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Community Forum

Ask questions, share knowledge, and help others
⭐ 0
All General ❓ Help 📐 Math ⚡ Functions 🐛 Bug Reports 🏆 Weekly Challenge 💡 Feature Wishes 🎨 Showcase 🎓 Tutorials 🛒 Marketplace 🤖 AI Generated
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Answers

Markdown-style formatting supported

Create New Post

Integration Hub

Universal pipeline engine — dynamically chain any functions across all sources
—
Total Functions
—
Categories
—
Connections
—
Sources

Pipeline Builder

Auto-Compose

Catalog — All Functions by Category

Compatibility Explorer — What can connect to what?

👥 Collaborate

Real-time sessions with screen-sharing, chat & feature sync

Direct Connection (WiFi)

Peer-to-peer, no server needed. Same network or copy-paste codes.

Status: Disconnected

Create Room

Room Name
Password (optional)

Join Room

Room ID or Name
Password (if required)

Public Rooms

Loading public rooms…
∑ Session Enter ⏎ evaluate · ↑↓ history · ans1 ans2 reuse · Ctrl+K focus
Insert
›
📝 Scratchpad ▾

Functions

⌕
Math help Temp programs Clear Model: —

⊞ Matrix Lab

Interactive matrix computation & visualisation

Matrix Builder

Rows
Cols

Operations on Matrix A

Matrix B (same size)

Result

— run an operation —

Linear System Ax=b

Vector b (comma-separated)

Expression Evaluator

Enter a matrix expression

Heatmap Visualisation

◉ Statistics Studio

Descriptive stats, distributions, hypothesis tests, regression

Dataset Input

Data (comma-separated numbers)

Descriptive Statistics

N
—
Mean
—
Median
—
Std Dev
—
Min
—
Max
—
Skewness
—
Kurtosis
—
IQR
—
Mode
—

Histogram

Bins

Two-Sample Tests

Sample A
Sample B

Linear Regression

X values
Y values

Distribution Calculator

Distribution
Param 1
Param 2
x

◈ Finance Calculator

Loans, investments, options, DCF, bonds, portfolio analysis

Loan / Mortgage

Principal ($)
Annual rate (%)
Term (years)
Payments/year

Compound Interest / Investment

Initial ($)
Annual rate (%)
Years
Monthly contrib ($)

Bond Pricing

Face value ($)
Coupon rate (%)
Years to maturity
YTM / mkt rate (%)

ROI / CAGR

Initial value
Final value
Years

Black-Scholes Options

Spot price S
Strike K
Time (years)
Risk-free rate r
Volatility σ

NPV / IRR (DCF)

Discount rate (%)
Cash flows (year 0,1,2… comma-separated)

Retirement Planner

Current savings ($)
Monthly contrib ($)
Annual return (%)
Years to retire
Retire annual need ($)
Inflation (%)

Currency Rates (live)

Amount
From
To

⇌ Unit Converter

Convert between hundreds of units — or write unit-aware expressions directly

⚡ Unit-Aware Expression Evaluator

Type any expression with mixed units — e.g. 5 km + 3 mi, 60 kg * 9.81 m/s^2, 100 degC to degF

Quick Convert

Value
From unit
To unit

Length

Mass

Temperature

°C
°F
K
°R (Rankine)
°De (Delisle)

Area

Volume

Speed

Energy / Power

Joules
kWh
Cal (kcal)
BTU
eV
ft·lbf

Digital Storage

📏 Unit Conversions

Comprehensive unit conversion across 20+ categories
Category
Value
From
→
To

All Conversions

🪐 Solar System Simulator

N-body gravitational simulation — place bodies, set velocities, watch orbits evolve

Playback

Speed: 1.0x
G constant: 1.0
Time: 0.00  |  Bodies: 0  |  Energy: —

Add Body

Name
Mass
X
Y
Vx
Vy
Color

Bodies

Interaction

Click to select bodies, drag canvas to pan

Collision Detection

Simulation auto-pauses on collision

Save / Load

Save/load custom arrangements to browser storage
Scroll to zoom · Drag to pan · Click to select

