math-foundations-ai999x — offline, verified math foundations solver with step-by-step answers

Install · Capability guide · Evaluation · AI999X on Hugging Face · Website · Support: Buy Me a Coffee

License: Apache-2.0 · Author: Ibnul Jaif Farabi (creator and engineer) · Organization: AI999X · Version 1.0.0

Field: Mathematics & Statistics · Tasks: Question Answering, Token Classification, Text Classification, Sentence Similarity, Feature Extraction

math-foundations-ai999x is an offline, deterministic Python solver that answers math foundations questions with verified results from 119 certified capabilities, gives 625 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

math-foundations-ai999x is a compact, deterministic math foundations engine for Python. It reads a question in plain English (or a typed call), picks a certified math foundations capability, computes the answer exactly, checks it, and returns the result with its steps, units, LaTeX and assumptions — or an honest refusal that says what is missing. It runs fully offline on any CPU, needs no GPU, no API key and no internet, and is built to give zero wrong answers: when it is not sure, it refuses instead of guessing.

Highlights

• Verified, not generated. Every answer is computed by a certified math foundations capability and checked (substituted back, dimension-checked or compared with an independent computation) before it is returned.
• Zero wrong answers by design. Seven explicit outcomes — SOLVED, NEEDS_INPUT, AMBIGUOUS, INVALID_INPUT, OUT_OF_SCOPE, UNRECOGNIZED_FORMAT, INTERNAL_ERROR — never a silent guess. Every one of its 625 documented answers is reproduced exactly on every tested platform.
• Plain-English questions and a guaranteed typed interface. ask() reads documented question forms; solve(capability_id, **inputs) always solves a supported task with valid inputs.
• Explains itself. Steps, formulas in LaTeX, units, assumptions and the model used, ready for notebooks, tutoring apps and homework checkers.
• Small and portable. Pure Python (sympy, mpmath); identical answers on Windows and Linux (Python 3.10, 3.11, 3.12, 3.13, 3.14).
• Coverage: arithmetic, algebra, functions, exponentials and logarithms, trigonometry, precalculus and complex numbers.

Model overview

Property	Value
Package	math-foundations-ai999x 1.0.0 (import math_foundations_ai999x)
Type	Deterministic neuro-symbolic math foundations engine (rule-based reading + certified symbolic capabilities)
Capabilities	19 certified derivations, 100 certified procedures
Knowledge pack	version 1.0.2
Golden answers	625 documented answers, identical on every tested platform
Platforms tested	Windows (Python 3.13), Linux x86-64 (Python 3.10, 3.11, 3.12, 3.13, 3.14)
Runtime	CPU only, offline; no GPU, no network, no API key
License	Apache-2.0
Creator / engineer	Ibnul Jaif Farabi
Organization	AI999X

Quickstart

pip install math-foundations-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/math-foundations-ai999x/resolve/main/dist/math_foundations_ai999x-1.0.0-py3-none-any.whl

import math_foundations_ai999x as math_foundations

math_foundations.ask("Solve x^2 - 5x + 6 = 0.").show()
# -> SOLVED — x1 = 2, x2 = 3, discriminant = 1  (quadratic_roots)
math_foundations.ask("In triangle ABC, a = 8, b = 10 and A = 40 degrees. Solve the triangle.").show()
# -> SOLVED — count = 2, B1 = 53.4641, C1 = 86.5359, c1 = 12.423  (law_of_sines_ssa)
math_foundations.ask("Compute (3 + 2i) / (1 - i).").show()
# -> SOLVED — re = 0.5, im = 2.5  (complex_arithmetic)
math_foundations.ask("Find the coefficient of x^3 in (2x - 1)^5.").show()
# -> SOLVED — coefficient = 80, binomial = 10  (binomial_coefficient_term)
math_foundations.solve("de_moivre_power", a=1, b=1, n=8).show()
# -> SOLVED — re = 16, im = 0, modulus = 16, argument_deg = 0  (de_moivre_power)
math_foundations.ask("Who created you?").message
# -> math-foundations-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

Command line: math-foundations-ai999x "<question>", math-foundations-ai999x --capabilities. The outputs above were produced by running this exact code on math-foundations-ai999x 1.0.0.

How every answer is verified

1. Reading — the question is turned into typed quantities (value, unit, kind, role) and one asked quantity; every stated number must be used or the question is refused.
2. Planning — only certified capabilities may answer; a capability applies only under the conditions the wording states (for example a pendulum, a liquid at rest, boiling).
3. Computing — exact symbolic or high-precision numeric evaluation.
4. Checking — dimensions, substitution back into the defining equation, and agreement between independent readings; any disagreement becomes a refusal, never a number.

Capabilities are certified before they ship: each must reproduce hand-worked reference cases, reject a deliberately wrong variant (a control), and leave every benchmark answer unchanged.

