Module 2: Mathematical Language and Symbols

• Mathematics

  A language with its own symbols, syntax, and rules
• Time allotment
• The story of “The Tower of Babel” illustrates the importance of language
• Language is one of the most important things among people because it has an important role in communication
• Mathematical language

  A system of communication using symbols
• Mathematics is essential in communicating important ideas
• Mathematical symbols

  Representations of mathematical ideas
• Five added by three is eight can be translated as 5 + 3 = 8
• Characteristics of Mathematical Language

  • Precise
  • Concise
  • Powerful
• Vocabulary vs. Sentences

  Mathematics has vocabulary (words) and rules for combining these words into complete thoughts (sentences)
• Mathematics vocabulary must be learned correctly
• Mathematical language is a major contributor to overall comprehension
• Mathematical language is vital for the development of Mathematics proficiency
• Mathematical language enables communication of mathematical knowledge with precision
• Comparison of Natural Language and Mathematical Language

  • Expressions
  • Sentences
• Mathematical sentence

  A correct arrangement of mathematical symbols that states a complete thought
• Sentences can be true or false
• Conventions in languages

  Languages have conventions that help readers distinguish between different types of expressions
• Expression


  A correct arrangement of mathematical symbols used to represent a mathematical object of interest
• Commonly used symbols in mathematics

  • Union
  • Intersection
  • Element
  • Not an element of
  • A set of
  • Subset
  • Not a subset of
  • Ellipses
  • Conjunction
  • Disjunction
  • Negation
  • Implies
  • If and only if
  • For all
  • There exist
  • Therefore
  • Such that
  • End of proof
  • Congruence / equivalent
  • Variables
• Basic Operations and Relational Symbols

  • Addition
  • Subtraction
  • Multiplication
  • Division
  • Composition of function
  • Equal sign
• Congruence / equivalent


  A is equivalent to B
• Congruence


  a is congruent to b modulo n
• Lower case variables


  • a
  • b
  • c
  • …
  • z
• Basic Operations and Relational Symbols

  • Addition; Plus sign
  • Subtraction; minus sign
  • Multiplication sign
  • Division sign; divides
  • Composition of function
  • Equal sign
  • Not equal to
  • Greater than
  • Less than
  • Greater than or equal to
  • Less than or equal to
  • Binary operation
• Addition


  • 3 + 2
• Subtraction

  • 3 - 2
• Multiplication

  • 4 • 3
  • (4)(3)
• Division

  • 10  5
  • 5 | 10
• Equal sign

  • 5 = 5
  • 3 + 2 = 2 + 3
• Not equal to

  • 3  4
• Greater than


  • 10  5
• Less than

  • 5  10
• Greater than or equal to


  • 10  5
• Less than or equal to

  • 5  10
• Set of natural numbers / whole numbers set (with zero)


  ℕ0 = {0,1,2,3,4,...}
• Set of natural numbers / whole numbers set (without zero)


  ℕ1 = {1,2,3,4,5,...}
• Set of integer numbers


  ℤ = {...-3,-2,-1,0,1,2,3,...}
• Set of rational numbers


  ℚ = {x | x=a/b, a,b∈ℤ and b≠0}
• Set of real numbers


  ℝ = {x | -∞ < x <∞}