⚛ Physics Toolkit

Constants, mechanics, electromagnetism, thermodynamics, relativity, quantum

Physical Constants (click to copy)

Kinematics

Choose equation

Electromagnetism

Thermodynamics

Special Relativity

Velocity v (m/s)
Rest mass m₀ (kg)
Proper time τ (s)
Rest length L₀ (m)

Quantum Mechanics

Wave Physics

Frequency f (Hz)
Wave speed v (m/s)

∞ Number Theory

Primes, factorisation, sequences, modular arithmetic, cryptography

Primality & Factorisation

Number n

Sieve of Eratosthenes

Find primes up to

GCD / LCM / Bezout

a
b

Modular Arithmetic

base
exponent
modulus

Integer Sequences

N terms
Sequence

Continued Fractions

Number (decimal or expression)

Chinese Remainder Theorem

Remainders (comma-sep)
Moduli (pairwise coprime)

RSA Key Generation (educational)

Prime p
Prime q

⊻ Logic & Bit Operations

Boolean algebra, truth tables, bitwise ops, base conversion, gates

Interactive Bit Manipulator (32-bit)

Decimal
Hex
Binary (space-grouped OK)
Octal

Bitwise Operations

A (decimal)
B (decimal)

Truth Table Generator

Boolean expression (use A,B,C and &&, ||, !, ^)
Variables (comma-sep)

Base Converter

Value
Input base

Logic Gate Simulator

Input A
Input B

Checksum / Hash (simple)

⚄ Probability & Simulation

Combinatorics, random sampling, dice, Monte Carlo, Markov chains

Combinatorics Calculator

n
k / r

Dice Roller

Number of dice
Sides
Rolls

Birthday Problem

People n
Days/year

Monte Carlo π Estimator

Points to sample

Random Sampling

Population (comma-separated or range a–b)
Sample size

Conditional Probability / Bayes

P(A)
P(B|A)
P(B|¬A)

Markov Chain Simulator

Transition matrix (rows comma-sep, rows semicolons)
Initial state
Steps

Expected Value & Variance

Values (comma-sep)
Probabilities (comma-sep, must sum to 1)

Waves & Optics

Quantum Mechanics

Classical Mechanics

Fluid Mechanics

Chemistry

Periodic table, stoichiometry, thermochemistry, equilibrium, pH

Periodic Table

Click an element for details.

Molar Mass Calculator

Chemical formula

pH / pOH Calculator

Type
Value
Param 2 (Ka/Kb, conc.)

Stoichiometry / Dilution

Thermochemistry

Gas Laws

Solution Chemistry

Electrochemistry

Equilibrium

Atomic Structure

△ Geometry Studio

2D & 3D shapes, coordinate geometry, transformations, curves

Shape Calculator

Shape

Coordinate Geometry

x₁
y₁
x₂
y₂

Polygon (arbitrary)

Vertices as x1,y1;x2,y2;… (closed)

Geometry Canvas

Transformations

Points (x,y;…)
Param 1
Param 2

Conic Sections

Conic type
h
k
a/r
b/p

Function Plotter

Multi-function, implicit curves, asymptotes, tangent lines, derivative overlay, parametric, polar
x: to y: to
∫ from to

ℂ Complex Plane Explorer

Argand diagram, transformations, Mandelbrot/Julia sets

Complex Number z

Re(z)
Im(z)

Operations

Re(w)
Im(w)

Fractal Mode

Max iterations

∂ ODE Solver + Phase Portrait

Vector fields, solution curves, multiple numerical methods

Equation dy/dx = f(x,y)

f(x,y) — use x, y, math functions
x₀
y₀
x end
Step h
Method

2nd Order: y'' = f(x,y,y')

f(x,y,v) where v=y'
y₀
y'₀

Phase Portrait (dy/dx = f)

x range
y range
Click canvas to add solution curves

∿ Fourier Series Builder

Live harmonic synthesis, DFT/FFT, waveform reconstruction

Harmonics (drag sliders)

