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Mathematics & Logic

Mathematical concepts, statistics, and logical reasoning

9 categories90 words

All 9 Categories

Mathematical Foundations

Core concepts and structures in mathematics

Algebra & Equations

Terms for algebraic structures and equation solving

Calculus & Analysis

Concepts of change, limits, and continuous mathematics

Geometry & Topology

Shapes, spaces, and their properties

Set Theory & Logic

Foundational concepts of mathematical reasoning

Probability Fundamentals

Core concepts of chance and uncertainty

Distributions & Models

Common probability distributions and statistical models

Statistical Inference

Drawing conclusions from data

Regression & Correlation

Analyzing relationships between variables

All 90 Words in Mathematics & Logic

Complete vocabulary list for easy reference and copy-paste.

🔢Mathematical Foundations10

Term	Definition	Example
axiom	a statement accepted as true without proof, serving as a starting point for reasoning	"Euclid's fifth axiom about parallel lines sparked centuries of mathematical inquiry."
theorem	a statement that has been proven true based on axioms and other theorems	"The Pythagorean theorem relates the sides of a right triangle."
lemma	a proven proposition used as a stepping stone to a larger result	"The proof relied on a clever lemma about prime factorization."
corollary	a proposition that follows directly from a proven theorem	"As a corollary, all equilateral triangles are also equiangular."
conjecture	a proposition believed to be true but not yet proven	"The Riemann conjecture remains one of mathematics' greatest unsolved problems."
proof	a logical argument establishing the truth of a statement	"Her elegant proof reduced the problem to a simple contradiction."
postulate	a statement assumed true as a basis for reasoning	"Einstein's postulate that light speed is constant revolutionized physics."
hypothesis	a proposed explanation or assumption to be tested	"The null hypothesis assumes no significant difference between groups."
proof by contradiction	proving a statement by showing its negation leads to an impossibility	"The irrationality of √2 is elegantly shown by proof by contradiction."
induction	proving a statement for all natural numbers by establishing a base case and inductive step	"Mathematical induction proves the formula holds for every positive integer."

➗Algebra & Equations10

Term	Definition	Example
variable	a symbol representing an unknown or changeable quantity	"Let x be the variable representing the unknown price."
coefficient	a numerical factor multiplying a variable	"In 3x², the coefficient is 3."
polynomial	an expression of variables and coefficients using only addition, subtraction, and multiplication	"x³ + 2x² - 5x + 1 is a cubic polynomial."
quadratic	involving the second power of a variable	"The quadratic formula solves any equation of the form ax² + bx + c = 0."
factorization	expressing a number or expression as a product of its factors	"The factorization of x² - 9 is (x+3)(x-3)."
matrix	a rectangular array of numbers arranged in rows and columns	"Matrix multiplication is not commutative."
determinant	a scalar value computed from a square matrix	"A matrix is invertible only if its determinant is non-zero."
eigenvalue	a scalar by which an eigenvector is scaled under a linear transformation	"The eigenvalues reveal the matrix's fundamental scaling behaviors."
inverse	an element that combines with another to produce an identity element	"The inverse of 5 under multiplication is 1/5."
identity	an element that leaves other elements unchanged under an operation	"Zero is the additive identity; one is the multiplicative identity."

📈Calculus & Analysis10

Term	Definition	Example
derivative	the instantaneous rate of change of a function	"The derivative of position with respect to time is velocity."
integral	the accumulation of quantities, or the area under a curve	"The integral of velocity over time gives displacement."
limit	the value a function approaches as the input approaches some value	"The limit of 1/x as x approaches infinity is zero."
asymptote	a line that a curve approaches but never touches	"The function has a horizontal asymptote at y = 0."
convergence	the property of approaching a definite value or state	"The series 1 + 1/2 + 1/4 + ... converges to 2."
divergence	the property of failing to approach any finite limit	"The harmonic series diverges to infinity."
continuous	having no breaks, jumps, or discontinuities	"A continuous function can be drawn without lifting the pen."
differential	an infinitesimally small change in a variable	"The differential dx represents an infinitesimal change in x."
inflection point	a point where a curve changes from concave to convex or vice versa	"At the inflection point, the second derivative equals zero."
extremum	a maximum or minimum value of a function	"Finding extrema is essential in optimization problems."

📐Geometry & Topology10

Term	Definition	Example
euclidean	relating to ordinary flat geometry based on Euclid's axioms	"In Euclidean space, the angles of a triangle sum to 180 degrees."
topology	the study of properties preserved under continuous deformation	"Topologically, a coffee mug and a donut are equivalent."
manifold	a space that locally resembles Euclidean space	"The surface of a sphere is a two-dimensional manifold."
isomorphism	a structure-preserving mapping between two mathematical objects	"The groups are isomorphic if there exists a bijective homomorphism between them."
symmetry	invariance under a transformation	"A circle has infinite rotational symmetry."
congruent	having exactly the same size and shape	"Two triangles are congruent if all corresponding sides and angles are equal."
tangent	touching a curve at exactly one point	"The tangent line represents the instantaneous slope of the curve."
perpendicular	at right angles to a line or surface	"The walls are perpendicular to the floor."
dimension	the number of independent coordinates needed to specify a point	"We live in three spatial dimensions."
tessellation	a pattern of shapes that fit together without gaps or overlaps	"Hexagons form a natural tessellation in honeycombs."

