Operators
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Arithmetic Operators
- Addition (+)
- Adds two quantities together.
- Basic arithmetic, summing values.
- Adds two quantities together.
- Subtraction (-)
- Subtracts one quantity from another.
- Basic arithmetic, finding the difference between values.
- Subtracts one quantity from another.
- Multiplication (*)
- Multiplies two quantities together.
- Repeated addition, scaling, area calculation.
- Multiplies two quantities together.
- Division (/)
- Divides one quantity by another.
- Sharing equally, finding rates, calculating proportions.
- Divides one quantity by another.
- Exponentiation (^ or **)
- Raises a base to a power.
- Compound interest, growth/decay, geometric progression.
- Raises a base to a power.
- Modulus (%)
- Finds the remainder of a division operation.
- Determining divisibility, cyclic patterns, hashing algorithms.
- Finds the remainder of a division operation.
- Floor Division (//)
- Divides one quantity by another, rounding down to the nearest integer.
- Integer division, partitioning, array indexing in programming languages.
- Divides one quantity by another, rounding down to the nearest integer.
Relational Operators
- Equality (==)
- Checks if two quantities are equal.
- Comparison, conditional statements, database queries.
- Checks if two quantities are equal.
- Inequality (!=)
- Checks if two quantities are not equal.
- Comparison, conditional statements, filtering data.
- Checks if two quantities are not equal.
- Greater Than (>)
- Determines if one quantity is greater than another.
- Ranking, sorting, conditional statements.
- Determines if one quantity is greater than another.
- Less Than (<)
- Determines if one quantity is less than another.
- Ranking, sorting, conditional statements.
- Determines if one quantity is less than another.
- Greater Than or Equal To (>=)
- Determines if one quantity is greater than or equal to another.
- Ranking, sorting, conditional statements.
- Determines if one quantity is greater than or equal to another.
- Less Than or Equal To (<=)
- Determines if one quantity is less than or equal to another.
- Ranking, sorting, conditional statements.
- Determines if one quantity is less than or equal to another.
Logical Operators
- AND (&&)
- Returns true if both conditions are true.
- Conditional statements, filtering data.
- Returns true if both conditions are true.
- OR (||)
- Returns true if at least one condition is true.
- Conditional statements, filtering data.
- Returns true if at least one condition is true.
- NOT (!)
- Negates the value of a condition.
- Conditional statements, filtering data.
- Negates the value of a condition.
Set Theory Operators
- Union (∪)
- Combines two sets into one, containing all unique elements.
- Set operations, combining data sets.
- Combines two sets into one, containing all unique elements.
- Intersection (∩)
- Finds the common elements between two sets.
- Set operations, filtering data.
- Finds the common elements between two sets.
- Complement (')
- Contains all elements not in a given set.
- Set operations, filtering data.
- Contains all elements not in a given set.
- Subset (⊆)
- Checks if all elements of one set are in another set.
- Set operations, subset testing.
- Checks if all elements of one set are in another set.
- Superset (⊇)
- Checks if one set contains all elements of another set.
- Set operations, superset testing.
- Checks if one set contains all elements of another set.
- Empty Set (∅)
- A set containing no elements.
- Placeholder, indicating absence of data.
- A set containing no elements.
- Set Difference (-)
- Contains elements in one set but not in another.
- Set operations, data comparison.
- Contains elements in one set but not in another.
Vector and Matrix Operators
- Dot Product (· or ⋅)
- Multiplies corresponding components of two vectors and sums the results.
- Finding angles between vectors, projection calculations.
- Multiplies corresponding components of two vectors and sums the results.
- Cross Product (× or ⨯)
- Produces a vector perpendicular to two given vectors.
- Determining orientations, calculating torque.
- Produces a vector perpendicular to two given vectors.
- Matrix Multiplication (*)
- Multiplies corresponding elements of two matrices and sums the results.
- Transformations, solving systems of equations.
- Multiplies corresponding elements of two matrices and sums the results.
- Transpose (T or ')
- Swaps the rows and columns of a matrix.
- Matrix manipulation, solving linear equations.
- Swaps the rows and columns of a matrix.
- Determinant (det)
- Scalar value that can be computed from the elements of a square matrix.
- Volume scaling factor in linear transformations.
- Scalar value that can be computed from the elements of a square matrix.
- Inverse (⁻¹)
- Matrix that, when multiplied by the original matrix, results in the identity matrix.