FFT Analysis

Signal data (comma-separated samples)

Waveform Presets

Terms N
Period T

↗ Linear Algebra Visualiser

Drag vectors, animate transforms, eigenvectors, SVD

2×2 Transformation Matrix

a
b
c
d

Eigen Analysis

SVD Decomposition

3D Vector Operations

u
v
Drag vectors on canvas · matrix shown in red/blue

⌖ Numerical Methods

Root finding, integration, interpolation — animated step-by-step

Root Finding

f(x)
a (left bracket)
b / x₀ (initial)
Tolerance
Max iterations

Numerical Integration

f(x)
a
b
n (intervals)

Interpolation

Known points (x,y;x,y;…)
Evaluate at x
Method

Differentiation (numerical)

f(x)
at x =
h (step)

Cryptography

Classical ciphers, frequency analysis, Diffie-Hellman, modern hashes

Text Input / Output

Plaintext / Ciphertext
Key / Parameter

Classical Ciphers

Frequency Analysis

Hash Functions

Diffie-Hellman Key Exchange

Prime p
Generator g
Alice's secret a
Bob's secret b

Base Encoding

One-Time Pad

Signal Processing

Convolution, filters, z-transform, windowing, spectral analysis

Signal Generator

Waveform
Frequency (Hz)
Amplitude
Sample rate
Duration (s)
Phase (rad)

Convolution & Correlation

Signal h(t) — kernel (comma-sep)

FIR Filter Design

Filter type
Cutoff f₁ (normalised)
Cutoff f₂ (band)
Taps N
Window

Windowing Functions

z-Transform (common pairs)

◎ Graph Theory

Build graphs, BFS/DFS, shortest path, MST, Euler/Hamilton

Graph Input

Edges (u-v or u-v:weight, one per line)
Directed?
Layout

Algorithms

Source node
Target node

Graph Properties

Click nodes to select · drag to reposition

Adjacency Matrix

📄 Documents

Full-featured rich text editor with equations, sharing & forum publishing
📄
Sign in to access your documents
Rich text editor with LaTeX equations, sharing & forum publishing.

⌨ Code IDE

Python 3.11 (Pyodide) — full stdlib + packages
Not loaded

Programs

📦 Packages

🔗 Export as Custom Function

Export the current code as a custom function in the Computus library. The function appears under the Custom category in the function menu.

Function name
Category
Description
0 lines | 0 chars
Console Output
Python 3.11+ runtime (Pyodide v0.27.3). Press Run to start. Install packages from the package manager. Enable input() checkbox for interactive input support.

Fractal Explorer

Interactive fractal rendering & exploration

Fractal Type

Type
Max Iterations
Color Scheme
Resolution

View Bounds

Center Re
Center Im
Zoom
Julia C (Re)
Julia C (Im)

Presets

Fractal Dimensions Reference

Mandelbrot boundary: 2 | Koch curve: log(4)/log(3) ≈ 1.2619 | Sierpinski: log(3)/log(2) ≈ 1.585 | Cantor set: log(2)/log(3) ≈ 0.6309 | Menger sponge: log(20)/log(3) ≈ 2.727 | Barnsley fern: ≈ 1.45 | Dragon curve: 2

Rendered Fractal (click to zoom in)

Hover over the fractal to see coordinates

🌀 Chaos & Attractor Lab

Explore dynamical systems, bifurcations & strange attractors

Attractor Type

System
Steps
dt
Downsample

Lorenz Parameters

σ (sigma)
ρ (rho)
β (beta)
x₀
y₀
z₀

Attractor Visualization

Time Series

◯ Topology Explorer

Euler characteristic, Betti numbers, knot invariants & surfaces

Euler Characteristic Calculator

Vertices (V)
Edges (E)
Faces (F)

Polyhedra Reference

Knot Invariants

Knot

Knot Parametric Curve

Knot Type

Simplicial Complex Analyzer

Simplices (JSON)

