math_definitions.txt

Estimate-To find a number that is close to an exact amount.
Equation-An algebraic or numerical sentence that shows that two quantities areequal.
Front-end estimation-A method of estimating sums or differences by using the valueof the front digits of the numbers.
Solution-A value that, when substituted for the variable, makes an equation true.
Expression-A mathematical phrase or the part of a number sentence that combines numbers, operation signs,and sometimes variables, but doesn’t have an equal sign.
Inequality-A mathematical sentence that shows that two amounts are not equal.
Variable-A letter or symbol that stands for one or more numbers.
Commutative Property of Addition-The property that states that when the order of two addends is changed, the sum is the same.
Associative Property of Addition-The property that states that when the grouping of addends is changed, the sum is the same.
Population-The entire group of objects or individuals considered for a survey.
Random sample-A sample in which each subject in the overall population has an                           equal chance of being chosen.
Survey-A method of gathering information about a group.
Cumulative frequency-A running total of data.
Sample-A part of a population.
Outlier-A value separated from the rest of the data.
Mean-The average of a set of numbers, found by dividing the sum of the set by the   number of addends.
Scale-A series of numbers starting at zero and placed at fixed distances on a graph to help label the graph.
Median-The middle number in a set of data that are arranged in order.
Interval-The distance between one number and the next on the scale of a graph.
Mode-The number or numbers that occur most often in a setof data.
Stem-and-leaf plot-A table that shows groups of data arranged by place value.
Histogram-A bar graph that shows the number of times data occur within intervals.
Multiple-The product of a given whole number and another whole number.
Function-A relationship between two quantities in which one quantity depends on the other.
Compatiblenumbers-Numbers that are easy to compute mentally.
CommutativePropertyofMultiplication-The property that states that when the order of two factors is changed, the product is the same.
Evaluate-To find the value of a numerical or algebraic expression.
AssociativePropertyMultiplication-The property that states that the way factors are grouped does not change the product.
Distributiveproperty-The property that states that multiplying a sum by a number is the same as multiplying each addend in the sum by the number and then adding the products.
Divisible-Capable of being divided without a remainder.
Exponent-A number that shows how many times the base is used as a factor.
Greatestcommonfactor(GCF)-The greatest factor that twoor more numbers have in common.
Base-A number used as a repeated factor.
Commonmultiple-A number that is a multiple of two or more numbers.
Squarenumber-A product of a number and itself.
Leastcommonmultiple(LCM)-The least number, other than zero, that is a common multiple of two or more numbers.
PrimeNumber-A whole number that has exactly two factors: I and itself.
Simplestform-The form of a fraction in which the numerator and denominator have only I as a common factor.
Primefactorization-The process of factoring a composite number into its prime components, as with a factor tree, so that it is written asthe product of all its prime factors.
Mixednumber-A number that is made up of a whole number and a fraction.
Factortree-A diagram that shows the prime factors of a number.
Leastcommondenominator(LCD)-The least common multiple of two ormore denominators.
Equivalentfractions-Fractions that name the same number or amount.
Point-An exact location in space, usually represented by a dot.
Plane-A flat surface that extends without end in all directions.
Line-A straight path in a plane, extending in both directions with no endpoints.
Angle-A figure formed by two rays that meet at a common endpoint.
Ray-A part of line; it begins at one endpoint and extends forever in one direction.
Parallellines-Lines in a plane that do not intersect.
Linesegment-A part of a line between two endpoints.
Intersectinglines-Lines that cross each other at exactly one point.
Perpendicular lines-Twolines that intersect to form right angles.
Regular polygon-A polygon in which all sides are congruent and all angles are congruent.
Degree-A unit for measuring angles or temperature.
Circle-A closed figure with all points on the figure the same distance from the center point.
Protractor-A tool used for measuring or drawing angles.
Radius-A line segment with one endpoint at the center of a circle and the other endpoint on the circle.
Polygon-a closed plane figure formed by three or more line segments.
Diameter-A line segment that passes through the center of a circle and has its endpoints on the circle.
Chord-A line segment with endpoints on a circle.
Similar-Having the same shape but not necessarily the same size.
Compass-A tool used to construct circles and arcs.
Corresponding angles-Angles that are in the same position in different plane figures.
Central angle-An angle formed by two radii of a circle that meet at its center.
Corresponding sides-Sides that are in the same position in different plane figures.
Congruent-Having the same size and shape.
Line symmetry-The property of a figure that can be separated by a line into two congruent parts.
Leg-In a right triangle, either of two sides that form the right angle.
Isosceles triangle-A triangle with exactly two congruent sides.
Hypotenuse-In a right triangle, the side opposite the right angle; the longest side in a right triangle.
Scalene triangle-A triangle with no congruent sides.
Figurate numbers-Numbers that can be represented by geometric figures.
Equilateral triangle-A triangle with three congruent sides.
Triangular numbers-Numbers that can be represented by triangular figures.
Trapezoid-A quadrilateral with exactly one pair of parallel sides.
Reflection-A movement of a figure to a new position by flipping it over a line; a flip.
Rhombus-a parallelogram with congruent sides.
Rotation-A movement of a figure to a new position by turning it around a fixed point; a turn.
Parallelogram-A quadrilateral whose opposite sides are parallel and congruent.
Transformation-The movement of a figure to a new position by a translation, reflection, or rotation.
Translation-A movement of a figure to a new position along a straight line; a slide.
Tessellation-A repeating pattern of closed figures that covers a surface with no gaps and no overlaps.
Polyhedron-A solid figure with faces that are polygons.
Integers-The set of whole numbers and their opposites.
Pyramid-A solid figure with a polygon base and all other faces are triangles that meet at a common vertex.
Negative integer-Any integer less than zero.
Prism-A solid figure that has two congruent, polygon-shaped bases and whose other faces are all rectangles.
Positive integer-Any integer greater than zero.
Base-A polygon’s side or a solid figure’s face by which the figure is measured or named.
Opposites-Two numbers that are the same distance, but in opposite directions, from zero on a number line.
Absolute value-The distance of a number from zero on a number line.
X-axis-The horizontal number line on a coordinate plane.