Capabilities

19 certified derivations, 100 certified procedures. A few of them:

• Derivations: area of rectangle; associative property of addition; cramer rule x solution; determinant cramer two variables; difference of cubes factorization; difference of squares; distributive property verification; elimination method identity; elimination solution two variables; linear equation solution formula; partial fraction decomposition linear factors; perfect square trinomial; product of sums; quadratic vertex form conversion; radical equation squaring; square of difference; substitution method identity; substitution solution two variables; sum of cubes factorization
• Procedures: Absolute value inequality |ax + b| compared with c; Add rational expressions with same denominator; Amount remaining after radioactive decay (half-life); Area of a triangle from its three sides (Heron); Arithmetic with complex numbers; Arithmetic with fractions; Asymptotes of a linear rational function; Average rate of change between two points; Average rate of change of a function between two inputs; Centre and radius of a circle from its general equation; Coefficient of a term in a binomial expansion; Combinations and permutations; Composition of two linear functions; Compound interest; Continuous exponential growth or decay; Degrees to radians; Determinant and inverse of a 2 by 2 matrix; Determinant of 2x2 matrix; Discriminant of a quadratic; Distance between two points; Distance from a point to a line; Divide a quantity in a given ratio; Domain of a square-root function; Dot product and angle between two vectors …

math_foundations_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 100 formulas

Show every formula, by topic

Matrices and linear systems

Determinant of 2x2 matrix

$$\text{det}=ad-bc\backslash text\{det\}\; =\; a\; d\; -\; b\; c$$

Matrix scalar multiple definition

$$b_{11}=s{a}_{11},b_{12}=s{a}_{12},b_{21}=s{a}_{21},b_{22}=s{a}_{22}b\_\{11\}\; =\; s\; a\_\{11\},\backslash quad\; b\_\{12\}\; =\; s\; a\_\{12\},\backslash quad\; b\_\{21\}\; =\; s\; a\_\{21\},\backslash quad\; b\_\{22\}\; =\; s\; a\_\{22\}$$

Rational expressions

Add rational expressions with same denominator

$$\text{sum num}={\text{num}}_1+{\text{num}}_2,\text{sum den}=\text{den}\backslash text\{sum\; num\}\; =\; \backslash text\{num\}\_\{1\}\; +\; \backslash text\{num\}\_\{2\},\backslash quad\; \backslash text\{sum\; den\}\; =\; \backslash text\{den\}$$

Multiply rational expressions

$$\text{prod num}={\text{num}}_1{\text{num}}_2,\text{prod den}={\text{den}}_1{\text{den}}_2\backslash text\{prod\; num\}\; =\; \backslash text\{num\}\_\{1\}\; \backslash text\{num\}\_\{2\},\backslash quad\; \backslash text\{prod\; den\}\; =\; \backslash text\{den\}\_\{1\}\; \backslash text\{den\}\_\{2\}$$

Algebra / absolute value

Absolute value inequality |ax + b| compared with c

Algebra / factoring

Factor a quadratic over the integers

$$a{x}^2+bx+c=g(px+q)(rx+s)ax^2+bx+c\; =\; g\backslash ,(px+q)(rx+s)$$

Algebra / inequalities

Quadratic inequality

Algebra / polynomial division

Polynomial long division

Synthetic division and the remainder theorem

$$p(x)=(x-r)q(x)+p(r)p(x)\; =\; (x\; -\; r)\backslash ,q(x)\; +\; p(r)$$

Algebra / polynomial equations

Rational roots of a polynomial (rational root theorem)

$$a_n{x}^n+\cdots +a_0=0,x=\frac{p}{q}\Rightarrow p\mid a_0,q\mid a_{n}a\_n\; x^n\; +\; \backslash dots\; +\; a\_0\; =\; 0,\backslash \; x\; =\; \backslash tfrac\{p\}\{q\}\; \backslash Rightarrow\; p\; \backslash mid\; a\_0,\backslash \; q\; \backslash mid\; a\_n$$

Algebra / radical equations

Radical equation sqrt(ax + b) = cx + d

$$\sqrt{ax+b}=cx+d\Rightarrow ax+b=(cx+d{)}^2,cx+d\ge 0\backslash sqrt\{ax+b\}\; =\; cx+d\; \backslash ;\backslash Rightarrow\backslash ;\; ax+b\; =\; (cx+d)^2,\backslash \; cx+d\; \backslash ge\; 0$$

Algebra / rational equations

Rational equation a/(x + b) = c/(x + d)

$$\frac{a}{x+b}=\frac{c}{x+d}\Rightarrow x=\frac{cb-ad}{a-c}\backslash frac\{a\}\{x+b\}\; =\; \backslash frac\{c\}\{x+d\}\; \backslash ;\backslash Rightarrow\backslash ;\; x\; =\; \backslash frac\{cb\; -\; ad\}\{a\; -\; c\}$$

Algebra / rational expressions

Partial fractions (two distinct linear factors)

$$\frac{px+q}{(x-r_1)(x-r_2)}=\frac{A}{x-r_1}+\frac{B}{x-r_2},A=\frac{p{r}_1+q}{r_1-r_2}\backslash frac\{px+q\}\{(x-r\_1)(x-r\_2)\}\; =\; \backslash frac\{A\}\{x-r\_1\}\; +\; \backslash frac\{B\}\{x-r\_2\},\backslash quad\; A\; =\; \backslash frac\{pr\_1+q\}\{r\_1-r\_2\}$$