- Solving systems of linear equations, finding inverses.
- Matrix that, when multiplied by the original matrix, results in the identity matrix.
- Trace (Tr)
- Sum of the elements on the main diagonal of a square matrix.
- Characteristic polynomial, calculating exponentials of matrices.
- Sum of the elements on the main diagonal of a square matrix.
- Adjoint (adj)
- Matrix obtained by taking the transpose of the cofactor matrix of a given square matrix.
- Finding inverses of matrices, solving systems of linear equations.
- Matrix obtained by taking the transpose of the cofactor matrix of a given square matrix.
- Kronecker Product (⊗)
- Produces a block matrix from two given matrices.
- Quantum mechanics, signal processing, and image processing.
- Produces a block matrix from two given matrices.
Calculus Operators
- Derivative (d/dx or ∂/∂x)
- Measures the rate at which a quantity changes with respect to another.
- Rate of change, optimization, slope of curves.
- Measures the rate at which a quantity changes with respect to another.
- Integral (∫)
- Computes the area under a curve.
- Accumulation of quantities, probability density functions.
- Computes the area under a curve.
- Gradient (grad or ∇)
- Vector that points in the direction of the greatest rate of increase of a scalar field.
- Optimization, direction of steepest ascent.
- Vector that points in the direction of the greatest rate of increase of a scalar field.
- Laplacian (∇²)
- Scalar operator that measures the divergence of the gradient of a scalar field.
- Heat flow, diffusion equations.
- Scalar operator that measures the divergence of the gradient of a scalar field.
- Divergence (div)
- Measures the extent to which a vector field diverges from or converges towards a point.
- Fluid flow, flux of a vector field.
- Measures the extent to which a vector field diverges from or converges towards a point.
- Curl (curl)
- Measures the rotation of a vector field.
- Fluid dynamics, electromagnetic fields.
- Measures the rotation of a vector field.
- Partial Differential (∂²/∂x²)
- Derivative with respect to multiple independent variables.
- Heat conduction, wave equations.
- Derivative with respect to multiple independent variables.
- Limit (lim)
- Describes the behavior of a function as its input approaches a certain value.
- Continuity, asymptotic behavior.
- Describes the behavior of a function as its input approaches a certain value.
- Taylor Series Expansion
- Representation of a function as an infinite sum of terms.
- Approximation of functions, numerical analysis.
- Representation of a function as an infinite sum of terms.
Trigonometric Operators
- Sine (sin)
- Ratio of the length of the side opposite an angle to the length of the hypotenuse.
- Modeling periodic phenomena, waveforms.
- Ratio of the length of the side opposite an angle to the length of the hypotenuse.
- Cosine (cos)
- Ratio of the length of the adjacent side to the length of the hypotenuse.
- Modeling periodic phenomena, oscillations.
- Ratio of the length of the adjacent side to the length of the hypotenuse.
- Tangent (tan)
- Ratio of the length of the side opposite an angle to the length of the side adjacent to the angle.
- Slope of lines, angular velocity.
- Ratio of the length of the side opposite an angle to the length of the side adjacent to the angle.
- Secant (sec)
- Reciprocal of the cosine function.
- Light rays, sound waves.
- Reciprocal of the cosine function.
- Cosecant (csc)
- Reciprocal of the sine function.
- Antennas, vibrations.
- Reciprocal of the sine function.
- Cotangent (cot)
- Reciprocal of the tangent function.
- Electrical circuits, mechanical systems.
- Reciprocal of the tangent function.
- Arcsine (arcsin)
- Inverse of the sine function.
- Solving trigonometric equations, calculating angles.
- Inverse of the sine function.
- Arccosine (arccos)
- Inverse of the cosine function.
- Solving trigonometric equations, calculating angles.
- Inverse of the cosine function.
- Arctangent (arctan)
- Inverse of the tangent function.
- Solving trigonometric equations, calculating angles.
- Inverse of the tangent function.
- Hyperbolic Sine (sinh)
- Analog of the trigonometric sine function for hyperbolic functions.
- Modeling exponential growth, catenary curves.
- Analog of the trigonometric sine function for hyperbolic functions.
- Hyperbolic Cosine (cosh)
- Analog of the trigonometric cosine function for hyperbolic functions.
- Modeling exponential growth, catenary curves.
- Analog of the trigonometric cosine function for hyperbolic functions.