Coordinates-The numbers in a ordered pair.
Y-axis-The vertical number line on a coordinate plane.
Precision-A property of measurement that is related to the unit of measure used; the smaller the unit of measure used, the more precise the measurement.
Coordinate plane-A plane formed by two intersecting and perpendicular number lines called axes.
Perimeter-The distance around a closed plane figure.
Circumference-The distance around a circle.
Area-The number of square units needed to cover a surface.
Group-A set equipped with an operation that combines any two elements to form a third element, satisfying closure, associativity, identity, and invertibility.
Ring-A set equipped with two binary operations, addition and multiplication, satisfying properties analogous to those of integers.
Field-A set on which addition, subtraction, multiplication, and division are defined and satisfy the usual arithmetic properties.
Vector Space-A set of vectors where vector addition and scalar multiplication are defined and satisfy certain axioms.
Eigenvalue-A scalar associated with a linear transformation that stretches or shrinks vectors without changing their direction.
Eigenvector-A non-zero vector that changes by only a scalar factor under a linear transformation.
Manifold-A topological space that locally resembles Euclidean space and allows for calculus to be performed.
Metric Space-A set where a distance (metric) is defined between elements.
Norm-A function that assigns a non-negative length or size to vectors in a vector space.
Inner Product-A generalization of the dot product that allows definition of angles and lengths in vector spaces.
Orthogonal-Two vectors are orthogonal if their inner product is zero, meaning they are perpendicular in some sense.
Basis-A set of linearly independent vectors that span the entire vector space.
Dimension-The number of vectors in a basis for a vector space.
Linear Transformation-A mapping between vector spaces that preserves vector addition and scalar multiplication.
Isomorphism-A bijective mapping between structures that preserves their operations and relations.
Homomorphism-A mapping between algebraic structures that preserves operations.
Kernel-The set of elements that map to the identity element under a given mapping.
Image-The set of output values produced by a function or mapping.
Surjective-A function that maps its domain onto its entire codomain.
Injective-A function where each element of the codomain is mapped by at most one element of the domain.
Bijective-A function that is both injective and surjective, establishing a one-to-one correspondence.
Differentiable-A function is differentiable at a point if its derivative exists at that point.
Integral-The accumulation of quantities, often representing areas under curves.
Gradient-A vector consisting of partial derivatives, pointing in the direction of greatest increase of a function.
Hessian-A square matrix of second-order partial derivatives of a scalar-valued function.
Jacobian-A matrix of first-order partial derivatives describing the rate of change of a vector-valued function.
Laplace Transform-An integral transform that converts a function of time into a function of complex frequency.
Fourier Transform-A mathematical transform that decomposes functions into oscillatory components.
Convolution-An operation on two functions producing a third function expressing how the shape of one is modified by the other.
Autocorrelation-A measure of the similarity of a signal with a delayed version of itself.
Characteristic Polynomial-A polynomial derived from a matrix, used to find eigenvalues.
Determinant-A scalar value that can be computed from the elements of a square matrix, providing information about the matrix’s invertibility.
Trace-The sum of the diagonal elements of a square matrix.
Rank-The maximum number of linearly independent rows or columns in a matrix.
Nullity-The dimension of the kernel of a matrix or linear transformation.
Singular Value Decomposition-A factorization of a matrix into three specific matrices, used in signal processing and statistics.
Principal Component Analysis-A technique for reducing the dimensionality of data by transforming to new variables that maximize variance.
Markov Chain-A stochastic process with the property that the future state depends only on the current state.
Probability Density Function-A function that describes the relative likelihood of a continuous random variable taking on a given value.
Cumulative Distribution Function-A function giving the probability that a random variable is less than or equal to a given value.
Moment-A quantitative measure related to the shape of a function's graph, often used in statistics.
Variance-A measure of the spread of a set of values around their mean.
Covariance-A measure of how much two random variables change together.
Correlation Coefficient-A standardized measure of the strength and direction of association between two variables.
Random Variable-A variable representing a numerical outcome of a random phenomenon.
Expectation-The weighted average of all possible values of a random variable.
Law of Large Numbers-A theorem stating that the sample average converges to the expected value as the sample size grows.
Central Limit Theorem-A fundamental theorem stating that the sum of many independent random variables tends toward a normal distribution.
Hypothesis Test-A statistical method for testing a proposed hypothesis about a population parameter.
Confidence Interval-A range of values, derived from sample data, that is likely to contain a population parameter.
Bayes' Theorem-A formula that describes the probability of an event based on prior knowledge of related conditions.
Likelihood-A function of parameters given observed data, used in statistical inference.
Maximum Likelihood Estimation-A method for estimating parameters by maximizing the likelihood function.
Regression-A statistical technique for modeling the relationship between variables.
Least Squares Method-A standard approach to minimizing the sum of the squares of the differences between observed and estimated values.
Gradient Descent-An optimization algorithm for finding the minimum of a function by iteratively moving in the direction of steepest descent.
Convex Function-A function whose domain forms a convex set and whose graph lies below the line segment connecting any two points on the graph.
Optimization-The process of finding the best solution to a problem within a given set of constraints.
Lagrange Multiplier-A strategy for finding the local maxima and minima of a function subject to equality constraints.
Duality-A concept where optimization problems can be expressed in a dual form, often providing deeper insight or alternative solutions.
Linear Programming-An optimization technique for maximizing or minimizing a linear objective function subject to linear constraints.
Integer Programming-An optimization technique where some or all variables are constrained to be integers.
Combinatorics-The branch of mathematics dealing with counting, arrangement, and combination of objects.
Graph Theory-The study of graphs, mathematical structures used to model pairwise relations between objects.
Tree-A connected acyclic graph with no cycles.
Spanning Tree-A subgraph that connects all vertices of a graph without forming any cycles.
Eulerian Path-A trail in a graph that visits every edge exactly once.
Hamiltonian Path-A path in a graph that visits each vertex exactly once.
Adjacency Matrix-A matrix representing the connections between vertices in a graph.
Incidence Matrix-A matrix that shows the relationship between vertices and edges in a graph.
Planar Graph-A graph that can be drawn on a plane without edges crossing.