Algebra / sequences and series

Sum of an infinite geometric series

Algebra / systems of linear equations

System of three linear equations

$$A\mathbf{x}=\mathbf{d},\mathbf{x}=A^{-1}\mathbf{d}A\backslash ,\backslash mathbf\{x\}\; =\; \backslash mathbf\{d\},\backslash quad\; \backslash mathbf\{x\}\; =\; A^\{-1\}\backslash mathbf\{d\}$$

Arithmetic / fractions

Arithmetic with fractions

$$\frac{a}{b}\pm \frac{c}{d}=\frac{ad\pm cb}{bd},\frac{a}{b}\cdot \frac{c}{d}=\frac{ac}{bd},\frac{a}{b}÷\frac{c}{d}=\frac{ad}{bc}\backslash frac\{a\}\{b\}\; \backslash pm\; \backslash frac\{c\}\{d\}\; =\; \backslash frac\{ad\; \backslash pm\; cb\}\{bd\},\backslash quad\; \backslash frac\{a\}\{b\}\backslash cdot\backslash frac\{c\}\{d\}\; =\; \backslash frac\{ac\}\{bd\},\backslash quad\; \backslash frac\{a\}\{b\}\; \backslash div\; \backslash frac\{c\}\{d\}\; =\; \backslash frac\{ad\}\{bc\}$$

Arithmetic / order of operations

Evaluate an arithmetic expression (order of operations)

$$\text{( )}\to a^b\to \times ÷\to +-\backslash text\{(\; )\}\; \backslash ;\backslash to\backslash ;\; a^\{b\}\; \backslash ;\backslash to\backslash ;\; \backslash times\backslash ,\backslash div\; \backslash ;\backslash to\backslash ;\; +\backslash ,-$$

Arithmetic / percentages

Percent change (increase or decrease)

$$\mathrm{\Delta }\mathrm{\%}=\frac{\mathrm{n}\mathrm{e}\mathrm{w}-\mathrm{o}\mathrm{l}\mathrm{d}}{\mathrm{o}\mathrm{l}\mathrm{d}}\cdot 100\backslash Delta\backslash \%\; =\; \backslash frac\{\backslash mathrm\{new\}-\backslash mathrm\{old\}\}\{\backslash mathrm\{old\}\}\backslash cdot\; 100$$

Percent of a number

$$\frac{p}{100}x\backslash frac\{p\}\{100\}\backslash ,x$$

What percent one number is of another

$$p=100\frac{\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}{\mathrm{w}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{e}}p\; =\; 100\backslash ,\backslash frac\{\backslash mathrm\{part\}\}\{\backslash mathrm\{whole\}\}$$

Whole from a part and its percentage

$$w=\frac{100\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}{p}w\; =\; \backslash frac\{100\backslash ,\backslash mathrm\{part\}\}\{p\}$$

Arithmetic / ratios and proportions

Divide a quantity in a given ratio

$${\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t}}_i=\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\cdot \frac{r_i}{\sum _j{r}_j}\backslash mathrm\{part\}\_i\; =\; \backslash mathrm\{total\}\backslash cdot\backslash frac\{r\_i\}\{\backslash sum\_j\; r\_j\}$$

Solve a proportion

$$\frac{n_1}{d_1}=\frac{n_2}{d_2}⟺n_1{d}_2=n_2{d}_1\backslash frac\{n\_1\}\{d\_1\}\; =\; \backslash frac\{n\_2\}\{d\_2\}\; \backslash iff\; n\_1\; d\_2\; =\; n\_2\; d\_1$$

Unit rates and the better buy

$$\text{unit rate}=\frac{\text{amount}}{\text{quantity}}\backslash text\{unit\; rate\}\; =\; \backslash frac\{\backslash text\{amount\}\}\{\backslash text\{quantity\}\}$$

Arithmetic / rounding

Round to decimal places

$$\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}(x,n)=\mathrm{sgn}(x)\frac{\lfloor \mathrm{\mid }x\mathrm{\mid }{10}^n+\frac{1}{2}\rfloor }{{10}^n}\backslash mathrm\{round\}(x,\; n)\; =\; \backslash operatorname\{sgn\}(x)\backslash ,\backslash frac\{\backslash lfloor\; \backslash vert\; x\backslash vert\; \backslash ,10^\{n\}\; +\; \backslash tfrac12\; \backslash rfloor\}\{10^\{n\}\}$$

Round to significant figures

$$\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}(x,n-1-\lfloor {\log }_{10}\mathrm{\mid }x\mathrm{\mid }\rfloor )\backslash mathrm\{round\}\backslash big(x,\backslash \; n\; -\; 1\; -\; \backslash lfloor\; \backslash log\_\{10\}\backslash vert\; x\backslash vert\; \backslash rfloor\backslash big)$$