3D Visualization

Fundamental Groups Reference

S¹ → Z (infinite cyclic) | T² → Z×Z | S² → {e} (trivial) | RP² → Z/2Z | Klein bottle → ⟨a,b | aba⁻¹b⟩ | Möbius strip → Z

Data Science

Pandas-like DataFrame with statistics & visualization
Import View Data Clean Transform Statistics Visualize Export

Import Data

Paste CSV
Paste JSON
Manual Entry
Columns (comma-sep)
Rows

Data Table

Import data to begin

Chart

Output

Ready

Neural Network

Design, train, and visualize neural networks from scratch

Network Architecture

Layer Type
Units / Filters
Activation
L1 Regularization
L2 Regularization
Dropout Rate

Training Configuration

Optimizer
Loss Function
Learning Rate
Epochs
Batch Size
Momentum (SGD)
Beta1/Beta2 (Adam)
Ready

Preset Architectures

Training Data

Input (JSON array of arrays)
Target (JSON array of arrays)

Network Architecture Visualization

Training Loss

Prediction

Input (JSON array)
No prediction yet

Weight Visualization

Model Import / Export

CAS+ Step-by-Step

Computer Algebra System with detailed solution steps

Equation Solving (Step-by-Step)

Expression / Equation
Variable

Differentiation (Step-by-Step)

Expression
Variable

Integration (Step-by-Step)

Expression
Variable
Definite?

Algebra (Step-by-Step)

Expression

Linear System Solver

Matrix A (rows as arrays)
Vector b

Apply Transformation Rule

Expression
Rule

Solution Hints

Expression

Limit Calculator (Step-by-Step)

f(x)
x →
Direction

Series Expansion (Taylor / Laurent)

f(x)
About x₀
Order

Polynomial Long Division

Numerator
Denominator

Partial Fraction Decomposition

Numerator P(x)
Denominator Q(x)

Inequality Solver

Inequality
Variable

Trigonometric Simplifier

Expression

Complex Number Operations

z₁ (a+bi)
z₂ (a+bi)

Matrix Operations (Step-by-Step)

Matrix A (rows semicolon-sep)
Matrix B (optional, for A·B)

Differential Equation Solver

Type
Equation / RHS
IC y(x₀) (optional)

Vector Calculus

Scalar f(x,y,z)
x₀
y₀
z₀

Substitution & Pattern Match

Expression
Substitution (e.g. x=u+1)

Logical Expression Simplifier

Boolean expression (use ∧, ∨, ¬, →, variables a..z)

Series Convergence Tests

General term aₙ

Unit Converter

Value
From
To

Solution Steps

Enter an expression and choose an operation

Visual Solution

Run a CAS operation to see its visual solution (graph / table / sign table)

Formula Database

Search

Quick Reference

📓 Notebook

Jupyter-style cell-based computation

Spreadsheet PRO

Parser-based engine · 122 sheet functions + full Computus bridge (5,900+ builtins · math.js · your functions) · fill · charts · undo/redo
A1 fx
sort by Right-click cells & tabs · drag ⌞ of selection to fill · F2 edit · Ctrl+Z/Y/C/X/V/F
ƒ Function reference
📈 Chart from selection

🔬 Simulation & Modeling

Physics, population, epidemiology, circuits, economics

Projectile

v0 (m/s)
Angle

Pendulum

L (m)
Angle

Spring-Mass

Mass
k
x0

Circuit RLC

V
R
C
L

Population (Logistic)

P0
r
K
tMax

SIR Model

N
beta
gamma
I0

Economics

GDP
Growth
Inflation
Unemploy

△ Interactive Geometry Lab

Euclidean constructions · measurements · transformations · coordinate geometry
(—, —)
BASICS CONSTRUCT MEASURE TRANSFORM GRID
Click to place a point
Properties
Select an object to inspect it.
Objects (0)

Advanced Statistics

ANOVA, Bayesian, Time Series, PCA

One-Way ANOVA

Bayesian Inference

Prior
Sensitivity
FPR

Time Series AR(1)

phi

PCA

CAD Pro

Professional 2D technical drawing — layers, snap, dimensioning, DXF/SVG export

Draw

Edit

Measure

Symbols

History

X: 0.00 Y: 0.00 (mm) Tool: Line — click first point 0 objects · 0 selected · zoom 100%

Layers

Properties

No selection. Use Select tool to pick an object.

