Mathematical Language Basics

Mathematical Language

• Mathematical language uses precise and concise symbols to avoid ambiguity.
• It relies on precise definitions for avoiding misunderstandings.
• Mathematical symbols include numbers, letters, binary operations, and special characters.
• Examples include one-half (1/2), seven times another number (7y), a number between -3 and 8 (-3 < n < 8), the difference between two consecutive numbers (x+1 - x = 1).
• The square of a whole number is a non-negative integer (a² ≥ 0).
• The sum of a number and 6 (n + 6).
• The product of two numbers (axb).
• Eight less than a number (n-8).
• Five more than four times a number (4x + 5).
• The sum of the squares of two numbers (x² + y²).
• One-eighth of a number (n/8).
• Three less than twice a number (2n - 3).
• A number is eight less than (n - 8)
• Eight is less than a number (-8 < n)
• Eight less a number (8 – n).

Importance of Mathematical Language

• Understanding expressed ideas is crucial.
• Communication with others is facilitated.
• Mathematical language is precise, concise, and powerful.

Problems with English Language

• English sometimes lacks precision.
• It can be ambiguous and not concise.

Mathematical Expressions

• Mathematical expressions are any combination of numbers.
• There is no equal sign in a mathematical expression.

Examples: Mathematical Sentences

• The sentences demonstrate subject, predicate, and complete thought.

Connectives

• Conjunction: P∧Q (P and Q) is true if and only if both P and Q are true.
• Disjunction: P∨Q (P or Q) is true if P is true, Q is true, or both are true.
• Implication: P→Q (If P then Q) is false only if P is true and Q is false.
• Biconditional: P↔Q (P if and only if Q) is true if both P and Q have the same truth value.
• Rules include truth tables for determining truth values of expressions involving “and,” “or,” “if,” and “if and only if.”

Logic

• Logic is crucial in determining the validity of arguments.
• Proposition: Expresses a complete thought.

Linguistic Concepts (Page 4)

• Variables: Characteristics or attributes that can hold different values.
• Data/Data Set: Values assumed by variables or collections of such values.
• Population: The complete group of subjects being studied.
• Sample: A subset of the population selected for study.
• Census/Survey: Methods for gathering data from a population or a sample.
• Quantitative Variables: Numerical values that can be ordered or ranked; examples include age, weight, and height.
• Qualitative Variables: Non-numerical values that can be placed into categories; examples include gender, blood type, or favorite color.
• Discrete Variables: Variables that can be counted.
• Continuous Variables: Variables that can take on an infinite number of values.
• Independent Variable: The cause; a variable that influences the dependent variable
• Dependent Variable: The effect; the variable that is influenced by the independent variable.
• Nuisance or Extraneous Variables: Unwanted influences on the dependent variable.
• Subject Variables: Measurable subject characteristics which are not changed by the experiment/study
• Manipulated/Experimental Variables: Variables the experimenter controls/changes which could cause changes in the subjects/participants.
• Levels of Measurement:
  • Nominal: Categories with no inherent order (e.g., religion, gender).
  • Ordinal: Categorical data with an implied order (e.g., level of education).
  • Interval: Numerical data with equal intervals between values but no true zero (e.g., temperature in Celsius or Fahrenheit).
  • Ratio: Numerical data with equal intervals and a true zero point (e.g., height, weight).

Inferential Statistics

• Inferential statistics generalizes from samples to populations.
• Estimation and hypothesis-testing are conducted to predict relationships.

Ordinary Levels of Measurement

• Nominal: Categorical, no order.
• Ordinal: Categorical, with an order.
• Interval: Numerical, equal intervals; no true zero.
• Ratio: Numerical, equal intervals; true zero.

Types of Reasoning

• Inductive reasoning: general conclusions from specific examples.
• Deductive reasoning: conclusion from applying general assumptions or procedures.

Statistics

• The science of collecting, analyzing, and making inferences from data.
• Descriptive statistics: Describes data (e.g., mean, median, standard deviation).
• Inferential statistics: makes inferences and projections about characteristics.