Arithmetic / scientific notation

Scientific notation

Complex numbers

Product and modulus of complex numbers

$$\mathrm{Re}=ac-bd,\mathrm{Im}=ad+bc,{\text{modulus}}^2=(a^2+b^2)(c^2+d^2)\backslash operatorname\{Re\}\; =\; a\; c\; -\; b\; d,\backslash quad\; \backslash operatorname\{Im\}\; =\; a\; d\; +\; b\; c,\backslash quad\; \backslash text\{modulus\}^\{2\}\; =\; \backslash left(a^\{2\}\; +\; b^\{2\}\backslash right)\; \backslash left(c^\{2\}\; +\; d^\{2\}\backslash right)$$

Complex numbers / De Moivre's theorem

Power of a complex number (De Moivre)

$${[r(\cos \theta +i\sin \theta )]}^n=r^n(\cos n\theta +i\sin n\theta )\backslash left[r(\backslash cos\backslash theta\; +\; i\backslash sin\backslash theta)\backslash right]^n\; =\; r^n(\backslash cos\; n\backslash theta\; +\; i\backslash sin\; n\backslash theta)$$

Complex numbers / arithmetic

Arithmetic with complex numbers

$$\frac{a+bi}{c+di}=\frac{(ac+bd)+(bc-ad)i}{c^2+d^2}\backslash frac\{a+bi\}\{c+di\}\; =\; \backslash frac\{(ac+bd)\; +\; (bc-ad)i\}\{c^2+d^2\}$$

Complex numbers / polar form

Modulus, argument and polar form of a complex number

$$z=r(\cos \theta +i\sin \theta ),r=\sqrt{a^2+b^2}z\; =\; r(\backslash cos\backslash theta\; +\; i\backslash sin\backslash theta),\backslash quad\; r\; =\; \backslash sqrt\{a^2+b^2\}$$

Complex numbers / roots

The n-th roots of a complex number

$$w_k=\sqrt[n]{r}(\cos \frac{\theta +2\pi k}{n}+i\sin \frac{\theta +2\pi k}{n})w\_k\; =\; \backslash sqrt[n]\{r\}\backslash left(\backslash cos\backslash frac\{\backslash theta\; +\; 2\backslash pi\; k\}\{n\}\; +\; i\backslash sin\backslash frac\{\backslash theta\; +\; 2\backslash pi\; k\}\{n\}\backslash right)$$

Conic sections

Centre and radius of a circle from its general equation

$${(t-h)}^2+{(w-k)}^2-{\text{rad}}^2=t^2+w^2+d_{d}t+e_{e}w+f_f\backslash left(t\; -\; h\backslash right)^\{2\}\; +\; \backslash left(w\; -\; k\backslash right)^\{2\}\; -\; \backslash text\{rad\}^\{2\}\; =\; t^\{2\}\; +\; w^\{2\}\; +\; d\_\{d\}\; t\; +\; e\_\{e\}\; w\; +\; f\_\{f\}$$

Coordinate plane

Distance between two points

$${\text{dist}}^2={(x_2-x_1)}^2+{(y_2-y_1)}^2\backslash text\{dist\}^\{2\}\; =\; \backslash left(x\_\{2\}\; -\; x\_\{1\}\backslash right)^\{2\}\; +\; \backslash left(y\_\{2\}\; -\; y\_\{1\}\backslash right)^\{2\}$$

Counting and probability

Combinations and permutations

$$\left(\genfrac{}{}{0px}{}{n}{k}\right)=\frac{n!}{k!(n-k)!},P(n,k)=\frac{n!}{(n-k)!}\backslash binom\{n\}\{k\}\; =\; \backslash frac\{n!\}\{k!\backslash ,(n-k)!\},\backslash quad\; P(n,k)\; =\; \backslash frac\{n!\}\{(n-k)!\}$$

Exponential functions

Compound interest

$$\ln \left(\text{amount}\right)=\ln \left(\text{principal}\right)+nt\ln (1+\frac{r}{n})\backslash ln\backslash left(\backslash text\{amount\}\backslash right)\; =\; \backslash ln\backslash left(\backslash text\{principal\}\backslash right)\; +\; n\; t\; \backslash ln\backslash left(1\; +\; \backslash frac\{r\}\{n\}\backslash right)$$

Exponential models

Doubling time of exponential growth

$$k{t}_d=\ln \left(2\right)k\; t\_\{d\}\; =\; \backslash ln\backslash left(2\backslash right)$$

Exponentials and logarithms / exponential equations

Exponential equation ab^(cx + d) = k*

$$x=\frac{{\log }_b(k\mathrm{/}a)-d}{c}x\; =\; \backslash frac\{\backslash log\_b(k/a)\; -\; d\}\{c\}$$

Exponentials and logarithms / growth and decay

Amount remaining after radioactive decay (half-life)

$$N(t)=N_0{\left(\frac{1}{2}\right)}^{t\mathrm{/}T}N(t)\; =\; N\_0\; \backslash left(\backslash tfrac12\backslash right)^\{t/T\}$$

Continuous exponential growth or decay

$$N(t)=N_0{e}^{kt}N(t)\; =\; N\_0\backslash ,e^\{kt\}$$

Doubling time at a percentage growth rate

$$T=\frac{\ln 2}{\ln (1+\frac{r}{100})}T\; =\; \backslash frac\{\backslash ln\; 2\}\{\backslash ln\backslash left(1\; +\; \backslash frac\{r\}\{100\}\backslash right)\}$$