Quick Help

Coordinate input
100,50 — absolute
@50,0 — relative
@100<45 — polar (dist<angle)
Mouse
Left: draw/select · Right: cancel tool
Wheel: zoom · Middle-drag: pan
Keys
Esc: cancel · Del: delete · Ctrl+Z: undo
Ctrl+A: select all · Ctrl+S: save project

LaTeX Editor

Write LaTeX with live KaTeX preview

Documents

Function Docs

Browse all functions

Select a function to view docs.

Tutorials

Interactive step-by-step lessons

Themes

Choose from 10+ themes
Light
Dark
Solarized
Dracula
Nord
Monokai
Ocean
Rose
Forest
Sunset

Plugins

Extend Computus

Install Plugin

Installed

Calculators

Mortgage, BMI, retirement, tax, resistor, carbon

Mortgage

Principal $
Rate
Years

BMI

Weight kg
Height m

Retirement

Age
Target
$/mo
Return
Saved $

Tax

Income $

Resistor

Band1
Band2
Mult
Tol

Carbon

Driving mi/yr
Flights
kWh
Gas

Math Art

Spirographs, L-systems, tessellations, mandalas, and more

Algorithm Visualiser

Sorting · searching · pathfinding — watch algorithms think
Ready comparisons 0 writes 0 time 0.0s

Competitive Math

Practice problems
Score: 0/0
Select a problem

Version History

Snapshots and rollback

Files

0 files

No file selected

No file

🌐 3D Plotter

Interactive 3D surface, parametric, vector field & contour plots

Plot Type

Surface z = f(x,y)

Expression
x min
x max
y min
y max
Resolution
Colorscale

Quick Presets

Analysis

Plot a surface to see analysis…

📊 Advanced Graphing Calculator

Multi-mode plotting · sliders · trace · roots · integrals · Taylor/Fourier · regressions · inequalities · animations
x: 0 y: 0
x:[-10,10] y:[-10,10]

Functions

Sliders (a, b, c, d, t)

View

Tools

Crosshair (trace point)

Analysis Output

No analysis run yet.

Quick Reference

Drag: pan · Wheel: zoom · T: trace · +/-: zoom · R: reset
Sliders: use a,b,c,d,t in expressions (e.g. a*sin(b*x+c))
Implicit: type "x^2+y^2-25" (RHS assumed 0)
Polar: r=θ/2 (use theta or t for θ)
Parametric: two functions paired: f1=x(t), f2=y(t)
Piecewise: f(x)=if(x<0,-x,x^2) — use if(cond,a,b)
Tangent/Normal: set x₀ in the crosshair box above, then click
Log-Log / Semi-Log: re-scales axes — toggle on/off
3D Fourier: animates Fourier epicycles for f₁
Bode: treats f₁ as s-domain transfer H(s); plots |H|, ∠H
Nyquist: polar plot of H(jω) over ω ∈ [10⁻², 10²]

⚡ Optimizer

Gradient descent, Nelder-Mead, L-BFGS, linear programming, constrained & global optimization

Unconstrained Minimization

Objective f(x) — single variable or f([x0,x1,…])
Starting x (or x0,x1,…)
Learning rate (GD)
Max iterations
Tolerance

Multivariable — f(x,y)

Objective f(x,y)
x₀
y₀

Linear Programming (Simplex)

Objective coefficients (maximize cᵀx)
Constraint rows Ax ≤ b (one per line: a1,a2 ≤ b)

Constrained Optimization

Minimize f(x,y)
Subject to g(x,y) = 0 (equality constraint)
And h(x,y) ≤ 0 (inequality, optional)
x₀
y₀

