Mathematics in the Modern World

NU LiPa Mathematics in the Modern World

• This course covers mathematics in the modern world
• Topics include Mathematical Language and Symbols, and the Four Basic Concepts of Mathematics
• Learning outcomes include discussing the language, symbols, and conventions of mathematics
• Explaining the nature of mathematics as a language
• Performing operations on mathematical expressions correctly
• Acknowledging that mathematics is a useful language

Mathematical Language

• Language is a system for communicating ideas or feelings using symbols, sounds, or marks
• Mathematical language uses technical terms and conventions peculiar to mathematical discourse
• Mathematical language employs a highly specialized symbolic notation for formulas

Characteristics of Mathematical Language

• Precise: Able to make very fine distinctions. Mathematical symbols are used based on their meaning and purpose
• Concise: Able to say things briefly. Complex English sentences can be shortened using mathematical symbols
• Powerful: Able to express complex thoughts with relative ease. Mathematical language facilitates critical thinking, analysis, and reasoning

Examples of commonly used symbols

• Hindu-Arabic numerals (0-9)
• Symbols for operations (+, -, ×, ÷)
• Symbols representing values (w, x, y, z, etc.)
• Other special symbols (=, <, >)

Expression versus Sentences

• Expression (mathematical expression): A finite combination of symbols that's well-formed according to context-dependent rules
• Expression (mathematical expression): Does not state a complete thought; cannot be deemed true or false
• Sentence (mathematical sentence): An analogue of an English sentence, a correct arrangement of mathematical symbols expressing a complete thought
• A mathematical sentence can be evaluated as true, false, or sometimes true/false

Parts of an Expression

• Terms: Expressions are built from terms
• Coefficient: The numerical factor in a term
• Variable: Symbols representing unknown quantities
• Constant: Numerical values in a term that do not change

Examples of Expressions

• Simple (e.g., 8x – 5) → linear polynomial
• More complex (e.g., 7x² + 4x – 10) → quadratic polynomial
• Complex (e.g., x²⁄4 –12 ) → rational fraction
• Complex (e.g. ∫𝑥^2 𝑑𝑥 )

Common Words and Phrases in Math

• Addition (+): plus, sum, more than, increased by, added to, greater than, total
• Subtraction (-): difference, subtract, less than, take away, decreased by
• Multiplication (×): product of, times, twice, thrice, factor, multiplied by
• Division (÷): divided by, quotient, split, share, distribute

Binary Operations for Functions

• Sum: f(x) + g(x)
• Difference: f(x) - g(x)
• Product: f(x) * g(x)
• Quotient: f(x) / g(x)

Types of Sets

• Empty/Null Set: A set with no elements (Ø or {})
• Finite Set: A set with a limited or countable number of elements
• Infinite Set: A set with an unlimited or uncountable number of elements
• Unit/Singleton Set: A set containing only one element
• Equal Sets: Sets containing precisely the same elements
• Equivalent Sets: Sets containing the same number of elements
• Universal Set: The set containing all the elements under consideration (denoted by U)
• Subsets: A set is considered a subset of another set if all its elements are also contained within the latter; denoted by ⊂
• Power Set: The collection of all possible subsets of a set; denoted by P(A) or Power(A)

Set Operations

• **Union (∪)$: A ∪ B$ - Includes elements from both sets $A$ and $B$
• **Difference (-): A-B$ – Includes elements from $A$ that are not in $B$
• **Intersection (∩): A ∩ B$ – Includes only elements that are in both sets$A$ and $B$
• **Complement (')$: A' $ - Includes elements from the universal set $U$ that are not in $A$

Order of Operations (PEMDAS/BODMAS)

• Parentheses/Brackets first
• Exponents/Orders next
• Multiplication and Division (left-to-right)
• Addition and Subtraction (left-to-right)

Relations and Functions

• Relation: A rule that pairs elements of one set (domain) to one or more elements in another set (range) forming ordered pairs

• Function: A specific type of relation where each element in the domain maps to precisely one element in the range.

  • Types of Functions: One-to-one, one-to-many, many-to-one, many-to-many