Chromatic Number-The minimum number of colors needed to color a graph so that no two adjacent vertices share the same color.
Topology-The study of properties of space that are preserved under continuous deformations.
Homeomorphism-A continuous deformation of one topological space into another that preserves topological properties.
Compactness-A property indicating that a space is limited in extent and closed under limit operations.
Connectedness-A property indicating that a space is in one piece and cannot be separated into disjoint parts.
Fundamental Group-An algebraic structure that describes the basic shape, or holes, of a topological space.
Covering Space-A topological space that maps onto another space in a locally homeomorphic way.
Homology-A method of associating a sequence of abelian groups or modules with a topological space.
Cohomology-A mathematical tool for studying topological spaces that refines homology.
Differential Geometry-The study of geometry using calculus and differential equations.
Geodesic-The shortest path between two points in a curved space.
Riemannian Metric-A way of measuring lengths and angles on a curved surface.
Curvature-A measure of how much a geometric object deviates from being flat.
Differential Equation-An equation involving derivatives of a function.
Partial Differential Equation-A differential equation involving partial derivatives of multivariable functions.
Ordinary Differential Equation-A differential equation involving functions of a single variable and their derivatives.
Boundary Value Problem-A differential equation together with a set of additional constraints called boundary conditions.
Stochastic Process-A collection of random variables representing a process evolving over time.
Brownian Motion-A continuous-time stochastic process with applications in physics and finance.
Martingale-A stochastic process with the property that its expected future value, given all past information, equals its present value.
Ergodic Theory-The study of dynamical systems with an invariant measure and related problems of long-term average behavior.
Chaos Theory-The study of systems that exhibit sensitive dependence on initial conditions.
Fractal-A complex geometric shape that can be split into parts, each of which is a reduced-scale copy of the whole.
Number Theory-The branch of mathematics devoted to the study of integers and integer-valued functions.
Prime Number-A natural number greater than 1 that has no positive divisors other than 1 and itself.
Greatest Common Divisor-The largest integer that divides two numbers without leaving a remainder.
Modular Arithmetic-A system of arithmetic for integers where numbers wrap around upon reaching a certain value, called the modulus.
Diophantine Equation-A polynomial equation for which integer solutions are sought.
Cryptography-The practice and study of techniques for securing communication and information.
Elliptic Curve-A smooth, projective, algebraic curve used in number theory and cryptography.
Fourier Series-A way to represent a function as a sum of periodic components and to recover the signal from those components.
Wavelet Transform-A mathematical transform that decomposes a function into wavelets, useful for analyzing localized variations of power.
Zeta Function-A special function of great significance in number theory and mathematical analysis, particularly the Riemann zeta function.
Gamma Function-A function that extends the factorial function to complex and real number arguments.
Addition-The process of finding the total or sum by combining two or more numbers.
Subtraction-The process of finding the difference between two numbers.
Multiplication-A mathematical operation to find the total of a number added to itself repeatedly.
Division-The process of splitting a number into equal parts.
Number-A basic mathematical unit used for counting, measuring, and labeling.
Digit-Any of the ten symbols from 0 to 9 used to write numbers.
Even Number-A whole number that is divisible by 2.
Odd Number-A whole number that is not divisible by 2.
Prime Number-A whole number greater than 1 that has exactly two factors, 1 and itself.
Composite Number-A whole number greater than 1 that has more than two factors.
Factor-A number that divides another number exactly.
Multiple-The product of a number and any whole number.
Square Number-A number that is the product of a number multiplied by itself.
Cube Number-A number that is the product of a number multiplied by itself twice.
Square Root-A value that, when multiplied by itself, gives the original number.
Cube Root-A value that, when multiplied by itself twice, gives the original number.
Fraction-A way to represent a part of a whole or a division of quantities.
Numerator-The top number in a fraction representing how many parts are considered.
Denominator-The bottom number in a fraction showing the total number of parts.
Decimal-A number that includes a decimal point to represent parts of a whole.
Percentage-A way of expressing a number as a fraction of 100.
Ratio-A comparison of two quantities by division.
Proportion-An equation stating that two ratios are equal.
Variable-A symbol used to represent an unknown value.
Expression-A combination of numbers, variables, and operations without an equals sign.
Equation-A mathematical statement that two expressions are equal.
Inequality-A mathematical statement comparing two expressions using <, >, ≤, or ≥.
Solution-The value or set of values that make an equation or inequality true.
Estimate-To find a number that is close to an exact amount.
Rounding-Reducing a number to a nearby, simpler value.
Absolute Value-The distance of a number from zero on the number line.
Mean-The sum of values divided by the number of values; also called average.
Median-The middle value in a set of numbers ordered from least to greatest.
Mode-The number that appears most frequently in a data set.
Range-The difference between the highest and lowest values in a set.
Data-A collection of facts, numbers, or measurements.
Graph-A diagram that shows the relationship between sets of data.
Bar Graph-A graph using bars to represent data quantities.
Line Graph-A graph that uses lines to show changes over time.
Pie Chart-A circular graph divided into slices to show parts of a whole.
Histogram-A graph showing the frequency of data within equal intervals.
Coordinate Plane-A two-dimensional surface formed by a horizontal and a vertical number line.
X-Axis-The horizontal line on a coordinate plane.
Y-Axis-The vertical line on a coordinate plane.
Origin-The point where the x-axis and y-axis intersect on a coordinate plane, represented by (0,0).
Point-A location in space with no size, represented by a dot.
Line-A straight one-dimensional figure extending infinitely in both directions.
Line Segment-A part of a line with two endpoints.
Ray-A part of a line that starts at a point and extends infinitely in one direction.
Angle-The space between two intersecting lines or rays, measured in degrees.
Right Angle-An angle that measures exactly 90 degrees.
Acute Angle-An angle that measures less than 90 degrees.
Obtuse Angle-An angle that measures more than 90 degrees but less than 180 degrees.
Straight Angle-An angle that measures exactly 180 degrees.
Triangle-A polygon with three sides and three angles.
Equilateral Triangle-A triangle with all sides and angles equal.
Isosceles Triangle-A triangle with two sides of equal length.
Scalene Triangle-A triangle with all sides of different lengths.