Global Optimization

Function f(x) (may have multiple local minima)
Search min
Search max

Convergence History

∇ PDE Solver

Finite-difference solvers for heat, wave, Laplace, Burgers, and Schrödinger equations

Equation

Parameters

Diffusivity α
Grid points Nx
Time steps Nt
Domain x: [0, L]
Time end T

Initial Condition u(x,0)

Boundary conditions

Numerical Scheme

Solution at t

t = T

Energy & Conservation

Run solver to see…

∬ Interval Arithmetic

Rigorous computation with guaranteed error bounds — never lose track of precision

Interval Calculator

Expression (use interval notation: [a,b])

Interval Functions

Significance Arithmetic

Value with uncertainty
Significant digits

Continued Fractions

Number to expand
Max terms

Interval Visualization

Precision Tracking Chain

Enter a computation chain to track error propagation

∑ Unified CAS Engine

Symbolic algebra, calculus, ODE/PDE solving, Risch integration, pattern matching, assumption-based simplification — all CAS engines in one
SymPy (Pyodide): initializing...

Symbolic Input

Variable

Assumptions

Assume about variable (e.g. "x > 0", "n is integer")

Pattern Matching

Pattern (use _ for wildcard)
Replacement

ODE Solver

ODE (e.g. y'' + y = 0)
Initial y(0)
Initial y'(0)
Range [0, T]
Steps

Symbolic Result (LaTeX)

ODE Solution Plot

Step-by-Step Solution

🌐 Wikidata Explorer

Search 100M+ entities · run SPARQL · compare · map · browse curated topics — powered by Wikidata

🔎 Search

📚 Browse Topics

🕐 Recent

⭐ Saved

📖 User Manual

The complete guide to every Computus app, feature, shortcut and hidden corner

Contents

🔌

Provider Preset

Choose a preset to auto-fill the URL and model, or configure manually.
🔑

Credentials

Your API key is stored only in this browser's local storage (unencrypted — anyone using this device/profile could view it) and is sent directly from your browser to the provider you choose, never to Computus servers. ⚠️ Anything you type into the AI chat is sent to that provider — avoid pasting personal data or secrets. See the privacy notes under ◎ About & Data.
💬

System Prompt

Customise how the AI behaves. It always has access to your functions, sessions, and files.
⚙

Precision & Modes

∫

Numerical Integration

Settings for numeric integration and root-finding routines.
🖥

Display

Auto-render LaTeX Render results as mathematical notation
Show computation time Display milliseconds taken per evaluation
Thousands separator Format large numbers with commas
Auto-close parentheses Automatically close open brackets on evaluate
Implicit multiplication Treat 2x as 2*x automatically
📜

History

🔘

Custom Buttons & Pinned Functions

Manage your custom calculator buttons and pinned functions.
0 custom buttons 0 pinned functions
🔢

Arithmetic Engine

Choose how Computus handles numeric computation. This affects ALL functions across every category.
Warn on precision loss Display a warning when regular arithmetic loses precision
Auto-detect large numbers Automatically switch to BigNumber for numbers > 15 digits
📐

Number Formatting

Trailing zeros Show trailing zeros in fixed format (1.200 instead of 1.2)
Show BigNum exponent Display BigNumber results in exponential form when large
🔄

Rounding & Overflow

Overflow to Infinity Show Infinity for values exceeding double range
Underflow to zero Show 0 for values below smallest representable positive number
🖥

Calculator Display

Show input echo Repeat the expression above the result
Color-coded results Different colors for different result types
Show result type badge Display "BigNumber", "Complex", etc. badge next to results
Truncate long results Show "..." for results exceeding max length
📊

Chart & Graph Display

Show grid lines Display grid on charts and graphs
Show chart legend Display legend on charts
Smooth curves Apply curve smoothing to plots
Interactive 3D rotation Allow mouse-drag rotation on 3D plots
Show axis labels Display x, y, z axis labels on plots
Show tick marks Display tick marks on chart axes
📝

Input Display

Syntax highlighting Color-code calculator input as you type
Autocomplete functions Show function name suggestions while typing
Show matching brackets Highlight matching parentheses/brackets
🔤