- Hyperbolic Tangent (tanh)
- Analog of the trigonometric tangent function for hyperbolic functions.
- Modeling logistic growth, saturation phenomena.
- Analog of the trigonometric tangent function for hyperbolic functions.
- Inverse Hyperbolic Sine (arsinh)
- Inverse of the hyperbolic sine function.
- Solving hyperbolic equations, calculating areas.
- Inverse of the hyperbolic sine function.
- Inverse Hyperbolic Cosine (arcosh)
- Inverse of the hyperbolic cosine function.
- Solving hyperbolic equations, calculating areas.
- Inverse of the hyperbolic cosine function.
- Inverse Hyperbolic Tangent (artanh)
- Inverse of the hyperbolic tangent function.
- Solving hyperbolic equations, calculating areas.
- Inverse of the hyperbolic tangent function.
Probability and Statistics Operators
- Expected Value (E)
- Mean value of a random variable.
- Estimating outcomes, decision-making.
- Mean value of a random variable.
- Variance (Var)
- Measure of how spread out the values in a data set are around the mean.
- Assessing risk, quantifying variability.
- Measure of how spread out the values in a data set are around the mean.
- Standard Deviation (σ or SD)
- Square root of the variance.
- Measuring dispersion, error bars in plots.
- Square root of the variance.
- Probability (P)
- Measure of the likelihood of an event occurring.
- Assessing uncertainty, predicting outcomes.
- Measure of the likelihood of an event occurring.
- Conditional Probability (P(A|B))
- Probability of event A occurring given that event B has already occurred.
- Bayesian inference, probability trees.
- Probability of event A occurring given that event B has already occurred.
- Joint Probability (P(A ∩ B))
- Probability of both events A and B occurring simultaneously.
- Probability distribution functions, contingency tables.
- Probability of both events A and B occurring simultaneously.
- Marginal Probability (P(A))
- Probability of a single event occurring regardless of other events.
- Summing joint probabilities, independence testing.
- Probability of a single event occurring regardless of other events.
- Cumulative Distribution Function (CDF)
- Function that maps the probability of a random variable being less than or equal to a certain value.
- Probability distribution functions, hypothesis testing.
- Function that maps the probability of a random variable being less than or equal to a certain value.
- Probability Density Function (PDF)
- Function that describes the likelihood of a random variable taking on a particular value.
- Modeling continuous random variables, integration.
- Function that describes the likelihood of a random variable taking on a particular value.
- Covariance
- Measure of the joint variability of two random variables.
- Assessing correlation, portfolio management.
- Measure of the joint variability of two random variables.
- Correlation
- Measure of the strength and direction of the relationship between two random variables.
- Analyzing relationships, predictive modeling.
- Measure of the strength and direction of the relationship between two random variables.
Number Theory Operators
- Greatest Common Divisor (gcd)
- Largest positive integer that divides each of the integers.
- Simplifying fractions, reducing fractions to lowest terms.
- Largest positive integer that divides each of the integers.
- Least Common Multiple (lcm)
- Smallest positive integer that is divisible by each of the integers.
- Finding common multiples, adding fractions with unlike denominators.
- Smallest positive integer that is divisible by each of the integers.
- Prime Counting Function (π)
- Count of the number of prime numbers less than or equal to a given number.
- Analyzing distribution of primes, cryptography.
- Count of the number of prime numbers less than or equal to a given number.
- Euler's Totient Function (φ)
- Count of the positive integers less than n that are coprime to n.
- Cryptography, number theory algorithms.
- Count of the positive integers less than n that are coprime to n.
- Mobius Function (μ)
- Arithmetic function defined on the positive integers.
- Analyzing number theory functions, combinatorics.
- Arithmetic function defined on the positive integers.
Special Functions
- Gamma Function (Γ)
- Generalization of the factorial function to complex numbers.
- Analyzing divergent series, statistical distributions.
- Generalization of the factorial function to complex numbers.
- Beta Function (B)
- Function defined by an integral.
- Statistical distributions, probability density functions.
- Function defined by an integral.
- Riemann Zeta Function (ζ)
- Analytic function that generalizes the sum of the infinite series.
- Analyzing prime numbers, number theory.
- Analytic function that generalizes the sum of the infinite series.
- Bessel Functions (J, Y, I, K)
- Solutions to Bessel's differential equation.
- Modeling oscillations, heat conduction.