Quadrilateral-A polygon with four sides and four angles.
Rectangle-A quadrilateral with opposite sides equal and all angles 90 degrees.
Square-A rectangle with all sides of equal length.
Parallelogram-A quadrilateral with opposite sides parallel and equal in length.
Rhombus-A parallelogram with all sides of equal length.
Trapezoid-A quadrilateral with at least one pair of parallel sides.
Polygon-A closed two-dimensional figure with straight sides.
Circle-A set of points that are all the same distance from a central point.
Radius-The distance from the center of a circle to any point on its edge.
Diameter-A line segment that passes through the center of a circle and connects two points on its edge.
Circumference-The distance around the edge of a circle.
Area-The measure of the space inside a two-dimensional figure.
Perimeter-The distance around the boundary of a two-dimensional figure.
Volume-The amount of space occupied by a three-dimensional object.
Surface Area-The total area of the surface of a three-dimensional object.
Cube-A three-dimensional shape with six equal square faces.
Rectangular Prism-A three-dimensional shape with six rectangular faces.
Sphere-A three-dimensional shape where all points are equidistant from the center.
Cylinder-A three-dimensional shape with two parallel circular bases connected by a curved surface.
Cone-A three-dimensional shape with a circular base and a point called the apex.
Pyramid-A three-dimensional shape with a polygonal base and triangular faces meeting at a point.
Probability-The measure of how likely an event is to occur.
Experiment-A procedure to observe and record outcomes of a random process.
Outcome-A possible result of an experiment.
Event-A set of outcomes in an experiment.
Sample Space-The set of all possible outcomes of an experiment.
Independent Events-Events where the outcome of one does not affect the outcome of another.
Dependent Events-Events where the outcome of one affects the outcome of another.
Permutation-An arrangement of items in a specific order.
Combination-A selection of items without regard to order.
Function-A relation where each input has exactly one output.
Input-The value that is put into a function.
Output-The result from applying a function to an input.
Slope-The measure of the steepness of a line.
Intercept-The point where a line crosses the x-axis or y-axis.
Pattern-A repeated or regular arrangement of numbers, shapes, or other elements.
Sequence-An ordered list of numbers following a specific pattern or rule.
Term-A single number, variable, or product in an expression or sequence.
Operation-A mathematical process such as addition, subtraction, multiplication, or division.
Property-A characteristic that applies to numbers or operations, like the commutative or associative property.
Order of Operations-The rules that define the correct sequence to evaluate expressions.
Unit-A standard quantity used for measurement.
Measurement-The process of determining the size, length, or amount of something.
Time-A measurement of the ongoing sequence of events.
Money-A system used to measure the value of goods and services.
Temperature-A measure of how hot or cold something is.
Weight-A measure of the force of gravity on an object.
Length-A measure of distance from one point to another.
Capacity-The amount a container can hold.
Speed-The rate of movement or progress.
Associative Property-A property stating that the grouping of numbers does not change their sum or product.
Commutative Property-A property stating that the order of numbers does not change their sum or product.
Distributive Property-A property showing how multiplication affects addition or subtraction inside parentheses.
Identity Property-A property where adding zero or multiplying by one does not change a number.
Inverse Property-A property that uses opposites to cancel out a number.
Zero Property of Multiplication-A property stating that any number multiplied by zero equals zero.
Algebra-A branch of mathematics dealing with symbols and rules for manipulating those symbols.
Arithmetic-The branch of mathematics that deals with basic operations like addition, subtraction, multiplication, and division.
Numeral-A symbol or group of symbols representing a number.
Ordinal Number-A number that shows position or order in a list.
Cardinal Number-A number that shows quantity or "how many".
Whole Number-A non-negative integer including zero.
Natural Number-A positive integer starting from 1 and continuing indefinitely.
Integer-A whole number that can be positive, negative, or zero.
Rational Number-A number that can be written as a fraction or a ratio of two integers.
Irrational Number-A number that cannot be expressed as a fraction and has an infinite non-repeating decimal part.
Real Number-Any number that can be found on the number line, including rational and irrational numbers.
Imaginary Number-A number that involves the square root of a negative number.
Complex Number-A number that has both a real and an imaginary part.
Exponent-A number that shows how many times the base is multiplied by itself.
Power-The result of a base raised to an exponent.
Base-The number that is multiplied by itself in repeated multiplication.
Scientific Notation-A method of writing very large or very small numbers using powers of ten.
Order of Magnitude-A way of expressing numbers on a logarithmic scale, often in powers of ten.
Prime Factorization-Expressing a number as a product of its prime factors.
Greatest Common Factor-The largest number that divides two or more numbers without a remainder.
Least Common Multiple-The smallest number that is a multiple of two or more numbers.
Divisibility-The ability of one number to be divided by another without leaving a remainder.
Remainder-The amount left over after division.
Quotient-The result of division.
Dividend-The number being divided in division.
Divisor-The number that divides another number.
Scaling-The process of increasing or decreasing numbers by multiplying or dividing.
Proportional Relationship-A relationship where two quantities increase or decrease at the same rate.
Direct Variation-A relationship between two variables where one is a constant multiple of the other.
Inverse Variation-A relationship where one variable increases as the other decreases.
Percent Increase-The percent by which a number grows relative to its original value.
Percent Decrease-The percent by which a number reduces relative to its original value.
Simple Interest-Interest calculated only on the original amount invested or loaned.
Compound Interest-Interest calculated on both the initial principal and the accumulated interest.
Savings-Money set aside for future use.
Debt-Money that is owed to others.
Budget-A plan for managing income and expenses.
Income-Money received, especially on a regular basis, for work or through investments.
Expense-Money spent on goods and services.
Tax-A compulsory financial charge imposed by a government.
Tip-An extra amount of money given for service.
Discount-A reduction in price.
Markup-The amount added to the cost of goods to cover expenses and profit.
Balance-The amount of money available in a financial account.
Deposit-Money placed into a financial account.
Withdrawal-Money taken out of a financial account.
Transaction-Any financial activity involving money.
Currency-The system of money used in a country.
Exchange Rate-The value of one currency compared to another.