Font & Typography

Overrides the selector above. Use any CSS font-family value.
14px
0.010em
1.50
Live Preview
Computus — The Math Engine
sin(π/4) = 0.7071… · ∑ · ∫ · ∂ · ∇
The quick brown fox jumps over the lazy dog. 1234567890
📤

Data Export

Export your Computus data for backup or migration.
Include session variables Export variable scope with sessions
Include AI categories Export AI-created function categories
Include files Export file explorer contents
Include notepad data Export notepad contents
Include Python programs Export saved Python programs
📥

Data Import

Import previously exported Computus data.
Merge on import Merge imported data with existing (vs replace)
Validate on import Validate data structure before importing
☁

Cloud Sync

Auto-sync to Firebase Automatically sync data when signed in
Sync on startup Pull latest data from cloud on app launch
📋

Operation Logging

Control which operations are logged to session log files. All logs are stored in the File Explorer.
Enable logging Log operations to session history files
Log calculator operations Log all calculator evaluations
Log function calls Log all built-in function executions
Log tool operations Log tool view operations (plotter, converter, etc.)
Log AI interactions Log AI chat messages and responses
Log Python execution Log Python code executions
Log file operations Log file create/edit/delete operations
Log notepad operations Log notepad create/save operations
📏

Log Detail Level

Include timestamps Add timestamps to all log entries
Include execution time Log how long each operation took
Log arithmetic mode Tag log entries with current arithmetic mode
⚡

Computation

Lazy evaluation Only compute results when displayed
Cache computation results Cache results for identical inputs (memoization)
Parallel computation Use Web Workers for heavy computations
💾

Storage

Auto-save Automatically save state on changes
Compress state data Compress state before storage to save space
🔧

Function System

Allow user-defined functions Enable custom function creation
Allow AI-created categories Enable AI to propose function categories
Sandbox user functions Run user-defined functions in a sandboxed scope
🐍

Python Engine

Auto-load on startup Load Python runtime when app starts
Auto-install common packages Auto-install numpy, scipy, matplotlib on first run
/* COMPUTUS_WIKIDATA_SETTINGS_V1 */
🌐

Knowledge Engine

Wikidata lookups Answer "who is X", "capital of Y", "atomic mass of Z", etc. by querying Wikidata. Falls back automatically when a query isn't a math expression.
Auto-fallback on math errors If a non-math query fails to evaluate, automatically try Wikidata before showing an error
♿

Visual Accessibility

High contrast mode Increase contrast for better readability
Large cursor Increase cursor size for visibility
Dyslexia-friendly font Use OpenDyslexic font throughout
Colorblind mode Adjust colors for color vision deficiency
🔊

Audio & Haptic

Sound effects Play sounds for button clicks and errors
Error sound Play a distinctive sound on errors
Screen reader support Add ARIA labels and live regions
⌨

Input Accessibility

Sticky keys Keep modifier keys active after press
Slow keys Require key to be held for a moment before registering
Voice input Enable voice-to-text for calculator input
🚀

Rendering

GPU acceleration Use GPU for canvas rendering when available
Lazy-load views Only render views when they become active
Debounce input Wait before processing rapid input changes
🧠

Memory

Auto garbage collect Periodically free unused memory
Clear plot cache on view switch Free Plotly/Chart.js memory when switching views
📡

Network

Retry failed requests Automatically retry failed API calls
Cache API responses Cache identical API requests
⌨

Keyboard Shortcuts

These shortcuts work anywhere in Computus.
EvaluateEnter
Clear inputEscape
Previous expression↑
Next expression↓
Go to CalculatorCtrl+1
Go to AI ChatCtrl+2
Go to ChartsCtrl+3
Go to FilesCtrl+4
Go to SettingsCtrl+5
New sessionCtrl+N
Session ManagerCtrl+Shift+S
Focus calculatorCtrl+K
Insert πCtrl+P
∑

