Operators

Arithmetic Operators

• Addition (+)
  • Adds two quantities together.
    • Basic arithmetic, summing values.

• Subtraction (-)
  • Subtracts one quantity from another.
    • Basic arithmetic, finding the difference between values.

• Multiplication (*)
  • Multiplies two quantities together.
    • Repeated addition, scaling, area calculation.

• Division (/)
  • Divides one quantity by another.
    • Sharing equally, finding rates, calculating proportions.

• Exponentiation (^ or **)
  • Raises a base to a power.
    • Compound interest, growth/decay, geometric progression.

• Modulus (%)
  • Finds the remainder of a division operation.
    • Determining divisibility, cyclic patterns, hashing algorithms.

• Floor Division (//)
  • Divides one quantity by another, rounding down to the nearest integer.
    • Integer division, partitioning, array indexing in programming languages.

Relational Operators

• Equality (==)
  • Checks if two quantities are equal.
    • Comparison, conditional statements, database queries.

• Inequality (!=)
  • Checks if two quantities are not equal.
    • Comparison, conditional statements, filtering data.

• Greater Than (>)
  • Determines if one quantity is greater than another.
    • Ranking, sorting, conditional statements.

• Less Than (<)
  • Determines if one quantity is less than another.
    • Ranking, sorting, conditional statements.

• Greater Than or Equal To (>=)
  • Determines if one quantity is greater than or equal to another.
    • Ranking, sorting, conditional statements.

• Less Than or Equal To (<=)
  • Determines if one quantity is less than or equal to another.
    • Ranking, sorting, conditional statements.

Logical Operators

• AND (&&)
  • Returns true if both conditions are true.
    • Conditional statements, filtering data.

• OR (||)
  • Returns true if at least one condition is true.
    • Conditional statements, filtering data.

• NOT (!)
  • Negates the value of a condition.
    • Conditional statements, filtering data.

Set Theory Operators

• Union (∪)
  • Combines two sets into one, containing all unique elements.
    • Set operations, combining data sets.

• Intersection (∩)
  • Finds the common elements between two sets.
    • Set operations, filtering data.

• Complement (')
  • Contains all elements not in a given set.
    • Set operations, filtering data.

• Subset (⊆)
  • Checks if all elements of one set are in another set.
    • Set operations, subset testing.

• Superset (⊇)
  • Checks if one set contains all elements of another set.
    • Set operations, superset testing.

• Empty Set (∅)
  • A set containing no elements.
    • Placeholder, indicating absence of data.

• Set Difference (-)
  • Contains elements in one set but not in another.
    • Set operations, data comparison.

Vector and Matrix Operators

• Dot Product (· or ⋅)
  • Multiplies corresponding components of two vectors and sums the results.
    • Finding angles between vectors, projection calculations.

• Cross Product (× or ⨯)
  • Produces a vector perpendicular to two given vectors.
    • Determining orientations, calculating torque.

• Matrix Multiplication (*)
  • Multiplies corresponding elements of two matrices and sums the results.
    • Transformations, solving systems of equations.

• Transpose (T or ')
  • Swaps the rows and columns of a matrix.
    • Matrix manipulation, solving linear equations.

• Determinant (det)
  • Scalar value that can be computed from the elements of a square matrix.
    • Volume scaling factor in linear transformations.

• Inverse (⁻¹)
  • Matrix that, when multiplied by the original matrix, results in the identity matrix.
    • Solving systems of linear equations, finding inverses.

• Trace (Tr)
  • Sum of the elements on the main diagonal of a square matrix.
    • Characteristic polynomial, calculating exponentials of matrices.

• 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.

• Kronecker Product (⊗)
  • Produces a block matrix from two given matrices.
    • Quantum mechanics, signal processing, and image processing.

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.

• Integral (∫)
  • Computes the area under a curve.
    • Accumulation of quantities, probability density functions.

• Gradient (grad or ∇)
  • Vector that points in the direction of the greatest rate of increase of a scalar field.
    • Optimization, direction of steepest ascent.

• Laplacian (∇²)
  • Scalar operator that measures the divergence of the gradient of a scalar field.
    • Heat flow, diffusion equations.