Inequality Symbols-Symbols like <, >, ≤, and ≥ used to compare numbers.
Interval-A range of numbers between two endpoints.
Domain-The set of all possible input values of a function.
Range (Function)-The set of all possible output values of a function.
Continuous Function-A function with no breaks, jumps, or holes in its graph.
Discrete Function-A function with distinct and separate values.
Linear Equation-An equation whose graph is a straight line.
Quadratic Equation-An equation where the highest degree of the variable is two.
Polynomial-An expression consisting of variables, coefficients, and exponents combined using addition, subtraction, and multiplication.
Binomial-A polynomial with two terms.
Trinomial-A polynomial with three terms.
Degree of a Polynomial-The highest exponent of the variable in a polynomial.
Root of an Equation-A solution that makes the equation equal to zero.
Axis of Symmetry-A line that divides a figure or graph into two identical halves.
Vertex (Graph)-The highest or lowest point on a graph of a quadratic function.
Slope-Intercept Form-A way of writing the equation of a line as y = mx + b.
Standard Form (Equation)-A way of writing an equation where terms are ordered by degree.
Factoring-Rewriting an expression as a product of its factors.
Completing the Square-A method of solving quadratic equations by creating a perfect square trinomial.
Quadratic Formula-A formula used to find the solutions of a quadratic equation.
Inequality Graph-A graph showing the solutions of an inequality.
Function Notation-A way to write functions using symbols like f(x).
Transformation (Graph)-A change to a graph’s position, shape, or size.
Reflection-A flip of a graph over an axis.
Translation (Graph)-A shift of a graph horizontally or vertically.
Dilation-A stretching or shrinking of a graph.
Rotation-Turning a figure around a point by a certain angle.
Congruent Figures-Figures that have the same shape and size.
Similar Figures-Figures that have the same shape but may differ in size.
Scale Factor-A number used to multiply the dimensions of a figure to create a similar figure.
Pythagorean Theorem-A formula relating the sides of a right triangle: a² + b² = c².
Distance Formula-A formula for finding the distance between two points on a coordinate plane.
Midpoint Formula-A formula to find the point exactly halfway between two points.
Parallel Lines-Lines in a plane that never intersect.
Perpendicular Lines-Lines that intersect at a right angle.
Tessellation-A pattern of shapes that fit perfectly together without gaps or overlaps.
Symmetry-A balanced arrangement of equivalent parts on opposite sides of a line or point.
Rotational Symmetry-A figure that looks the same after some rotation.
Line of Symmetry-A line that divides a figure into two matching parts.
Fractal-A geometric figure that can be split into parts, each of which is a smaller copy of the whole.
Algorithm-A step-by-step procedure used for solving problems.
Statistics-The study of collecting, analyzing, and interpreting data.
Probability Model-A mathematical representation of a random process.
Random Sample-A subset of a population chosen without bias.
Survey-A method of gathering information from a group of people.
Experiment (Statistics)-A procedure used to test a hypothesis by collecting data.
Simulation-An experiment that imitates a real-world process using models or computers.
Theoretical Probability-The probability based on reasoning or calculations rather than experiments.
Experimental Probability-The probability based on the results of actual experiments.
Venn Diagram-A diagram showing relationships among sets using overlapping circles.
Set-A collection of distinct objects or numbers.
Subset-A set containing some or all elements of another set.
Universal Set-The set containing all elements under consideration in a given context.
Union (Set)-The set of all elements that are in either of two sets.
Intersection (Set)-The set of elements that are common to two sets.
Complement (Set)-The set of elements not in a given set but in the universal set.
Supervised Learning-A machine learning technique where the model learns from labeled data.
Unsupervised Learning-A machine learning technique where the model finds patterns in unlabeled data.
Semi-Supervised Learning-A learning method using a small amount of labeled data with a large amount of unlabeled data.
Reinforcement Learning-A learning technique where an agent learns by interacting with an environment and receiving rewards or penalties.
Classification-A task of predicting discrete class labels.
Regression-A task of predicting continuous numeric values.
Clustering-A technique to group similar data points without labeled responses.
Overfitting-A modeling error where a model learns noise instead of the underlying pattern.
Underfitting-A modeling error where the model is too simple to capture data patterns.
Bias-An error due to incorrect assumptions in the learning algorithm.
Variance-An error due to excessive sensitivity to training data fluctuations.
Generalization-The ability of a model to perform well on unseen data.
Cross-Validation-A technique to assess how the results of a model will generalize to an independent dataset.
K-Fold Cross-Validation-A method of cross-validation that splits the data into k subsets and trains k models.
Train-Test Split-A method for evaluating a model by dividing data into training and testing sets.
Feature-An individual measurable property or characteristic of a phenomenon being observed.
Feature Engineering-The process of creating new input features from existing data.
Feature Selection-The process of selecting the most relevant features for use in model training.
Normalization-Scaling data to have zero mean and unit variance or within a specific range.
Standardization-Scaling features to have a mean of zero and a standard deviation of one.
Dimensionality Reduction-The process of reducing the number of features in a dataset.
Principal Component Analysis (PCA)-A dimensionality reduction technique using orthogonal transformation.
Singular Value Decomposition (SVD)-A matrix factorization technique used in dimensionality reduction.
Autoencoder-A neural network used for unsupervised feature learning and dimensionality reduction.
Nearest Neighbor-A method that predicts based on the closest training data points.
K-Nearest Neighbors (KNN)-A classification method using the k closest data points for prediction.
Linear Regression-A regression algorithm modeling the relationship between variables linearly.
Logistic Regression-A classification algorithm for binary or multiclass classification.
Decision Tree-A tree-based algorithm for classification and regression tasks.
Random Forest-An ensemble learning method combining multiple decision trees.
Gradient Boosting-An ensemble method that builds models sequentially to correct previous errors.
XGBoost-An optimized implementation of gradient boosting for speed and performance.
LightGBM-A gradient boosting framework designed for speed and efficiency on large datasets.
CatBoost-A gradient boosting library optimized for categorical features.
Support Vector Machine (SVM)-A supervised learning model for classification and regression using hyperplanes.
Kernel Trick-A technique used in SVMs to enable nonlinear classification using kernel functions.
