MATHEMATICS IN THE MODERN WORLD
LESSON 1: TRANSFORMATION, FRACTALS, FIBONACCI, GOLDEN RATIO
PATTERNS
Logic pattern
- Ability to discover meaningful patterns in strange and unpredictable situations
Number pattern
- Sequence of numbers formed in accordance with a definite rule
Geometric pattern
- Represented by geometrical figures i.e. polygons, isometric shapes
Tiling
- combination of geometric patterns
Word pattern
- Represented by jumbled words; analyze the hidden logic in it
Alan Turing
- English mathematician ; decrypted Russian codes during WW2
SYMMETRY
Translation
- No change of shape and size
- Objects moves up, down, diagonally
Reflection
- Mirror image
- Has line of reflection/mirror line
Rotation/Turn
- Has a point about which the rotation is made and an angle that says how far to rotate
Dilation
- Transformation which changes object size
FRACTALS
Fractal
- object/quantity that displays self-similarity
Fibonacci Sequence
- Set of numbers starting with 1 or 0, followed by a one, and proceeds based on the rule
that each number (called a Fibonacci number) is equal to the sum of the two preceding
numbers
- E.g. 0, 1, 1, 2, 3, 5, 8
Tessellations
- Pattern made of identical shapes that fit together with no gaps and do not overlap
LOGIC
Logic
- Formal systematic study of the principles of valid inference and correct reasoning
- Invented by Aristotle
- Attributed to George Boole and Augustus de Morgan
Proposition
- Declarative sentence that is complete
- Cannot be ended with exclamation point and question mark
- Either true or false, but not both simultaneously
- Basic building block of logic
Symbol
Operator
Words Commonly Used
-
Negation
Not
^
Conjunction
And
v
Disjunction
Or
Arrow right
Conditional
If, then
Arrow both sides
Biconditional
If and only if
~
Propositions are represented by letters p, q, r, P, Q, etc.
The negation of P is denoted ~P
Example:
P: Today is Monday
~P: Today is not Monday
Q: There is a quiz next Friday
~Q: There is no quiz next Friday
R: 2 is an odd number
~R: 2 is not an odd number
P: She is wealthy
Q: She is happy
Write each of the following symbolic statements:
1. ~(p^q) = ~p^~q
- She is not wealthy and she is not happy.
2. ~p^q
- She is not wealthy but she is happy.
3. ~(pvq) = ~pv~q
- She is neither wealthy nor happy.
P: A student misses lecture
Q: A student studies
R: A student fails
(q^~p) -> ~r
If a student studies and does not miss a lecture, then the student will not fail.
S = The stromboli is hot
L = The lasagna is cold
P = The pizza will be delivered
The stromboli is hot and the pizza will not be delivered.
S^~P
If the lasagna is cold, then the pizza will be delivered.
L -> P
Either the lasagna is cold or the pizza won’t be delivered.
L v ~P
If the pizza won’t be delivered, then both the stromboli is hot and the lasagna is cold.
~P -> (S^L)
The lasagna is isn’t cold if and only if the stromboli isn’t hot
~L <-> ~S
Example: Find the converse, inverse, and contrapositive of the given conditional.
Given: If it rains, then I buy a new umbrella.
Converse:
If I buy a new umbrella, then it is raining.
Inverse:
If it does not rain, then I will not buy a new umbrella.
Contrapositive:
If I do not buy a new umbrella, then it is not raining.
CONJUNCTION (P^Q) (and, but)
P
Q
P^Q
T
T
T
T
F
F
F
T
F
F
F
F
P
Q
PvQ
T
T
T
T
F
T
F
T
T
F
F
F
P
Q
PvQ
T
T
T
T
F
F
F
T
T
F
F
T
DISJUNCTION (P v Q) (or)
CONDITIONAL (->)
BICONDITIONAL (<->)
P
Q
P <-> Q
T
T
T
T
F
F
F
T
F
F
F
T
Make a truth table
1.) (~p^q) v (p^~q)
p
q
~p
~p^q
~q
p^~q
v
T
T
F
F
F
F
F
T
F
F
F
T
T
T
F
T
T
T
F
F
T
F
F
T
F
T
F
F
(p^~q)->[~(p^q)]
p
q
~q
p(^~q)
->
~(p^q)
p^q
T
T
F
F
T
F
T
T
F
T
T
T
T
F
F
T
F
F
T
T
F
F
F
T
F
F
T
F
Tautology
- If all are true in the final answer
~(~p^~q) <-> ~[(p^~q)]
p
q
~p
~q
(~p^~q)
~(~p^~q)
<->
~(p^~q)
(p^~q)
T
T
F
F
F
T
T
T
F
T
F
F
T
F
T
F
F
T
F
T
T
F
F
T
T
T
F
F
F
T
T
T
F
F
T
F