Mathematics in the Modern World: Nature of Mathematics and Mathematical Language

Nature of Mathematics

• Definition and Scope:
  • Mathematics is the study of pattern and structure.
  • Many patterns and occurrences exist in nature, in our world, and in life; mathematics helps make sense of these patterns and occurrences.
  • Mathematics is a tool to quantify, organize, and control our world, predict phenomena, and make life easier for us.
  • Mathematics is fundamental to the physical and biological sciences, engineering, information technology, economics, and increasingly to the social sciences.
  • It provides a useful way to think about nature and our world.

Fibonacci and the Fibonacci Sequence

• Historical Background:

  • Fibonacci, also known as Leonardo Pisano, was an Italian mathematician from the Republic of Pisa.
  • He is considered to be "the most talented western mathematician of the Middle Ages".

• The Rabbit Breeding Model:

  • Fibonacci discovered his famous sequence while looking at how generations of rabbits breed over time:
  • Month 0: At the start, there is just one pair. Total: $1$ pair.
  • Month 1: After the first month, the initial pair mates, but have no young. Total: $1$ pair.
  • Month 2: After the second month, the initial pair give birth to a pair of babies. Total: $2$ pairs.
  • Month 3: After the third month, the initial pair give birth to a second pair, and their first-borns mate but have not yet given birth to any young. Total: $3$ pairs.
  • Month 4: After the fourth month, the initial pair give birth to another pair, and their first-born pair also produces a pair of their own. Total: $5$ pairs.
  • Month 5: After the fifth month, the initial pair give birth to another pair, their first-born pair produces another pair, and the second-born pair produce a pair of their own. Total: $8$ pairs.
  • The process continues indefinitely, forming the sequence: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, \dots$

• Sequences of Numbers:

  • The Fibonacci Sequence is the series of numbers: $0, 1, 1, 2, 3, 5, 8, 13, 21, 34, \dots$
  • The next number is found by adding up the two numbers before it:
  • The number $2$ is found by adding the two numbers before it ($1 + 1$).
  • The number $3$ is found by adding the two numbers before it ($1 + 2$).
  • The number $5$ is found by adding the two numbers before it ($2 + 3$).
  • Extended sequence terms:
  • $1; 1; 2; 3; 5; 8; 13; 21; 34; 55; 89; 144; 233; 377; 610; 987; 1597; 2584; 4181; 6765; 10946; 17711; 28657; 46368; 75025; 121393; 196418; 317811; 514229; 832040; 1346269; 2178309; 3524578; 5702887; 9227465; 14930352; \dots$
  • The first two numbers are $1$ and subsequent numbers are obtained by adding the previous two numbers.

• The Recurrence Rule:

  • $x_n = x_{n-1} + x_{n-2}$
  • Where:
  • $x_n$ is term number $n$.
  • $x_{n-1}$ is the previous term ($n - 1$).
  • $x_{n-2}$ is the term before that ($n - 2$).

The Golden Rule and Phi

• Definition and Mathematical Expression:

  • The Golden Rule (or Golden Ratio) is a special number found by dividing a line into two parts so that the longer part divided by the smaller part is also equal to the whole length divided by the longer part.
  • Symbolized using phi ($\phi$), after the 21st letter of the Greek alphabet.
  • Equation:     $\frac{a}{b} = \frac{a+b}{a} = 1.6180339887498948420\dots$

• Properties:

  • Phi is defined as the ratio of the circumference of a circle to its diameter.
  • Usually rounded off to $1.618$.
  • Also called the Golden mean, Golden section, or divine proportion.

• Occurrences in Art, Architecture, Nature, and Science:

  • Ancient Architecture: Great Pyramids and the Parthenon.
  • Nature and Science: Flower petals, seed beads, pinecones, tree branches, shells, spiral galaxies, hurricanes, DNA molecules, animal bodies, and fingers.

Mathematical Language and Symbols

• Mathematics as a Language:

  • Language is defined as a systematic means of communicating ideas or feelings by the use of conventionalized signs, sounds, gestures, or marks having understood meanings.
  • According to Dr. Burns, "the language of mathematics makes it easy to express the kinds of thoughts that mathematicians like to express."
  • Key Characteristics:
  1. Precise: Able to make very fine distinctions.
  2. Concise: Able to say things briefly.
  3. Powerful: Able to express complex thoughts with relative ease.