- Solutions to Bessel's differential equation.
- Legendre Polynomials (P)
- Orthogonal polynomials that arise in the solution of Laplace's equation.
- Quantum mechanics, celestial mechanics.
- Orthogonal polynomials that arise in the solution of Laplace's equation.
- Hermite Polynomials (H)
- Orthogonal polynomials that arise in the solution of the quantum harmonic oscillator.
- Quantum mechanics, statistical mechanics.
- Orthogonal polynomials that arise in the solution of the quantum harmonic oscillator.
- Laguerre Polynomials (L)
- Orthogonal polynomials that arise in the solution of the quantum hydrogen atom.
- Quantum mechanics, atomic physics.
- Orthogonal polynomials that arise in the solution of the quantum hydrogen atom.
Matrix Decomposition
- Singular Value Decomposition (SVD)
- Decomposes a matrix into singular vectors and singular values.
- Dimensionality reduction, image compression.
- Decomposes a matrix into singular vectors and singular values.
- Eigenvalue Decomposition
- Decomposes a matrix into eigenvectors and eigenvalues.
- Stability analysis, systems of ordinary differential equations.
- Decomposes a matrix into eigenvectors and eigenvalues.
- QR Decomposition
- Decomposes a matrix into an orthogonal matrix and an upper triangular matrix.
- Solving linear least squares problems, numerical stability.
- Decomposes a matrix into an orthogonal matrix and an upper triangular matrix.
- Cholesky Decomposition
- Decomposes a symmetric positive definite matrix into the product of a lower triangular matrix and its conjugate transpose.
- Solving linear systems, generating correlated random variables.
- Decomposes a symmetric positive definite matrix into the product of a lower triangular matrix and its conjugate transpose.
- LU Decomposition
- Decomposes a matrix into the product of a lower triangular matrix and an upper triangular matrix.
- Solving linear systems, matrix inversion.
- Decomposes a matrix into the product of a lower triangular matrix and an upper triangular matrix.
- Schur Decomposition
- Decomposes a matrix into a unitary matrix and an upper triangular matrix.
- Solving linear systems, stability analysis.
- Decomposes a matrix into a unitary matrix and an upper triangular matrix.
- Jordan Normal Form
- Canonical form of a matrix under a similarity transformation.
- Stability analysis, solving systems of linear differential equations.
- Canonical form of a matrix under a similarity transformation.
Other Mathematical Operators
- Summation (∑)
- Adds up a sequence of numbers.
- Finding totals, calculating areas under curves.
- Adds up a sequence of numbers.
- Product (∏)
- Multiplies a sequence of numbers.
- Finding products, calculating volumes of irregular shapes.
- Multiplies a sequence of numbers.
- Integral with Limits (∫ₐᵇ)
- Computes the area under a curve between two points.
- Finding areas, computing volumes, calculating work.
- Computes the area under a curve between two points.
- Partial Derivative (∂)
- Derivative of a function of several variables with respect to one variable, holding others constant.
- Multivariable calculus, optimization.
- Derivative of a function of several variables with respect to one variable, holding others constant.
- Infimum (inf)
- Greatest lower bound of a set.
- Optimization, analysis of functions.
- Greatest lower bound of a set.
- Supremum (sup)
- Least upper bound of a set.
- Optimization, analysis of functions.
- Least upper bound of a set.
- Absolute Value (|x|)
- Distance of a number from zero.
- Distance calculations, finding magnitudes.
- Distance of a number from zero.
- Factorial (!)
- Product of all positive integers less than or equal to a given number.
- Counting permutations, combinations.
- Product of all positive integers less than or equal to a given number.
- Round (round)
- Rounds a number to the nearest integer.
- Approximation, formatting numbers.
- Rounds a number to the nearest integer.
- Logarithm (log)
- Inverse operation to exponentiation.
- Scaling, orders of magnitude, solving exponential equations.
- Inverse operation to exponentiation.
- Greatest Integer Function (⌊x⌋)
- Greatest integer less than or equal to a given number.
- Floor function, truncating decimals.
- Greatest integer less than or equal to a given number.
- Ceiling Function (⌈x⌉)
- Smallest integer greater than or equal to a given number.
- Ceiling function, rounding up.
- Smallest integer greater than or equal to a given number.
- Signum Function (sgn)
- Returns the sign of a number.
- Determining direction, characterizing solutions.
- Returns the sign of a number.