Naive Bayes-A probabilistic classifier based on Bayes' theorem with the assumption of feature independence.
Gaussian Naive Bayes-A Naive Bayes classifier assuming Gaussian-distributed features.
Bernoulli Naive Bayes-A Naive Bayes classifier for binary/boolean features.
Multinomial Naive Bayes-A Naive Bayes classifier for count data or discrete features.
Neural Network-A computational model inspired by biological neural networks for learning tasks.
Deep Learning-A subset of machine learning involving deep neural networks with multiple layers.
Artificial Neural Network (ANN)-A neural network consisting of input, hidden, and output layers.
Convolutional Neural Network (CNN)-A deep learning model specialized in processing grid-like data such as images.
Recurrent Neural Network (RNN)-A type of neural network suited for sequential data.
Long Short-Term Memory (LSTM)-A type of RNN designed to learn long-term dependencies.
Gated Recurrent Unit (GRU)-A simplified version of LSTM for learning sequences.
Transformer-A deep learning architecture using self-attention mechanisms for sequential tasks.
Attention Mechanism-A technique that allows models to focus on specific parts of the input sequence.
Self-Attention-A method where a sequence pays attention to different parts of itself.
Multi-Head Attention-A mechanism combining multiple attention layers for better learning.
Positional Encoding-A technique to inject sequence order information into transformer models.
Activation Function-A function that introduces non-linearity into a neural network.
ReLU (Rectified Linear Unit)-A widely used activation function that outputs zero for negative inputs.
Sigmoid-An activation function that maps values to the range (0,1).
Tanh-An activation function that maps values to the range (-1,1).
Softmax-An activation function that turns logits into probabilities for classification tasks.
Loss Function-A function that measures the difference between predicted and actual values.
Mean Squared Error (MSE)-A common loss function for regression tasks.
Cross-Entropy Loss-A loss function used for classification tasks.
Hinge Loss-A loss function used mainly for "maximum-margin" classifiers like SVMs.
Optimizer-An algorithm that updates the model parameters to minimize loss.
Stochastic Gradient Descent (SGD)-A basic optimization algorithm that updates model parameters based on gradients.
Momentum-An optimization technique that accelerates gradient descent by using past updates.
Adam-An optimization algorithm that combines momentum and adaptive learning rates.
Learning Rate-A hyperparameter that controls the step size during optimization.
Learning Rate Decay-A technique that reduces the learning rate over time to improve convergence.
Early Stopping-A method to stop training when the model performance on validation data stops improving.
Regularization-Techniques used to prevent overfitting by penalizing complex models.
L1 Regularization-A regularization technique that encourages sparsity by adding the absolute value of weights.
L2 Regularization-A regularization method that discourages large weights by adding the squared value of weights.
Dropout-A regularization technique where random neurons are ignored during training.
Batch Normalization-A technique to normalize the activations of each layer in a neural network.
Mini-Batch Gradient Descent-A method that splits the dataset into small batches for training.
Epoch-One complete pass through the entire training dataset.
Batch Size-The number of samples processed before the model updates its weights.
Hyperparameter Tuning-The process of finding the best set of hyperparameters for a model.
Grid Search-A method for hyperparameter tuning that exhaustively searches over a specified parameter grid.
Random Search-A method for hyperparameter tuning that selects random hyperparameter combinations.
Bayesian Optimization-A probabilistic method for optimizing hyperparameters efficiently.
Ensemble Learning-Combining multiple models to improve predictive performance.
Bagging-A technique that trains multiple models independently and averages their predictions.
Boosting-A technique that trains models sequentially, each correcting the previous one’s errors.
Stacking-An ensemble method that combines predictions from multiple models using another model.
Outlier-A data point that differs significantly from others in a dataset.
Anomaly Detection-The process of identifying abnormal or unusual patterns in data.
Label Encoding-A technique that converts categorical labels into numeric values.
One-Hot Encoding-A technique that converts categorical variables into binary vectors.
SMOTE (Synthetic Minority Over-sampling Technique)-A technique for handling imbalanced datasets by oversampling the minority class.
ROC Curve-A graphical plot showing the performance of a binary classifier.
Precision-The ratio of correctly predicted positive observations to the total predicted positives.
Recall-The ratio of correctly predicted positive observations to all actual positives.
F1 Score-The harmonic mean of precision and recall.
Confusion Matrix-A table showing true positives, false positives, true negatives, and false negatives.
AUC-ROC-A metric that summarizes the ROC curve performance into a single value.
Mean Absolute Error (MAE)-A regression metric that measures the average magnitude of errors.
R-squared-A statistical measure of how well a model explains the variance of the outcome.
Gradient Descent-An optimization algorithm that minimizes a loss function by updating parameters in the opposite direction of the gradient.
Derivative-The rate at which a function changes with respect to its variable.
Differentiation-The process of finding the derivative of a function.
Function-A relation where each input has exactly one output.
Limit-The value a function approaches as the input approaches some value.
Slope-The measure of steepness of a line, equivalent to the derivative for linear functions.
Rate of Change-The speed at which one quantity changes with respect to another.
Instantaneous Rate of Change-The derivative at a specific point, representing the slope of the tangent.
Secant Line-A line intersecting two points on a curve.
Tangent Line-A line that touches a curve at a point without crossing it, with slope equal to the derivative at that point.
Differentiable Function-A function that has a derivative at each point in its domain.
Non-Differentiable Function-A function that lacks a derivative at certain points or intervals.
Continuity-A property of a function without breaks or holes, necessary for differentiability.
Critical Point-A point where the derivative is zero or undefined, often linked to maxima or minima.
Stationary Point-A point where the derivative equals zero.
Local Maximum-A point where the function reaches a peak locally.
Local Minimum-A point where the function reaches a valley locally.
Global Maximum-The highest point of a function over its domain.
Global Minimum-The lowest point of a function over its domain.
Monotonic Function-A function that is either entirely non-increasing or non-decreasing.
Increasing Function-A function whose derivative is positive over an interval.
Decreasing Function-A function whose derivative is negative over an interval.
Concavity-The direction in which a curve bends; related to the second derivative.
