symmetry.
Leonardo da Vinci’s Vitruvian Man- showing the proportions and symmetry of the human body.
Starfish- There are other types of symmetry depending on the number of sides or faces that are
symmetrical.
• Note that if you rotate the starfish, you can still achieve the sane appearance as
the original position. This is known as rotational symmetry.
• A more common way of describing rotational symmetry is by order of rotation.
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Order of Rotation
❖ A figure has a rotational symmetry of order n (n-fold rotational symmetry) if 1 𝑛 of a complete turn leaves the figure
unchanged.
❖ Formula: 𝐴𝑛𝑔𝑙𝑒 𝑜𝑓 𝑟𝑜𝑡𝑎𝑡𝑖𝑜𝑛 = 360^ / n
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Tiger’s Stripes and Hyena’s Spot’s
❖ Patterns are also exhibited in the external appearance of animals. The tiger looks and hyena’s spot, these seemingly
random designs are believed to be governed by mathematical equation.
❖ According to the theory of Allan Turing the chemical factors in the cell determine growth patterns, and influence factors
like hair color. There are two chemical processes: reaction and diffusion.
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The Sunflower
❖ Looking at the sunflower, there is a definite pattern of clockwise and counterclockwise arc of spirals extending outward
from the center of the flower.
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Flower Petals
❖ Flowers are easily considered as things of beauty.
❖ Flowers with five petals are said to be most common.
❖ Number of petals are all Fibonacci numbers.
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Population Growth
❖ Formula: 𝐴 = 𝑃𝑒^rt
❖ Where:
A is the size of the population after it grows.
P is the initial number of people, r is the rate of growth
t is time
e is the Euler’s constant with an approximate value of 2.718.
LESSON 2: SERIES AND SEQUENCE
Series
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the summation of the numbers
with plus sign
example: 2+4+6+8+10
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Two types of series:
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Arithmetic Series - the sum of a sequence in which each term is computed from the previous one by adding (or
subtracting) a constant. (Addition or subtraction)
Example: 2+4+6+8+10 (common difference is +2)
10+7+4+1 (common difference is -3)
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Geometric Series- is the sum of an infinite number of terms that have a constant ratio between successive terms.
(Multiplication or division)
Example: 3+9+27+81 (common ratio is ×3)
20+10+5+2.5 (common ratio is ÷2)
Sequence
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without operation
example: 2,4,6,8,10
Four types of sequence:
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Arithmetic Sequence- is an ordered set of numbers that have a common difference between each consecutive term.
Example: 2,4,6,8,10… (common difference is +2)
10,7,4,1… (common difference is -3)
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Geometric Sequence- each term is found by multiplying the previous term by a constant.
Example: 3,9,27,81… (common ratio is ×3)
20,10,5,2,5… (common ratio is ÷2)
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Harmonic Sequence- is formed by taking the reciprocals of every term in an arithmetic sequence.
Example: ½, ¼, 1/6, 1/8….
(Note: but not all 1/n is a harmonic sequence example: 1/3, 1/9, 1/27 is still a geometric sequence.
harmonic sequence is only applicable to arithmetic sequence)
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Fibonacci Sequence- each number is equal to the sum of the preceding two numbers.
-this sequence is infinite
Example: 0,1,1,2,3,5,8… (0+1=1, 1+1=2, 1+2=3…)
Leonardo of Pisa – Fibonacci
❖ He formulated the Fibonacci Sequence
❖ He is the greatest European mathematician in the middle age.
❖ Born on 1170, died on 1240.
❖ He also introduced the Arabic number system in Europe.
❖ Fibonacci Sequence was discovered after the reproduction of rabbit.
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Infinite sequence or series has ellipsis (…) if none then, the sequence or series if finite.
Formula in getting the nth term of Arithmetic Sequence:
an= a1 + (n−1) d
Where:
an= nth term of the sequence
(note: 1 is constant)
a1= first term of the sequence
n= number of the term
d= common difference
Example:
Find the 20th term of the sequence where; a1= 3 and d= 4
a20= 3 + (20-1) 4
a20= 3 + (19) 4
a20= 3 + 76
a20+ 79
Formula in getting the nth term of Geometric Sequence:
an= a1 ( r ) ^ n-1
Where:
an= nth term of the sequence
a1= first term of the sequence
n= number of the term
r= common ration
Example:
Find the 20th term of the sequence where; a1= 6 and d= 5
a20= 6 (5) ^ 20-1
a20= 6 (5) ^ 19
a20= 6 (2, 476, 099)
a20= 14, 856, 594
Fibonacci’s Sequence Formula
Fn= Fn-2 + Fn-1
Where:
Fn= nth term of the sequence
Fn-2= nth term in the sequence, 2 is constant
Fn-1= nth term in the sequence, 1 is constant
Binet’s Formula- finding the nth term of Fibonacci sequence.
Calcu:
Fn= nth term of the sequence
= 1. 618 (Phi or golden ratio)
= 0. 618
n= nth term of the sequence
(( 1.618)^n – (0. 618)^n)