Exponentials and logarithms / laws of logarithms

Laws of logarithms (product, quotient, power)

$${\log }_b\text{\hspace{0.17em}⁣}\left(M^p{N}^q\right)=p{\log }_{b}M+q{\log }_{b}N\backslash log\_b\backslash !\backslash left(M^\{p\}\; N^\{q\}\backslash right)\; =\; p\backslash log\_b\; M\; +\; q\backslash log\_b\; N$$

Exponentials and logarithms / logarithmic equations

Logarithmic equation log_b(cx + d) = k

$${\log }_b(cx+d)=k⟺cx+d=b^k\backslash log\_b(cx+d)\; =\; k\; \backslash iff\; cx\; +\; d\; =\; b^\{k\}$$

Exponentials and logarithms / logarithms

Evaluate a logarithm

$${\log }_{b}x=\frac{\ln x}{\ln b}\backslash log\_b\; x\; =\; \backslash frac\{\backslash ln\; x\}\{\backslash ln\; b\}$$

Exponents and logarithms

Solution of an exponential equation

$$x\ln \left(b\right)=\ln \left(y\right)x\; \backslash ln\backslash left(b\backslash right)\; =\; \backslash ln\backslash left(y\backslash right)$$

Factoring

Factoring a monic quadratic

$$(t+p)(t+q)=t^2+bt+c\backslash left(t\; +\; p\backslash right)\; \backslash left(t\; +\; q\backslash right)\; =\; t^\{2\}\; +\; b\; t\; +\; c$$

Functions

Average rate of change of a function between two inputs

$$\text{arc}(x_{\text{end}}-x_{\text{start}})=f_{x,end}-f_{x,start}\backslash text\{arc\}\; \backslash left(x\_\{\backslash text\{end\}\}\; -\; x\_\{\backslash text\{start\}\}\backslash right)\; =\; f\_\{x,end\}\; -\; f\_\{x,start\}$$

Composition of two linear functions

$$mt+k=a(ct+d)+bm\; t\; +\; k\; =\; a\; \backslash left(c\; t\; +\; d\backslash right)\; +\; b$$

Functions / composition

Value of a composite function

$$(f\circ g)(x_0)=f(g(x_0))(f\backslash circ\; g)(x\_0)\; =\; f\backslash big(g(x\_0)\backslash big)$$

Functions / domain

Domain of a square-root function

$$ax+b\ge 0,cx+d\mathrm{\ne }0ax\; +\; b\; \backslash ge\; 0,\backslash quad\; cx\; +\; d\; \backslash ne\; 0$$

Functions / evaluation

Evaluate a function at a number

$$f(x_0)f(x\_0)$$

Functions / inverse functions

Inverse of a linear function

$$f^{-1}(x)=\frac{x-b}{a}f^\{-1\}(x)\; =\; \backslash frac\{x\; -\; b\}\{a\}$$

Functions / lines

Distance from a point to a line

$$d=\frac{\mathrm{\mid }a{x}_0+b{y}_0+c\mathrm{\mid }}{\sqrt{a^2+b^2}}d\; =\; \backslash frac\{\backslash vert\; a\; x\_0\; +\; b\; y\_0\; +\; c\backslash vert\; \}\{\backslash sqrt\{a^2\; +\; b^2\}\}$$

Intercepts and slope of a line

$$ax+by=c:x_0=\frac{c}{a},y_0=\frac{c}{b},m=-\frac{a}{b}ax\; +\; by\; =\; c:\backslash quad\; x\_0\; =\; \backslash frac\{c\}\{a\},\backslash \; y\_0\; =\; \backslash frac\{c\}\{b\},\backslash \; m\; =\; -\backslash frac\{a\}\{b\}$$

Functions / piecewise functions

Evaluate a piecewise function

Functions / transformations

Image of a point under a graph transformation

$$(x_0,y_0)\mapsto (\frac{x_0}{B}+h,A{y}_0+k)(x\_0,\; y\_0)\; \backslash mapsto\; \backslash left(\backslash frac\{x\_0\}\{B\}\; +\; h,\backslash ;\; A\; y\_0\; +\; k\backslash right)$$

Inequalities

Solution of a linear inequality

$$a{x}_0+b=c,s\sqrt{a^2}=aa\; x\_\{0\}\; +\; b\; =\; c,\backslash quad\; s\; \backslash sqrt\{a^\{2\}\}\; =\; a$$

Linear and absolute value inequalities

Solution interval of |x - c| < d

$$\text{upper}-\text{lower}=2d,\text{upper}+\text{lower}=2c\backslash text\{upper\}\; -\; \backslash text\{lower\}\; =\; 2\; d,\backslash quad\; \backslash text\{upper\}\; +\; \backslash text\{lower\}\; =\; 2\; c$$

Solution set of |x - c| > d

$$\text{right limit}-\text{left limit}=2d,\text{right limit}+\text{left limit}=2c\backslash text\{right\; limit\}\; -\; \backslash text\{left\; limit\}\; =\; 2\; d,\backslash quad\; \backslash text\{right\; limit\}\; +\; \backslash text\{left\; limit\}\; =\; 2\; c$$