Concave Up-A curve where the second derivative is positive.
Concave Down-A curve where the second derivative is negative.
Point of Inflection-A point where the concavity of the function changes.
First Derivative Test-A method to determine local extrema using the first derivative.
Second Derivative Test-A method to classify critical points using the second derivative.
Higher-Order Derivative-Derivatives of a function beyond the first derivative.
Chain Rule-A rule for finding the derivative of a composite function.
Product Rule-A rule for differentiating the product of two functions.
Quotient Rule-A rule for differentiating the quotient of two functions.
Power Rule-A basic rule for finding the derivative of a power function.
Implicit Differentiation-A technique for finding derivatives of implicit functions.
Partial Derivative-The derivative of a function with respect to one variable while keeping others constant.
Gradient-A vector containing all partial derivatives of a function.
Directional Derivative-The rate of change of a function in a specified direction.
Jacobian Matrix-A matrix of first-order partial derivatives of a vector-valued function.
Hessian Matrix-A square matrix of second-order partial derivatives, used in optimization.
Parametric Derivative-The derivative of parametric equations with respect to their parameter.
Logarithmic Differentiation-A technique to differentiate functions involving logarithms.
Derivative of Exponential Functions-The derivative of functions involving exponential terms.
Derivative of Logarithmic Functions-The derivative of functions involving logarithms.
Derivative of Trigonometric Functions-Derivatives of sine, cosine, tangent, etc.
Derivative of Inverse Trigonometric Functions-Derivatives of arcsin, arccos, arctan, etc.
Implicit Function Theorem-A theorem ensuring the existence of implicit derivatives under certain conditions.
Mean Value Theorem-A theorem that guarantees the existence of a point with the instantaneous rate of change equal to the average rate.
Rolle’s Theorem-A special case of the Mean Value Theorem with equal function values at endpoints.
L’Hôpital’s Rule-A method for evaluating limits involving indeterminate forms using derivatives.
Newton’s Method-A root-finding algorithm using derivatives to approximate solutions.
Taylor Series-An infinite sum of terms representing a function using its derivatives at a point.
Maclaurin Series-A Taylor series centered at zero.
Linear Approximation-An estimation of function values near a point using the tangent line.
Differentials-An approximation of changes in function values using derivatives.
Optimization-The process of finding maximum or minimum values using derivatives.
Related Rates-A method of finding the rate at which related quantities change.
Implicit Curve-A curve defined by an equation where y is not explicitly solved for x.
Parametric Curve-A curve defined by parametric equations involving a parameter.
Curvature-A measure of how rapidly a curve changes direction.
Arc Length-The distance along a curve, often computed using derivatives.
Velocity-The derivative of position with respect to time.
Acceleration-The derivative of velocity; second derivative of position.
Jerk-The derivative of acceleration; third derivative of position.
Snap-The fourth derivative of position with respect to time.
Pop-The fifth derivative of position with respect to time.
Crackle-The sixth derivative of position with respect to time.
Torsion-A measure of the twisting of a space curve, related to derivatives.
Sensitivity Analysis-The study of how changes in input affect output, often using derivatives.
Gradient Descent-An optimization method that uses derivatives to minimize functions.
Backpropagation-A method for training neural networks using derivatives to compute gradients.
Sobolev Space-A function space that considers both functions and their derivatives.
Differential Equation-An equation involving derivatives of a function.
Ordinary Differential Equation (ODE)-A differential equation involving functions of a single variable.
Partial Differential Equation (PDE)-A differential equation involving multiple variables.
Separable Differential Equation-A type of ODE that can be solved by separating variables.
Homogeneous Differential Equation-A differential equation where each term is proportional to the dependent variable or its derivatives.
Exact Differential Equation-A differential equation that can be solved by finding an exact differential.
Integrating Factor-A function used to simplify solving certain differential equations.
Bernoulli Differential Equation-A type of nonlinear differential equation.
Euler’s Method-A numerical technique for solving ODEs using derivatives.
Runge-Kutta Method-A higher-order numerical method for solving differential equations.
Stiff Equation-A differential equation requiring specialized methods for stable numerical solutions.
Jacobi’s Method-A numerical method using derivatives for solving systems of linear equations.
Fourier Series-A way to represent functions as sums of sines and cosines, linked to derivatives.
Fourier Transform-A mathematical transform linking derivatives and frequency domain analysis.
Laplace Transform-A technique to solve differential equations by transforming them into algebraic equations.
Green’s Function-A method for solving inhomogeneous differential equations using derivatives.
Boundary Conditions-Constraints involving derivatives at the boundaries of a domain.
Cauchy Problem-A type of differential equation problem with specified initial conditions.
Initial Value Problem-A problem where the solution to a differential equation must satisfy initial conditions.
Variational Calculus-The study of optimization involving functionals and their derivatives.
Euler-Lagrange Equation-A fundamental equation of variational calculus involving derivatives.
Hamiltonian Mechanics-A reformulation of classical mechanics using derivatives in phase space.
Gradient Vector Field-A field defined by the gradient (derivatives) of a scalar function.
Divergence-A scalar measure of the magnitude of a vector field's source or sink, involving derivatives.
Curl-A measure of the rotation of a vector field, involving derivatives.
Laplace Operator-A differential operator equal to the divergence of the gradient, used in PDEs.
Schwarz’s Theorem-A result stating that mixed partial derivatives are equal under certain conditions.
Taylor Polynomial-A finite sum approximation of a function using its derivatives.
Cusp-A point where a function has a sharp point with an undefined derivative.
Corner-A point where the derivative changes abruptly, making it non-differentiable.
Vertical Tangent-A point where the slope of the tangent line approaches infinity.
Osculating Circle-A circle that best approximates a curve near a point, involving derivatives.
Envelopes-Curves that are tangent to each member of a family of curves, defined by derivatives.
Calculus-A branch of mathematics focusing on change and motion.
Differential Calculus-The study of how functions change using derivatives.
Integral Calculus-The study of accumulation of quantities and areas under curves.
Derivative-The rate of change of a function with respect to a variable.
Integral-The accumulation of quantities, representing area under a curve.
Limit-The value that a function approaches as the input approaches a specific point.
