Problem Solving, Mathematics in the Real World, and Mathematical Language

Vocabulary flashcards covering Inductive and Deductive Reasoning, Polya's Problem Solving, Sequences, Patterns in Nature, Set Theory, Binary Operations, and Mathematical Logic.

Inductive Reasoning

The process of making general conclusions based on specific patterns and making conjectures.

Conjecture

A conclusion or statement formulated through the use of inductive reasoning.

Deductive Reasoning

The process of making specific conclusions based on general principles, facts, rules, definitions, or properties.

George Polya

A Hungarian mathematician who immigrated to the United States in 1940, famous for his major contributions to problem solving.

Polya's Four-Step Process

A problem-solving framework consisting of four steps: 1. Understand the problem, 2. Devise a plan (Translate), 3. Carry out the plan (Solve), and 4. Look back (Check and interpret).

Difference Table

A table constructed to show the differences between successive terms of a sequence, used to discover patterns and predict subsequent terms.

Recursive Definition

A definition for a sequence that defines successive terms through the use of its preceding terms, such as $F_1 = 0$, $F_2 = 1$, and $F_n = F_{n-1} + F_{n-2}$ for $n \ge 3$.

Binet's Formula

An explicit formula used to calculate the $n\text{th}$ Fibonacci number directly using the expression $F_n = \frac{(1 + \sqrt{5})^n - (1 - \sqrt{5})^n}{2^n \sqrt{5}}$.

Arithmetic Sequence

A mathematical sequence in which each successive term originates by adding or subtracting a constant value to its predecessor.

Geometric Sequence

A mathematical sequence in which each successive term originates by multiplying or dividing its predecessor by a constant known as the common ratio.

Triangular Numbers

Numbers that can form an equilateral triangular dot pattern, calculated using the formula $X_n = \frac{n^2 + n}{2}$.

Square Numbers

Numbers that can form a square pattern, calculated using the formula $X_n = n^2$.

Cube Numbers

Numbers calculated using the formula $X_n = n^3$, producing terms such as $1$, $8$, $27$, and $64$.

Pascal's Triangle

A triangular array of numbers named after French mathematician Blaise Pascal made up of the powers of $11$, starting with $11^0$.

Tetrahedral Number

A number that can form a tetrahedral or triangular pyramid pattern, such as $1$, $4$, $10$, and $20$.

Recreational Mathematics

Mathematical puzzles and games, such as logic puzzles, Sudoku, magic squares, fractals, Rubik's cubes, and tangrams, that require deductive reasoning.

Sphere

A perfectly round geometrical object in three-dimensional space, such as the shape of a round ball.

Oblate Spheroid

A sphere flattened along the axis from pole to pole with a bulge around the equator, which closely describes the shape of the Earth.

Hexagon

A two-dimensional six-sided closed polygon that enables close packing to maximize space without gaps, used by bees in beehives.

Symmetric Figure

A figure that can be folded or divided into two identical halves.

Reflection Symmetry

A type of symmetry where the left half of a pattern is identical to the right half, also known as line symmetry or mirror symmetry.

Rotational Symmetry

A type of symmetry where a figure looks identical to its original form after a rotation of less than one full turn.

Translational Symmetry

A type of symmetry present in patterns where sliding an object across a given direction creates an identical repeating pattern.

Leonardo Pisano Bigollo

An Italian mathematician born in 1170 in Pisa, Italy, also known as Fibonacci, considered the most talented Western mathematician of the Middle Ages.

Golden Ratio

The ratio approximately equal to $1.618$ obtained by dividing a Fibonacci number by its preceding number as the terms increase.

Mathematical Language

A specialized language designed to express mathematical thoughts easily, characterized as precise, concise, and powerful.

Mathematical Expression

A name given to a mathematical object of interest (such as a number, set, function, matrix, or ordered pair) that does not state a complete thought.

Mathematical Sentence

A mathematical statement that states a complete thought and can be classified as true, false, or sometimes true/sometimes false.

Tabular Form

A method of describing a set by listing all of its elements between braces, also called the roster method.

Set-Builder Notation

A method of describing a set by stating a property or rule satisfied by all its elements within braces, also called the rule method.

Subset

A set $A$ where every element contained in $A$ is also an element of set $B$, denoted as $A \subseteq B$.

Proper Subset

A subset $A$ of set $B$ where set $B$ contains at least one element that is not in $A$, denoted as $A \subset B$.

Unit Set

A set that contains only one element.

Empty Set

A set that contains no elements, denoted by $\{\}$ or $\phi$.

Cardinal Number

A number used to measure or count the total number of elements in a given set, denoted as $n$.

Universal Set

The set containing all possible elements under discussion in a given context, denoted as $U$.

Disjoint Sets

Two sets $A$ and $B$ that have no elements in common and are mutually exclusive.

Complement of a Set

The set of all elements in the universal set $U$ that are not elements of set $A$, denoted as $A'$.

Cartesian Product

The set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$, denoted as $A \times B$.

Function

A relation in which each element of the first set (domain) corresponds to exactly one element of the second set (range).

Unary Operation

An operation performed on a single number to assign another number to it.

Binary Operation

An operation that joins two values to create a new value, such as addition, subtraction, multiplication, or division.

Closure Property

A property stating that performing a binary operation on any two elements of a set always produces an element within that same set.

Commutative Property

A binary operation property where changing the order of the operands does not change the result, such that $a * b = b * a$.

Associative Property

A binary operation property where changing the grouping of operands does not change the result, such that $(a * b) * c = a * (b * c)$.

Connective

A word or symbol (such as $\wedge$, $\vee$, $\rightarrow$, $\leftrightarrow$) used to join two sentences to produce a new sentence.

Quantifier

Words placed at the beginning of sentences to indicate scope, specifically universal quantifiers ("for all", $\forall$) and existential quantifiers ("there exists", $\exists$).

Negation

A new statement asserting the exact opposite truth value of an original statement, denoted using the tilde symbol $\sim$.

Variable

Any letter used to denote a mathematical object, state general principles, represent sequences of operations, or stand for unknown quantities.