• Divergence (div)
  • Measures the extent to which a vector field diverges from or converges towards a point.
    • Fluid flow, flux of a vector field.

• Curl (curl)
  • Measures the rotation of a vector field.
    • Fluid dynamics, electromagnetic fields.

• Partial Differential (∂²/∂x²)
  • Derivative with respect to multiple independent variables.
    • Heat conduction, wave equations.

• Limit (lim)
  • Describes the behavior of a function as its input approaches a certain value.
    • Continuity, asymptotic behavior.

• Taylor Series Expansion
  • Representation of a function as an infinite sum of terms.
    • Approximation of functions, numerical analysis.

Trigonometric Operators

• Sine (sin)
  • Ratio of the length of the side opposite an angle to the length of the hypotenuse.
    • Modeling periodic phenomena, waveforms.

• Cosine (cos)
  • Ratio of the length of the adjacent side to the length of the hypotenuse.
    • Modeling periodic phenomena, oscillations.

• 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.

• Secant (sec)
  • Reciprocal of the cosine function.
    • Light rays, sound waves.

• Cosecant (csc)
  • Reciprocal of the sine function.
    • Antennas, vibrations.

• Cotangent (cot)
  • Reciprocal of the tangent function.
    • Electrical circuits, mechanical systems.

• Arcsine (arcsin)
  • Inverse of the sine function.
    • Solving trigonometric equations, calculating angles.

• Arccosine (arccos)
  • Inverse of the cosine function.
    • Solving trigonometric equations, calculating angles.

• Arctangent (arctan)
  • Inverse of the tangent function.
    • Solving trigonometric equations, calculating angles.

• Hyperbolic Sine (sinh)
  • Analog of the trigonometric sine function for hyperbolic functions.
    • Modeling exponential growth, catenary curves.

• Hyperbolic Cosine (cosh)
  • Analog of the trigonometric cosine function for hyperbolic functions.
    • Modeling exponential growth, catenary curves.

• Hyperbolic Tangent (tanh)
  • Analog of the trigonometric tangent function for hyperbolic functions.
    • Modeling logistic growth, saturation phenomena.

• Inverse Hyperbolic Sine (arsinh)
  • Inverse of the hyperbolic sine function.
    • Solving hyperbolic equations, calculating areas.

• Inverse Hyperbolic Cosine (arcosh)
  • Inverse of the hyperbolic cosine function.
    • Solving hyperbolic equations, calculating areas.

• Inverse Hyperbolic Tangent (artanh)
  • Inverse of the hyperbolic tangent function.
    • Solving hyperbolic equations, calculating areas.

Probability and Statistics Operators

• Expected Value (E)
  • Mean value of a random variable.
    • Estimating outcomes, decision-making.

• Variance (Var)
  • Measure of how spread out the values in a data set are around the mean.
    • Assessing risk, quantifying variability.

• Standard Deviation (σ or SD)
  • Square root of the variance.
    • Measuring dispersion, error bars in plots.

• Probability (P)
  • Measure of the likelihood of an event occurring.
    • Assessing uncertainty, predicting outcomes.

• Conditional Probability (P(A|B))
  • Probability of event A occurring given that event B has already occurred.
    • Bayesian inference, probability trees.

• Joint Probability (P(A ∩ B))
  • Probability of both events A and B occurring simultaneously.
    • Probability distribution functions, contingency tables.

• Marginal Probability (P(A))
  • Probability of a single event occurring regardless of other events.
    • Summing joint probabilities, independence testing.

• 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.

• Probability Density Function (PDF)
  • Function that describes the likelihood of a random variable taking on a particular value.
    • Modeling continuous random variables, integration.

• Covariance
  • Measure of the joint variability of two random variables.
    • Assessing correlation, portfolio management.

• Correlation
  • Measure of the strength and direction of the relationship between two random variables.
    • Analyzing relationships, predictive modeling.

Number Theory Operators

• Greatest Common Divisor (gcd)
  • Largest positive integer that divides each of the integers.
    • Simplifying fractions, reducing fractions to lowest terms.