• Classification of Symbols:

  1. Numbers: Mathematical objects used to count, quantify, and label another object. These include elements of:
  • Real numbers ($\mathbf{R}$)
  • Rational numbers ($\mathbf{Q}$)
  • Irrational numbers ($\mathbf{Q}'$)
  • Integers ($\mathbf{Z}$)
  • Natural numbers ($\mathbf{N}$)
  1. Operation Symbols:
  • Addition ($+$)
  • Subtraction ($-$)
  • Multiplication ($\times$ or $\cdot$)
  • Division ($\div$ or $/$)
  • Exponentiation ($x^n$, where $x$ is the base and $n$ is the exponent)
  1. Relation Symbols:
  • Greater than or equal ($\ge$)
  • Less than or equal ($\le$)
  • Equal ($=$)
  • Not equal ($\neq$)
  • Similar ($\sim$): Figures that are the same shape, but not necessarily the same size.
  • Approximately equal ($\approx$): Two quantities that are close enough in value so the difference is insignificant in practical terms.
  • Congruent ($\cong$): Figures that are the exact same shape and size.
  1. Grouping Symbols:
  • Parentheses ($( )$)
  • Curly brackets or braces (${ }$)
  • Square brackets ($[ ]$)
  1. Variables: Symbols used when quantities take different values, usually letters of the alphabet.
  2. Set Theory Symbols:
  • Subset ($\subseteq$)
  • Union ($\cup$)
  • Intersection ($\cap$)
  • Element ($\in$)
  • Not element ($\notin$)
  • Empty set ($\emptyset$)
  1. Logic Symbols:
  • Implies ($\Rightarrow$)
  • Equivalent ($\Leftrightarrow$)
  • And ($\wedge$)
  • Or ($\vee$)
  • For all ($\forall$)
  • There exists ($\exists$)
  • Therefore ($\therefore$)
  1. Statistical Symbols:
  • Sample mean ($\bar{x}$)
  • Population mean ($\mu$)
  • Median ($\tilde{x}$)
  • Population standard deviation ($\sigma$)
  • Summation ($\Sigma$)
  • Factorial ($n!$)

Mathematical Expressions and Sentences

• Definitions and Distinction:

  • Mathematical Expression: Analogous to a "noun". It is defined as a mathematical phrase that comprises a combination of symbols that can designate numbers (constants), variables, operations, symbols of grouping, and other punctuation. It does not state a complete thought.
  • Mathematical Sentence: Makes a statement about two expressions. The two expressions use numbers, variables, or a combination of both. It uses symbols or words like equals, greater than, or less than, and it states a complete thought.

• Types of Sentences:

  • Open Sentence: A sentence that uses variables; thus it is not known whether or not the mathematical sentence is true or false.
  • Closed Sentence: A mathematical sentence that is known to be either true or false.

• Examples:

  1. $10$ is an odd number. -> Closed, False
  2. $4 + 5x = 9$ -> Open
  3. $10 - 1 = 7 + 2$ -> Closed, True
  4. $6 - x = 5$ -> Open
  5. The square root of $4x$ is $2$. -> Open

Translating Phrases to Mathematical Expressions or Sentences

• Operation and Relation Phrase Translations:
  • Addition ($+$): combined with, plus, the sum of, increased by, total, more than, added to
  • Subtraction ($-$): minus, the difference of, decreased by, fewer than, less than, subtracted from, less, take away
  • Multiplication ($\times$): twice (times 2), thrice (times 3), squared, cubed, times, the product of, multiplied by, of
  • Division ($\div$): divided by, the quotient of, half of, a third of, ratio, shared equally
  • Equal ($=$): Equals, Is, Is the same as, Yields, amount to
  • Less than or equal ($\le$): at most, not greater than
  • Greater than or equal ($\ge$): at least, not less than

Four Basic Concepts: Set, Relation, Function, and Binary Operation

• 1. Sets:

  • Definition: A set is a well-defined collection of distinct objects.
  • The objects in sets can be anything: numbers, letters, movies, people, animals, etc.
  • Each object belonging to a set is called the element or member of the set.
  • Example: The set $C$ of counting numbers less than $4$ has numbers $1, 2, 3$ as elements. The notation $\in$ indicates that a specific element belongs to a set ($1 \in C$); otherwise, $\notin$ is used ($0 \notin C$).
  • Methods of Specifying Sets:
  • Roster Method: The elements of the set are enumerated, separated by commas, and enclosed in a pair of braces (${ }$).
  • Rule Method: A phrase is used to describe all the elements in the set.
  • Definition of Terms:
  • Empty Set / Null Set: The set with no elements, denoted by $\emptyset$ or ${ }$.
  • Singleton Set: The set with only one element.
  • Universal Set: Contains all elements under consideration, denoted by $U$.
  • Finite Set: Consists of a finite number of elements.
  • Infinite Set: Consists of an infinite number of elements.
  • Equal Sets: Two sets $A$ and $B$ are equal ($A = B$) if $A$ and $B$ have exactly the same elements.
  • Equivalent Sets: Two sets $A$ and $B$ are equivalent if they have the same number of elements.
  • Subset: A set $A$ is a subset of a set $B$ ($A \subseteq B$) if and only if every element of $A$ is also an element of B$.\n * **Proper Subset**: If A \subseteq B$and$A eq B$, then$A$is a proper subset of$B$($A \subset B).\n * **Remarks**:\n 1. The null set is a proper subset of every set.\n 2. Any set is a subset of itself.\n 3. A set with n$elements has a total of$2^n subsets.\n * **Operations on Sets**:\n * Let A$and$B be two arbitrary sets:\n * **Union (A \cup B$)**: The set containing all elements which belong to either$A$or$B or to both.\n * **Intersection (A \cap B$)**: The set of all elements which are common to both$A$and$B$.
    • Complement ($A'$): The set of all elements which are in the universal set $U$ but not in A$.\n * **Worked Example**:\n * Consider sets U = {1, 2, 3, 4, 5}$,$A = {1, 5}$, and$B = {2, 3, 5}:\n 1. A \cup B = {1, 5} \cup {2, 3, 5} = {1, 2, 3, 5}\n 2. A \cap B = {1, 5} \cap {2, 3, 5} = {5}\n 3. B' = {2, 3, 5}' = {1, 4}\n 4. A \cup B' = {1, 5} \cup {1, 4} = {1, 4, 5}\n\n* **Cartesian Product**:\n * Let A$and$B$be two non-empty sets. The Cartesian product of sets$A$and$B$, denoted by$A \times B$, is the set of all ordered pairs$(a, b)$where$a \in A$and$b \in B$.
  • Worked Example:
  • Consider sets $U = \{1, 2, 3, 4, 5\}$, $A = \{1, 5\}$, and $B = \{2, 3, 5\}$:
    1. $A \times B = \{(1, 2), (1, 3), (1, 5), (5, 2), (5, 3), (5, 5)\}$
    2. $A \times A = \{(1, 1), (1, 5), (5, 1), (5, 5)\}$
    3. $B \times A = \{(2, 1), (2, 5), (3, 1), (3, 5), (5, 1), (5, 5)\}$

• 2. Relations and Functions:

  • Intuitively, a 'relation' is a relationship between sets of information (e.g., couple pairing, or pairing students' names with courses taken).
  • Mathematical Definition: A relation $R$ from set $X$ to set $Y$ is a subset of $X \times Y$. If $(x, y) \in R$, then $x$ is related to $y$ (or $y$ is in relation with $x$).
  • Domain: The domain of the relation $R$, denoted by $D(R)$, is the set of all first coordinates in the ordered pairs which belong to $R$:     $D(R) = \{x : x \in X, (x, y) \in R\}$
  • Image: The image of the relation $R$, denoted by $I(R)$, is the set of all second coordinates in the ordered pairs in $R$:     $I(R) = \{y : y \in Y, (x, y) \in R\}$

• 3. Binary Operations:
  • Let $S$ be a non-empty set. A binary operation $*$ on $S$ is a function from $S \times S$ into $S$ such that for $x, y \in S$, we have $x * y$ for $(x, y)$.
  • The image of $*$ is a subset of $S$. Thus, $S$ is closed under $*$.