Linear equations

Root of a linear equation

$$ax+b=ca\; x\; +\; b\; =\; c$$

Linear functions

Average rate of change between two points

$$m(x_2-x_1)=y_2-y_{1}m\; \backslash left(x\_\{2\}\; -\; x\_\{1\}\backslash right)\; =\; y\_\{2\}\; -\; y\_\{1\}$$

Line through two points

$$m{x}_1+k=y_1,m{x}_2+k=y_{2}m\; x\_\{1\}\; +\; k\; =\; y\_\{1\},\backslash quad\; m\; x\_\{2\}\; +\; k\; =\; y\_\{2\}$$

Slope of a perpendicular line

$$m{m}_{\text{perp}}=-1m\; m\_\{\backslash text\{perp\}\}\; =\; -1$$

y-intercept of a line from its slope and a point

$$y_1=m{x}_1+by\_\{1\}\; =\; m\; x\_\{1\}\; +\; b$$

Logarithms

Solution of a basic logarithmic equation

$$\ln \left(x\right)=c\ln \left(b\right)\backslash ln\backslash left(x\backslash right)\; =\; c\; \backslash ln\backslash left(b\backslash right)$$

Matrices

Determinant and inverse of a 2 by 2 matrix

$$\text{det}=ad-bc,a{i}_{11}+b{i}_{21}=1,a{i}_{12}+b{i}_{22}=0,c{i}_{11}+d{i}_{21}=0,c{i}_{12}+d{i}_{22}=1\backslash text\{det\}\; =\; a\; d\; -\; b\; c,\backslash quad\; a\; i\_\{11\}\; +\; b\; i\_\{21\}\; =\; 1,\backslash quad\; a\; i\_\{12\}\; +\; b\; i\_\{22\}\; =\; 0,\backslash quad\; c\; i\_\{11\}\; +\; d\; i\_\{21\}\; =\; 0,\backslash quad\; c\; i\_\{12\}\; +\; d\; i\_\{22\}\; =\; 1$$

Other equations

Solutions of an absolute value equation

$$a{x}_1+b=c,a{x}_2+b=-ca\; x\_\{1\}\; +\; b\; =\; c,\backslash quad\; a\; x\_\{2\}\; +\; b\; =\; -c$$

Polynomial equations / cubic and quartic equations

Roots of a cubic equation

$$a{x}^3+b{x}^2+cx+d=0a\; x^3\; +\; b\; x^2\; +\; c\; x\; +\; d\; =\; 0$$

Roots of a quartic equation

$$a{x}^4+b{x}^3+c{x}^2+dx+e=0a\; x^4\; +\; b\; x^3\; +\; c\; x^2\; +\; d\; x\; +\; e\; =\; 0$$

Polynomial equations / quadratic equations

Discriminant of a quadratic

$$\mathrm{\Delta }=b^2-4ac\backslash Delta\; =\; b^2\; -\; 4ac$$

Roots of a quadratic equation

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

Polynomials

Product of two binomials

$$p_2{t}^2+p_{1}t+p_0=(at+b)(ct+d)p\_\{2\}\; t^\{2\}\; +\; p\_\{1\}\; t\; +\; p\_\{0\}\; =\; \backslash left(a\; t\; +\; b\backslash right)\; \backslash left(c\; t\; +\; d\backslash right)$$

Precalculus / binomial theorem

Coefficient of a term in a binomial expansion

$$(ax+b{)}^n=\sum _{k=0}^n\left(\genfrac{}{}{0px}{}{n}{k}\right)a^k{b}^{n-k}{x}^k(ax\; +\; b)^n\; =\; \backslash sum\_\{k=0\}^\{n\}\; \backslash binom\{n\}\{k\}\; a^k\; b^\{n-k\}\; x^k$$

Precalculus / conic sections

Foci, axes and eccentricity of an ellipse

$$\frac{(x-h{)}^2}{A}+\frac{(y-k{)}^2}{B}=1,c^2=a^2-b^2,e=\frac{c}{a}\backslash frac\{(x-h)^2\}\{A\}\; +\; \backslash frac\{(y-k)^2\}\{B\}\; =\; 1,\backslash quad\; c^2\; =\; a^2\; -\; b^2,\backslash quad\; e\; =\; \backslash frac\{c\}\{a\}$$

Foci, eccentricity and asymptotes of a hyperbola

$$\frac{(x-h{)}^2}{a^2}-\frac{(y-k{)}^2}{b^2}=1,c^2=a^2+b^2\backslash frac\{(x-h)^2\}\{a^2\}\; -\; \backslash frac\{(y-k)^2\}\{b^2\}\; =\; 1,\backslash quad\; c^2\; =\; a^2\; +\; b^2$$

Focus and directrix of a parabola

$$(x-h{)}^2=4p(y-k):F=(h,k+p),y=k-p(x-h)^2\; =\; 4p(y-k):\backslash \; F\; =\; (h,\; k+p),\backslash \; y\; =\; k\; -\; p$$