🔗Set Theory & Logic10

Term	Definition	Example
set	a collection of distinct objects considered as a whole	"The set of prime numbers less than 10 is {2, 3, 5, 7}."
subset	a set whose elements are all contained in another set	"The even numbers are a subset of the integers."
union	the set containing all elements from two or more sets	"The union of {1,2} and {2,3} is {1,2,3}."
intersection	the set of elements common to two or more sets	"The intersection of {1,2,3} and {2,3,4} is {2,3}."
cardinality	the number of elements in a set	"The cardinality of {a, b, c} is 3."
bijection	a one-to-one correspondence between two sets	"A bijection proves two sets have equal cardinality."
predicate	a statement containing variables that becomes true or false when values are assigned	"The predicate 'x is even' is true when x = 4."
quantifier	a symbol specifying the quantity of specimens in a domain	"The universal quantifier ∀ means 'for all'."
tautology	a statement that is true under all possible interpretations	"'It will rain or it will not rain' is a tautology."
paradox	a statement that contradicts itself or defies intuition	"Russell's paradox shook the foundations of set theory."

🎲Probability Fundamentals10

Term	Definition	Example
probability	a measure of the likelihood that an event will occur	"The probability of rolling a six is 1/6."
stochastic	randomly determined; having a random probability distribution	"Stock prices follow a stochastic process."
random variable	a variable whose value is determined by chance	"The number of heads in 10 coin flips is a random variable."
expected value	the long-run average value of a random variable	"The expected value of a fair die roll is 3.5."
variance	a measure of how spread out a distribution is	"High variance indicates data points are far from the mean."
standard deviation	the square root of variance; a measure of dispersion	"Most data falls within two standard deviations of the mean."
distribution	a function showing all possible values and their probabilities	"Test scores often follow a normal distribution."
independent	events where one does not affect the probability of another	"Successive coin flips are independent events."
conditional probability	the probability of an event given that another has occurred	"The conditional probability of rain given clouds is higher than without."
Bayes' theorem	a formula for updating probabilities based on new evidence	"Bayes' theorem helps doctors interpret diagnostic test results."

📊Distributions & Models10

Term	Definition	Example
normal distribution	a symmetric bell-shaped probability distribution	"Heights in a population follow approximately a normal distribution."
binomial	relating to experiments with two possible outcomes	"The number of successes in n trials follows a binomial distribution."
Poisson distribution	a distribution modeling rare events over fixed intervals	"The number of calls per hour follows a Poisson distribution."
exponential	a distribution modeling time between events in a Poisson process	"Wait times often follow an exponential distribution."
uniform	a distribution where all outcomes are equally likely	"A fair die has a discrete uniform distribution."
skewness	a measure of asymmetry in a probability distribution	"Income distributions typically show positive skewness."
kurtosis	a measure of the 'tailedness' of a probability distribution	"High kurtosis indicates more extreme outliers."
multivariate	involving two or more random variables	"Multivariate analysis considers multiple variables simultaneously."
percentile	the value below which a given percentage of observations fall	"Scoring in the 95th percentile means outperforming 95% of test-takers."
outlier	an observation that lies far from other values	"The billionaire's income is an outlier in the dataset."

🧐Statistical Inference10

Term	Definition	Example
hypothesis testing	a procedure for deciding between competing claims about a population	"Hypothesis testing revealed the drug was more effective than placebo."
null hypothesis	the default assumption that there is no effect or difference	"We reject the null hypothesis if p < 0.05."
p-value	the probability of observing results as extreme as the data, assuming the null hypothesis	"A p-value of 0.03 suggests the result is unlikely due to chance."
confidence interval	a range of values likely to contain the true population parameter	"The 95% confidence interval for the mean is 45 to 55."
significance	the quality of being statistically unlikely to occur by chance	"The difference was statistically significant at the 0.01 level."
power	the probability of correctly rejecting a false null hypothesis	"We need a larger sample to achieve adequate statistical power."
type I error	incorrectly rejecting a true null hypothesis (false positive)	"Convicting an innocent person is analogous to a type I error."
type II error	failing to reject a false null hypothesis (false negative)	"Missing a real disease is a type II error in medical testing."
sample	a subset of a population selected for analysis	"A random sample of 1000 voters was surveyed."
bias	systematic error that skews results in a particular direction	"Selection bias can invalidate survey results."

📉Regression & Correlation10

Term	Definition	Example
correlation	a measure of the strength and direction of relationship between variables	"Height and weight show a positive correlation."
regression	a method for modeling the relationship between variables	"Linear regression predicts salary based on years of experience."
coefficient of determination	the proportion of variance explained by a model (R²)	"An R² of 0.85 means the model explains 85% of variance."
residual	the difference between observed and predicted values	"Analyzing residuals reveals patterns the model missed."
heteroscedasticity	non-constant variance in the error terms of a regression	"Heteroscedasticity violates key regression assumptions."
multicollinearity	high correlation among predictor variables	"Multicollinearity inflates standard errors of coefficients."
interpolation	estimating values within the range of known data	"We used interpolation to estimate the value at x = 5."
extrapolation	estimating values beyond the range of known data	"Extrapolating too far from the data is risky."
spurious correlation	an apparent relationship caused by a third variable or chance	"The correlation between ice cream sales and drowning is spurious."
causation	a relationship where one event directly produces another	"Correlation does not imply causation."

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