• Least Common Multiple (lcm)
  • Smallest positive integer that is divisible by each of the integers.
    • Finding common multiples, adding fractions with unlike denominators.

• Prime Counting Function (π)
  • Count of the number of prime numbers less than or equal to a given number.
    • Analyzing distribution of primes, cryptography.

• Euler's Totient Function (φ)
  • Count of the positive integers less than n that are coprime to n.
    • Cryptography, number theory algorithms.

• Mobius Function (μ)
  • Arithmetic function defined on the positive integers.
    • Analyzing number theory functions, combinatorics.

Special Functions

• Gamma Function (Γ)
  • Generalization of the factorial function to complex numbers.
    • Analyzing divergent series, statistical distributions.

• Beta Function (B)
  • Function defined by an integral.
    • Statistical distributions, probability density functions.

• Riemann Zeta Function (ζ)
  • Analytic function that generalizes the sum of the infinite series.
    • Analyzing prime numbers, number theory.

• Bessel Functions (J, Y, I, K)
  • Solutions to Bessel's differential equation.
    • Modeling oscillations, heat conduction.

• Legendre Polynomials (P)
  • Orthogonal polynomials that arise in the solution of Laplace's equation.
    • Quantum mechanics, celestial mechanics.

• Hermite Polynomials (H)
  • Orthogonal polynomials that arise in the solution of the quantum harmonic oscillator.
    • Quantum mechanics, statistical mechanics.

• Laguerre Polynomials (L)
  • Orthogonal polynomials that arise in the solution of the quantum hydrogen atom.
    • Quantum mechanics, atomic physics.

Matrix Decomposition

• Singular Value Decomposition (SVD)
  • Decomposes a matrix into singular vectors and singular values.
    • Dimensionality reduction, image compression.

• Eigenvalue Decomposition
  • Decomposes a matrix into eigenvectors and eigenvalues.
    • Stability analysis, systems of ordinary differential equations.

• QR Decomposition
  • Decomposes a matrix into an orthogonal matrix and an upper triangular matrix.
    • Solving linear least squares problems, numerical stability.

• 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.

• LU Decomposition
  • Decomposes a matrix into the product of a lower triangular matrix and an upper triangular matrix.
    • Solving linear systems, matrix inversion.

• Schur Decomposition
  • Decomposes a matrix into a unitary matrix and an upper triangular matrix.
    • Solving linear systems, stability analysis.

• Jordan Normal Form
  • Canonical form of a matrix under a similarity transformation.
    • Stability analysis, solving systems of linear differential equations.

Other Mathematical Operators

• Summation (∑)
  • Adds up a sequence of numbers.
    • Finding totals, calculating areas under curves.

• Product (∏)
  • Multiplies a sequence of numbers.
    • Finding products, calculating volumes of irregular shapes.

• Integral with Limits (∫ₐᵇ)
  • Computes the area under a curve between two points.
    • Finding areas, computing volumes, calculating work.

• Partial Derivative (∂)
  • Derivative of a function of several variables with respect to one variable, holding others constant.
    • Multivariable calculus, optimization.

• Infimum (inf)
  • Greatest lower bound of a set.
    • Optimization, analysis of functions.

• Supremum (sup)
  • Least upper bound of a set.
    • Optimization, analysis of functions.

• Absolute Value (|x|)
  • Distance of a number from zero.
    • Distance calculations, finding magnitudes.

• Factorial (!)
  • Product of all positive integers less than or equal to a given number.
    • Counting permutations, combinations.

• Round (round)
  • Rounds a number to the nearest integer.
    • Approximation, formatting numbers.

• Logarithm (log)
  • Inverse operation to exponentiation.
    • Scaling, orders of magnitude, solving exponential equations.

• Greatest Integer Function (⌊x⌋)
  • Greatest integer less than or equal to a given number.
    • Floor function, truncating decimals.

• Ceiling Function (⌈x⌉)
  • Smallest integer greater than or equal to a given number.
    • Ceiling function, rounding up.

• Signum Function (sgn)
  • Returns the sign of a number.
    • Determining direction, characterizing solutions.