Mathematics in the Modern World: Finals

Comprehensive vocabulary flashcards covering mathematical problem solving, reasoning, financial math, linear programming, information theory, election systems, and statistics.

Problem Solving

The process of identifying the difference between the actual and the desired state of affairs and then taking action to resolve that discrepancy.

Mathematical Problem

A specific statement requiring resolution and a task that involves either mathematical operations or logical constructions.

Method

The strategy, process, or reasoning used in reaching a mathematical solution.

Answer

The final value or conclusion resulting from a mathematical process.

Solution

The complete combination of the method (process) and the result (answer).

Mathematical Reasoning

The ability to analyze a situation, construct logical arguments, and justify conclusions.

Conditional Statement

A structured mathematical argument following the format: If (hypothesis/condition) → Then (conclusion/result).

Inductive Reasoning

The process of reaching a general conclusion by examining specific examples using a bottom-up approach.

Conjecture

A conclusion formed through inductive reasoning that is not yet proven and may or may not be correct.

Deductive Reasoning

The process of reaching a conclusion by applying general rules, principles, or established facts using a top-down approach.

Counterexample

A specific instance that contradicts a general statement, enough to disprove an entire claim.

Commutative Property

The algebraic property stating that the order of multiplication does not matter ($xy = yx$).

Polya’s Four-Step Framework

A structured problem-solving method consisting of: 1. Understand the Problem, 2. Devise a Plan, 3. Carry Out the Plan, and 4. Look Back.

System of Linear Equations

A set of two or more linear equations involving the same variables that are solved simultaneously to find values satisfying all equations.

Substitution Method

A technique used to solve systems of equations by expressing one variable in terms of another and substituting it into the second equation.

Base Variable Strategy

Selecting one unknown as a reference point and expressing all other unknowns in terms of it to simplify multi-variable problems.

Working Backward Strategy

A problem-solving method starting from the final result and applying inverse operations to determine the original value.

Interest

The cost of borrowing money or the payment earned from depositing or investing money, representing the "time value of money."

Principal

The original amount of money borrowed, invested, or deposited before interest is added.

Simple Interest

Interest calculated only on the original principal amount, defined by the formula $I = P i n$.

Compound Interest

Interest calculated on both the original principal and previously accumulated interest, resulting in exponential growth.

Compound Interest Formula

$A = P\times\text{ (}1+\frac{i}{n}\text{)}^{nt}$ where $A$ is final amount, $P$ is principal, $i$ is annual rate, $n$ is compounding periods per year, and $t$ is years.

Present Value (PV)

The current worth of a future amount of money given a specific interest rate, determined by discounting the future amount.

Compounding Period

The interval (e.g., annually, quarterly, monthly) after which interest is added to the principal.

Linear Programming

A mathematical optimization technique used to maximize profit or minimize cost while operating under specific constraints.

Objective Function

In linear programming, the mathematical expression representing the quantity researchers intend to maximize or minimize.

Constraints

Restrictions or limitations placed on a linear programming problem, such as budget, time, or labor limits.

Decision Variables

The unknown quantities, such as the number of different products to produce, that must be determined in linear programming.

Coding Theory

The study of encoding information into symbols to detect and correct errors during transmission or storage to protect data integrity.

Error Detection

The ability to recognize that transmitted data contains mistakes, such as a corrupted file during download.

Error Correction

The ability to automatically repair small data errors without the need for retransmission.

Cryptography

The study of securing information by transforming readable data into coded forms to prevent unauthorized access and ensure confidentiality.

Cipher

A system or algorithm used to transform readable text (plaintext) into coded text (ciphertext) using a specific rule or key.

Encryption

The process of converting readable information (plaintext) into a coded form (ciphertext).

Decryption

The process of converting coded text (ciphertext) back into readable plaintext.

Electoral System

A set of rules and procedures used to conduct elections, collect votes, and determine winners.

Suffrage

The right to vote in elections; in the Philippines, this requires being at least $18$ years old and meeting residency requirements.

Preference Ballot

A ballot where voters rank candidates from most preferred to least preferred, providing detailed information on voter opinions.

Preference Schedule

A table summarizing the voter rankings and identical ballots to organize complex voter data for mathematical analysis.

Plurality Method

A voting method that selects the candidate with the highest number of first-place votes, ignoring lower rankings.

Borda Count Method

A voting method that assigns points to ranking positions on each ballot, rewarding candidates with broad overall support.

Plurality-with-Elimination Method

Also called Instant Runoff Voting (IRV), it repeatedly removes the candidate with the fewest first-place votes until one candidate gains a majority.

Pairwise Comparisons Method

Also known as Copeland's Method, it evaluates candidates by comparing every candidate directly against every other candidate in head-to-head matches.

Condorcet Candidate

A candidate who defeats every other candidate in head-to-head comparisons, representing the strongest overall consensus choice.

Measure of Central Tendency

A statistical concept used to find a single value that represents the center or typical value of a data set.

Arithmetic Mean

The balanced center of a data set calculated by summing all values and dividing by the total count ($\text{Sum} / \text{Count}$).

Median

The literal middle value of a data set when numbers are sorted sequentially; for even data sets, it is the mean of the two central numbers.

Mode

The specific value that appears with the highest frequency in a data set.

Linear Regression

A statistical model that attempts to show the relationship between a dependent variable ($y\text{-axis}$) and an explanatory variable ($x\text{-axis}$) with a linear equation.

Linear Correlation Coefficient ($r$)

A statistic ranging from $-1$ to $1$ that determines the strength and direction of a linear relationship between two variables.

Data Ethics

A branch of ethics that evaluates moral issues concerning data gathering, algorithms, and practices to promote morally sound solutions.

Personally Identifiable Information (PII)

Sensitive data used to uniquely identify, contact, or locate an individual, such as medical records or survey results.