Precalculus / polar coordinates

Polar to rectangular coordinates

$$x=r\cos \theta ,y=r\sin \theta x\; =\; r\backslash cos\backslash theta,\backslash quad\; y\; =\; r\backslash sin\backslash theta$$

Rectangular to polar coordinates

$$r=\sqrt{x^2+y^2},\tan \theta =\frac{y}{x}r\; =\; \backslash sqrt\{x^2+y^2\},\backslash quad\; \backslash tan\backslash theta\; =\; \backslash frac\{y\}\{x\}$$

Precalculus / vectors

Dot product and angle between two vectors

$$\mathbf{u}\cdot \mathbf{v}=\mathrm{\mid }\mathbf{u}\mathrm{\mid }\mathrm{\mid }\mathbf{v}\mathrm{\mid }\cos \theta \backslash mathbf\; u\backslash cdot\backslash mathbf\; v\; =\; \backslash vert\; \backslash mathbf\; u\backslash vert\; \backslash vert\; \backslash mathbf\; v\backslash vert\; \backslash cos\backslash theta$$

Magnitude and direction of a vector

$$\mathrm{\mid }\mathbf{v}\mathrm{\mid }=\sqrt{x^2+y^2},\theta =\mathrm{atan2}(y,x)\backslash vert\; \backslash mathbf\; v\backslash vert\; =\; \backslash sqrt\{x^2\; +\; y^2\},\backslash quad\; \backslash theta\; =\; \backslash operatorname\{atan2\}(y,\; x)$$

Quadratic functions

Vertex form of a quadratic function

$$a{(t-h)}^2+k=a{t}^2+bt+ca\; \backslash left(t\; -\; h\backslash right)^\{2\}\; +\; k\; =\; a\; t^\{2\}\; +\; b\; t\; +\; c$$

Rational functions

Asymptotes of a linear rational function

$$c{v}_a+d=0,c{h}_a=ac\; v\_\{a\}\; +\; d\; =\; 0,\backslash quad\; c\; h\_\{a\}\; =\; a$$

Real numbers and algebraic expressions

Perimeter of a rectangle

$$p-2h=2wp\; -\; 2\; h\; =\; 2\; w$$

Sequences

Term and partial sum of a geometric sequence

$$g_n=g_1{q}^{n-1},s_n(1-q)=g_1(1-q^n)g\_\{n\}\; =\; g\_\{1\}\; q^\{n\; -\; 1\},\backslash quad\; s\_\{n\}\; \backslash left(1\; -\; q\backslash right)\; =\; g\_\{1\}\; \backslash left(1\; -\; q^\{n\}\backslash right)$$

Term and partial sum of an arithmetic sequence

$$a_n=a_1+(n-1)d,2s_n=n(a_1+a_n)a\_\{n\}\; =\; a\_\{1\}\; +\; \backslash left(n\; -\; 1\backslash right)\; d,\backslash quad\; 2\; s\_\{n\}\; =\; n\; \backslash left(a\_\{1\}\; +\; a\_\{n\}\backslash right)$$

Systems of equations

Solution of two linear equations in two unknowns

$$ax+by=c,dx+ey=fa\; x\; +\; b\; y\; =\; c,\backslash quad\; d\; x\; +\; e\; y\; =\; f$$

The coordinate plane and graphs

Midpoint of a line segment

$$2m_x=x_1+x_2,2m_y=y_1+y_{2}2\; m\_\{x\}\; =\; x\_\{1\}\; +\; x\_\{2\},\backslash quad\; 2\; m\_\{y\}\; =\; y\_\{1\}\; +\; y\_\{2\}$$

Trigonometry / angle measure

Degrees to radians

$${\theta }_{\mathrm{r}\mathrm{a}\mathrm{d}}={\theta }_{\mathrm{deg}}\cdot \frac{\pi }{180}\backslash theta\_\{\backslash mathrm\{rad\}\}\; =\; \backslash theta\_\{\backslash deg\}\backslash cdot\backslash frac\{\backslash pi\}\{180\}$$

Radians to degrees

$${\theta }_{\mathrm{deg}}={\theta }_{\mathrm{r}\mathrm{a}\mathrm{d}}\cdot \frac{180}{\pi }\backslash theta\_\{\backslash deg\}\; =\; \backslash theta\_\{\backslash mathrm\{rad\}\}\backslash cdot\backslash frac\{180\}\{\backslash pi\}$$

Trigonometry / identities

Trigonometric values from one value and the quadrant

$$\cos t=\pm \sqrt{1-{\sin }^{2}t},\sin 2t=2\sin t\cos t,\cos 2t={\cos }^{2}t-{\sin }^{2}t\backslash cos\; t\; =\; \backslash pm\backslash sqrt\{1\; -\; \backslash sin^2\; t\},\backslash quad\; \backslash sin\; 2t\; =\; 2\backslash sin\; t\backslash cos\; t,\backslash quad\; \backslash cos\; 2t\; =\; \backslash cos^2\; t\; -\; \backslash sin^2\; t$$

