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Mathematics in the Modern World Prelim Reviewer, Lecture notes of Mathematics

University of the East, Coloocan (UEC)Mathematics

Contains reviewers and lecture notes about various topics discussed during Preliminary term in Mathematics in the Modern World.

Typology: Lecture notes

2022/2023

Available from 06/10/2024

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GOLDEN RATIO
Two quantities are in the golden ratio if the ratio between the sum of those quantities and
the larger one is the same as the ratio between the larger one and the smaller.
Help us get the next number in the Fibonacci Sequence.
Limit of the ratios successive turns in the fibonacci sequence.
A mathematical constant approximately 1.6180339887
Divide all Fibonacci number to
Multiply the Golden Ratio to the Fibonacci Number = we get the next number in fibonacci sq
or 1.618 (the 13th number in the sequence)
the previous numbers to get
the golden ratio.
Golden ratio and Fibonacci numbers
Recall: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, ...
1.618 × 55 = 88.99 (89)
1.618 x 233 = 376.99 (377)
= 1.618
ϕ
1. Fn = Fn-1 • Fn +1 —> F13 = 144 • 377 = 233
2. Fn = Fn-2 + Fn-1 —> F13 = 89 + 144 = 233
FIBONACCI SEQUENCE
Discovered by Leonardo Pisano Bigollo Fibonacci, which means Leonardo of Pisa.
The sequence begins with zero. Each subsequent number is the sum of the two preceding
numbers.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34 , 5 5, 8 9, 144 ... .
Mathematics in the Modern World
The heart of mathematics is more than just numbers, numbers which many supposed to be
meaningless and uninteresting. It is part of our daily life. It is everywhere.
FINDING FIBONACCI NUMBERS
Example Problem: Find F13 or the 13th term
0, 1, 1, 2, 3, 5, 8, 13, 21, 34 , 5 5, 8 9, 144 ... .
F0, F 1, F2, F 3, F4 , F 5, F 6, F7 , F8, F 9, F 10, F1 1, F 12. . . .
Formulas & Solutions
GOLDEN RATIO PROBLEMS
a = width
b = length FORMULAS: W = GR x L L = GR x W
GR = L ÷ W
If u have a wood that is 0.75 m wide,
how long should u cut it such that the
Golden Ratio 1.618 is observed?
GR = 1.618
W = 0.75 m
L = ?
L = GR x W
1.618 x 0.75
= 1.2135m
1.2135 meters long should be cut so
that the Golden Ratio is still observed.
DO NOT SHARE, POST OR SELL. TNXX
♡
pf3

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GOLDEN RATIO

Two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller.

Help us get the next number in the Fibonacci Sequence. Limit of the ratios successive turns in the fibonacci sequence. A mathematical constant approximately 1.

Divide all Fibonacci number to

Multiply the Golden Ratio to the Fibonacci Number = we get the next number in fibonacci sq

or 1.618 (the 13th number in the sequence)

the previous numbers to get the golden ratio.

Golden ratio and Fibonacci numbers

Recall: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, ...

1.618 × 55 = 88.99 (89)

1.618 x 233 = 376.99 (377)

ϕ^ = 1.

  1. Fn = Fn-1 • Fn +1 —> F13 = 144 • 377 = 233
  2. Fn = Fn-2 + Fn-1 —> F13 = 89 + 144 = 233

FIBONACCI SEQUENCE

Discovered by Leonardo Pisano Bigollo Fibonacci, which means Leonardo of Pisa.

The sequence begins with zero. Each subsequent number is the sum of the two preceding numbers. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144....

Mathematics in the Modern World

The heart of mathematics is more than just numbers, numbers which many supposed to be meaningless and uninteresting. It is part of our daily life. It is everywhere.

FINDING FIBONACCI NUMBERS

Example Problem: Find F13 or the 13th term

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144.... F0, F1, F2, F3, F4, F5, F6, F7, F8, F9, F10, F11, F12.... Formulas & Solutions

GOLDEN RATIO PROBLEMS

a = width b = length

FORMULAS: W = GR x L^ L = GR x W GR = L ÷ W

If u have a wood that is 0.75 m wide,

how long should u cut it such that the

Golden Ratio 1.618 is observed?

GR = 1.

W = 0.75 m L =?

L = GR x W

1.618 x 0.

= 1.2135m

1.2135 meters long should be cut so

that the Golden Ratio is still observed.

DO NOT SHARE, POST OR SELL. TNXX ♡

C

C

ENGLISH vs MATHEMATICS

English has Noun and Sentences

Mathematics Language & Symbols

Mathematics, therefore, is the language of the sciences, business, economics, music, architecture, arts, and even politics.

Noun is a name given to an object. It can be a person, place or a thing. Sentences must state a complete thought. It can be true, false and sometimes true/sometimes false.

Mathematics has Expressions and Sentences Expression is a name given to mathematical object of interest. It can be a number, set, function, matrix and ordered pair. Sentences must state a complete thought. It can be true, false and sometimes true/sometimes false.

MATHEMATICAL LANGUAGE

precise which means it is able to make very fine distinctions or definitions among a set of mathematical symbols. concise because a mathematician can express otherwise long expositions or sentences briefly using the language of mathematics. powerful, that is, one can express complex thoughts with relative ease.

MATHEMATICAL SYMBOLS

the sum of

there exists

for every (for any)

element of (or member of)

not an element of (or not a member of)

subset of

if..,then

if and only if

set of real numbers

set of natural number

set of integers

set of rational numbers

infinity

Z

Q

N

R

MATHEMATICAL SENTENCES

The word "is" could mean equality, inequality, or membership in a set. Numbers can be CARDINAL (numbers are used for counting and answering the question "how many?", ORDINAL (tells the position of a thing in terms of first, second, third, etc.), NOMINAL (numbers are used only as a name, or to identify something, not as an actual value or position).

EXAMPLES:

A. For every real number x, there exists a real number y such that the sum of x and y is equal to 10. B. For every positive integer x, there exists a real number y such that the square of y is equal to x. C. For any real numbers x and y, the square of their sum is equal to the sum of their squares plus twice their product. D. There exist integers m and n such that m minus n is less than or equal to m plus n.

C. d.