Continuity-A property of a function where there are no breaks, jumps, or holes.
Differentiation-The process of finding the derivative of a function.
Antiderivative-A function whose derivative equals the original function.
Indefinite Integral-The general form of an antiderivative without limits of integration.
Definite Integral-The integral of a function over a specific interval.
Fundamental Theorem of Calculus-A theorem connecting differentiation and integration.
Rate of Change-The speed at which one quantity changes with respect to another.
Instantaneous Rate of Change-The derivative at a specific point in a function.
Slope-The measure of steepness of a line or curve.
Tangent Line-A line that touches a curve at one point with the same slope as the curve.
Secant Line-A line that intersects a curve at two points.
Critical Point-A point where the derivative is zero or undefined.
Maximum-The highest point of a function on an interval.
Minimum-The lowest point of a function on an interval.
Relative Maximum-A point higher than nearby points in a local region.
Relative Minimum-A point lower than nearby points in a local region.
Absolute Maximum-The highest point over the entire domain of a function.
Absolute Minimum-The lowest point over the entire domain of a function.
Inflection Point-A point where the concavity of a function changes.
Concavity-The direction in which a curve bends, determined by the second derivative.
Concave Up-A curve that opens upwards, with a positive second derivative.
Concave Down-A curve that opens downwards, with a negative second derivative.
Chain Rule-A rule for finding the derivative of composite functions.
Product Rule-A rule for differentiating the product of two functions.
Quotient Rule-A rule for differentiating the quotient of two functions.
Power Rule-A basic rule for differentiating power functions.
Implicit Differentiation-A method for differentiating functions not explicitly solved for one variable.
Partial Derivative-The derivative with respect to one variable in a multivariable function.
Gradient-A vector of partial derivatives representing the slope in multiple dimensions.
Directional Derivative-The rate of change of a function in a specific direction.
Higher-Order Derivative-Derivatives taken multiple times of a function.
Velocity-The derivative of position with respect to time.
Acceleration-The derivative of velocity with respect to time.
Jerk-The derivative of acceleration with respect to time.
Optimization-The process of finding maximum or minimum values of a function.
Critical Number-A value in the domain where the derivative is zero or undefined.
First Derivative Test-A test using the first derivative to determine relative extrema.
Second Derivative Test-A test using the second derivative to classify critical points.
L'Hôpital's Rule-A technique for evaluating limits of indeterminate forms.
Mean Value Theorem-A theorem stating that a function's derivative equals its average rate of change at some point.
Rolle’s Theorem-A special case of the Mean Value Theorem where endpoints have equal function values.
Newton's Method-An iterative technique to approximate roots using derivatives.
Related Rates-A method for finding rates of change in related variables.
Differential-A method to approximate changes in a function using derivatives.
Linear Approximation-An estimate of a function near a point using its tangent line.
Taylor Series-A series expansion of a function using its derivatives at a point.
Maclaurin Series-A Taylor series centered at zero.
Taylor Polynomial-A polynomial approximation of a function derived from its derivatives.
Arc Length-The length of a curve over a given interval.
Surface Area-The area of a surface generated by rotating a curve.
Volume of Revolution-The volume of a solid generated by rotating a region around an axis.
Disk Method-A method to compute the volume of a solid of revolution using disks.
Washer Method-A method to compute volume using hollow disks or washers.
Shell Method-A method for finding volume by summing cylindrical shells.
Substitution Rule-A method for evaluating integrals by substitution.
Integration by Parts-A technique for integrating products of functions.
Trigonometric Substitution-A method for evaluating integrals involving square roots using trig identities.
Partial Fraction Decomposition-A method for integrating rational functions by breaking them into simpler fractions.
Improper Integral-An integral with infinite limits or unbounded integrand.
Convergence-The property of an integral or series approaching a finite value.
Divergence-The property of an integral or series not approaching a finite value.
Parametric Equations-Equations expressing coordinates as functions of a parameter.
Polar Coordinates-A coordinate system using radius and angle instead of x and y.
Center of Mass-The weighted average position of mass in an object or region.
Moments-Quantities that measure the distribution of mass relative to an axis.
Work-The product of force and distance, often computed using integration.
Probability Density Function-A function representing probabilities over a continuous interval.
Double Integral-An integral over a two-dimensional region.
Triple Integral-An integral over a three-dimensional region.
Change of Variables-A technique for simplifying integrals by changing coordinates.
Jacobian-A determinant used in multivariable calculus for changing variables.
Green’s Theorem-A theorem relating a line integral around a closed curve to a double integral over the region it encloses.
Divergence Theorem-A theorem relating the flux of a vector field through a surface to a triple integral over a volume.
Stokes’ Theorem-A theorem relating a surface integral of a curl to a line integral around the boundary.
Vector Field-A function assigning a vector to each point in a space.
Line Integral-An integral over a curve, often representing work done by a force field.
Surface Integral-An integral over a surface, useful for calculating flux.
Flux-The amount of a field passing through a surface.
Curl-A measure of the rotation of a vector field.
Divergence-A measure of the outward flow of a vector field from a point.
Laplace Operator-A differential operator representing the divergence of the gradient.
Differential Equation-An equation involving derivatives of a function.
Separable Differential Equation-A differential equation that can be solved by separating variables.
Homogeneous Differential Equation-A differential equation where each term is a multiple of the dependent variable or its derivatives.
Linear Differential Equation-A differential equation where the dependent variable and its derivatives appear linearly.
Exact Differential Equation-A differential equation that can be solved by integrating an exact differential.
Integrating Factor-A function used to simplify certain differential equations.
Bernoulli Differential Equation-A type of nonlinear differential equation that can be solved using substitution.
Phase Line-A graphical tool for analyzing equilibrium points of differential equations.
Slope Field-A graphical representation of solutions to a differential equation.
Euler’s Method-A numerical method for approximating solutions of differential equations.
Runge-Kutta Method-A more accurate numerical method for solving differential equations.
Fourier Series-A representation of functions as sums of sines and cosines.
Fourier Transform-A mathematical technique for transforming functions into frequency space.
Laplace Transform-A technique for solving differential equations by transforming them into algebraic equations.