Human Formula — Interdisciplinary Scientific Formula Query and Interpretation

Algebra & Number Theory

root formula for a quadratic equation of one variable

x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

solves for the zeros of a quadratic polynomial; the discriminant determines the number of real roots.

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Algebra & Number Theory

Binomial Theorem

(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k

Expands binomial sums raised to integer powers into finite sums.

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Algebra & Number Theory

geometric sequence sum

S_n=a\frac{1-r^n}{1-r}

Calculate the sum of the first n terms whose leading term is a and whose common ratio is r.

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Algebra & Number Theory

Euler’s Theorem

a^{\varphi(n)}\equiv1\pmod n

An integer that is relatively prime to the modulus, and its power has a periodic structure in the modular sense.

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Calculus & Analysis

derivative definition

f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}

describes the instantaneous rate of change with the limit of the local difference quotient.

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Calculus & Analysis

Fundamental Theorem of Calculus

\int_a^b f(x)\,dx=F(b)-F(a),\quad F'=f

Calculates the definite integral using the boundary difference of the original function.

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Calculus & Analysis

chain rule The

\frac{d}{dx}f(g(x))=f'(g(x))g'(x)

composite function is equal to the product of the local change rates of each layer.

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Calculus & Analysis

Division points

\int_a^b u v'\,dx=[uv]_a^b-\int_a^b u'v\,dx

rewrites the product differential rule into an integral calculation method.

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Linear Algebra

eigenvalue equation

A\mathbf v=\lambda\mathbf v

finds nonzero vectors whose direction does not change in linear transformations.

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Linear Algebra

singular value decomposition

A=U\Sigma V^{*}

represents an arbitrary matrix with orthogonal directions and nonnegative stretch.

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Linear Algebra

linear least squares

\hat\beta=(X^{T}X)^{-1}X^{T}y

Find the coefficient that minimizes the sum of the squared errors of the predicted value and the observed value.

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Linear Algebra

Adjoint expression of matrix inverse

A^{-1}=\frac{\operatorname{adj}(A)}{\det(A)}

Use algebraic cofactors and determinants to express the inverse of a square matrix.

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Geometry & Topology

Pythagorean Theorem

a^2+b^2=c^2

The sum of the squares of the two right-angled sides of a right triangle is equal to the square of the hypotenuse.

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Geometry & Topology

volume of the ball

V=\frac43\pi r^3

Converts the radius of a sphere to a three-dimensional volume.

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Geometry & Topology

plane curve curvature

\kappa=\frac{\lvert x'y''-y'x''\rvert}{(x'^2+y'^2)^{3/2}}

quantifies the degree of local curvature of the parameter curve.

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Geometry & Topology

Polyhedral Euler formula

V-E+F=2

The number of vertices, edges and faces of a convex polyhedron satisfies a fixed relationship.

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Probability

Bayes’ Theorem

P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}

updates the probability of an event using observed evidence.

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Probability

discrete expectation

\mathbb E[X]=\sum_x x\,P(X=x)

averages possible values based on probability weighting. The

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Probability

Variance decomposition identity

\operatorname{Var}(X)=\mathbb E[X^2]-\mathbb E[X]^2

uses the second-order moment to measure the dispersion of random variables around the mean.

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Probability

Poisson distribution

P(X=k)=\frac{\lambda^k e^{-\lambda}}{k!}

describes the random event count in a fixed window.

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Statistics

unbiased sample variance

s^2=\frac1{n-1}\sum_{i=1}^n(x_i-\bar x)^2

estimates the population variance corrected with degrees of freedom.

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Statistics

t confidence interval for the mean

\bar x\pm t_{1-\alpha/2,n-1}\frac{s}{\sqrt n}

expresses the coverage of the population mean by a repeated sampling procedure.

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Statistics

Pearson correlation coefficient

r=\frac{\sum_i(x_i-\bar x)(y_i-\bar y)}{\sqrt{\sum_i(x_i-\bar x)^2\sum_i(y_i-\bar y)^2}}

measures the linear association between two sets of paired data.

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Statistics

chi-square goodness of fit statistic

\chi^2=\sum_i\frac{(O_i-E_i)^2}{E_i}

compares the observed frequency of a category with the model expected frequency.

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Human Formulas

root formula for a quadratic equation of one variable

Binomial Theorem

geometric sequence sum

Euler’s Theorem

derivative definition

Fundamental Theorem of Calculus

chain rule The

Division points

eigenvalue equation

singular value decomposition

linear least squares

Adjoint expression of matrix inverse

Pythagorean Theorem

volume of the ball

plane curve curvature

Polyhedral Euler formula

Bayes’ Theorem

discrete expectation

Variance decomposition identity

Poisson distribution

unbiased sample variance

t confidence interval for the mean

Pearson correlation coefficient

chi-square goodness of fit statistic

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