Trigonometry / oblique triangles

Law of cosines (three sides)

$$\cos A=\frac{b^2+c^2-a^2}{2bc}\backslash cos\; A\; =\; \backslash frac\{b^2\; +\; c^2\; -\; a^2\}\{2bc\}$$

Law of cosines (two sides and the included angle)

$$a^2=b^2+c^2-2bc\cos Aa^2\; =\; b^2\; +\; c^2\; -\; 2bc\backslash cos\; A$$

Law of sines (two angles and a side)

$$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\backslash frac\{a\}\{\backslash sin\; A\}\; =\; \backslash frac\{b\}\{\backslash sin\; B\}\; =\; \backslash frac\{c\}\{\backslash sin\; C\}$$

Law of sines, the ambiguous case (two sides and a non-included angle)

$$\sin B=\frac{b\sin A}{a},B_2={180}^{\circ }-B_1\backslash sin\; B\; =\; \backslash frac\{b\backslash sin\; A\}\{a\},\backslash quad\; B\_2\; =\; 180^\backslash circ\; -\; B\_1$$

Trigonometry / right triangles

Right triangle from a leg and the hypotenuse

$$b=\sqrt{c^2-a^2},\sin A=\frac{a}{c}b\; =\; \backslash sqrt\{c^2\; -\; a^2\},\backslash quad\; \backslash sin\; A\; =\; \backslash frac\{a\}\{c\}$$

Right triangle from an acute angle and a side

$$\sin A=\frac{\mathrm{o}\mathrm{p}\mathrm{p}}{\mathrm{h}\mathrm{y}\mathrm{p}},\cos A=\frac{\mathrm{a}\mathrm{d}\mathrm{j}}{\mathrm{h}\mathrm{y}\mathrm{p}},\tan A=\frac{\mathrm{o}\mathrm{p}\mathrm{p}}{\mathrm{a}\mathrm{d}\mathrm{j}}\backslash sin\; A\; =\; \backslash frac\{\backslash mathrm\{opp\}\}\{\backslash mathrm\{hyp\}\},\backslash \; \backslash cos\; A\; =\; \backslash frac\{\backslash mathrm\{adj\}\}\{\backslash mathrm\{hyp\}\},\backslash \; \backslash tan\; A\; =\; \backslash frac\{\backslash mathrm\{opp\}\}\{\backslash mathrm\{adj\}\}$$

Right triangle from its two legs

$$c=\sqrt{a^2+b^2},\tan A=\frac{a}{b}c\; =\; \backslash sqrt\{a^2\; +\; b^2\},\backslash quad\; \backslash tan\; A\; =\; \backslash frac\{a\}\{b\}$$

Trigonometry / triangle area

Area of a triangle from its three sides (Heron)

$$K=\sqrt{s(s-a)(s-b)(s-c)},s=\frac{a+b+c}{2}K\; =\; \backslash sqrt\{s(s-a)(s-b)(s-c)\},\backslash quad\; s\; =\; \backslash frac\{a+b+c\}\{2\}$$

Trigonometry / trigonometric equations

Solve a trigonometric equation over one full turn

$$\sin u=k\Rightarrow u=\arcsin k,{180}^{\circ }-\arcsin k(\mathrm{m}\mathrm{o}\mathrm{d}{360}^{\circ }),u=nx\backslash sin\; u\; =\; k\; \backslash Rightarrow\; u\; =\; \backslash arcsin\; k,\backslash \; 180^\backslash circ\; -\; \backslash arcsin\; k\; \backslash pmod\{360^\backslash circ\},\backslash quad\; u\; =\; n\; x$$

Trigonometry / unit circle

Exact trigonometric values on the unit circle

$$\sin {30}^{\circ }=\frac{1}{2},\sin {45}^{\circ }=\frac{\sqrt{2}}{2},\sin {60}^{\circ }=\frac{\sqrt{3}}{2}\backslash sin\; 30^\backslash circ\; =\; \backslash tfrac12,\backslash \; \backslash sin\; 45^\backslash circ\; =\; \backslash tfrac\{\backslash sqrt2\}\{2\},\backslash \; \backslash sin\; 60^\backslash circ\; =\; \backslash tfrac\{\backslash sqrt3\}\{2\}$$

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
• Questions outside its certified math foundations capabilities, and any question whose stated numbers it cannot all use — it refuses with the reason instead.
• High-stakes engineering, medical or legal decisions without independent review.

Limitations

• It answers what its certified capabilities cover (arithmetic, algebra, functions, exponentials and logarithms, trigonometry, precalculus and complex numbers); anything else is refused with a reason.
• ask() reads documented question forms and common phrasings; unusual wording may be refused — use solve() for guaranteed results.
• Results are as correct as the certified relations and their stated conditions; a certified relation is a checked model, not a claim about every physical situation.

Citation

@software{math_foundations_ai999x_1_0_0,
  title  = {math-foundations-ai999x: an offline, verified math foundations solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {1.0.0},
  url    = {https://huggingface.co/AI999X/math-foundations-ai999x}
}

Contact

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Created and engineered by Ibnul Jaif Farabi · AI999X · ai999x.com