Calculator Syntax Quick Reference

Examples of expressions the calculator understands.
Assignmentx := 5
Symbolic deriv.derive('x^2','x')
Numeric integralintNum('sin(x)',0,pi)
LimitlimitNum('sin(x)/x',0)
Solve linear sys.solveLin([[1,2],[3,4]],[5,6])
Complexcomplex(3,4)
Unit convertconvert('5 km','mi')
Matrix[[1,2],[3,4]]
Taylor seriestaylor('sin(x)',0,7)
Prime testisPrime(997)
Big factorialfactorial(1000)
Statisticsmean(1,2,3,4,5)
◎

About Computus

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Built-in fns
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Sessions
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User fns
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Files
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Storage used
64
Precision
Computus is The Math Engine — a powerful, privacy-first calculator running entirely in your browser.
Powered by math.js (arbitrary precision, symbolic algebra, linear algebra, units)
and KaTeX for beautiful mathematical typesetting.
Private by default: calculations, sessions, files and settings stay in this browser. Optional features that talk to the outside world: AI chat (your messages go to the AI provider you configure), and account / forum / marketplace (stored in our Firebase cloud so they can sync — only if you sign in). No ads, no analytics, no tracking.
Read the full privacy notes →
💾

Data Management

All sessions, custom functions, files, and settings are stored in your browser's IndexedDB/local storage on this device — they are not uploaded anywhere unless you explicitly sign in and use cloud sync, the forum, or the marketplace.
🔄 Transfer to Another Device
Bundle ALL your Computus data — every session, file, custom function, custom category, and setting — into a single .cpt file you can load on another device. The Computus program itself is NOT included.
🧩

Libraries & Licences

math.js 12.4.2Apache 2.0
Algebrite 1.4.0MIT
KaTeX 0.16.11MIT
Chart.js 4.4.3MIT
👤

Profile

Manage your account profile. Your username is visible to others in collaboration rooms and the Feature Market.
Unique identifier used in collab rooms, forum posts, and marketplace.
🔒

Security

Change your password and manage security options.
Two-Factor Authentication (2FA) Add an extra layer of security (coming soon)
⚙️

Preferences

Customize how your account behaves across Computus.
Start page Which view opens when Computus loads
Auto-save interval How often Computus saves state to IndexedDB
🔔

Notifications

Control which notifications and alerts you receive.
Desktop notificationsBrowser push notifications for collab invites and replies
Sound effectsPlay sounds for messages and errors
Collab room invitesNotify on room invite links
Marketplace updatesNotify when installed functions update
Forum repliesNotify when someone replies to your posts
☁️

Cloud Sync & Data

Manage cloud data sync via Firebase.
Auto-sync to cloudSync sessions, functions, and settings automatically
Sync custom functionsInclude user-defined functions in cloud backups
Sync sessionsInclude calculation sessions in cloud backups
Sync marketplace installsInclude marketplace functions in cloud backups
Never
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Activity & Usage

Review your account activity and usage statistics.
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Total calculations
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Functions created
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Sessions
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Collab rooms joined
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Forum posts
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Market installs
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🌐

Connected Devices

Manage devices signed into your account.
💻
Current Device
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This device
Sign out all other devicesForce sign-out on other devices
⚠️

Danger Zone

These actions are irreversible.
Reset all account settings
Restore all preferences to defaults
Delete account
Permanently delete your account and all cloud data
🎨

Custom Theme Editor

Create your own themes that control every visual aspect of Computus. Changes apply instantly via CSS custom properties — no page reload needed.
No custom themes yet. Create one below!
Captures all current CSS variables as a starting point you can then customize.

🧮 Miscellaneous Calculators

50 everyday calculators for finance, health, conversion, and more
Category
🧮

Select a calculator from the list

⏱ Session Manager

Create, switch, and inspect your calculation sessions

🔤 Advanced Algebra

Polynomial operations, systems of equations, inequalities, and more

Polynomial Roots

Enter polynomial in x (e.g. x^3 - 6x + 4)

Polynomial Factor

Enter polynomial to factor (e.g. x^2 - 4)

Expand Expression

Expand (e.g. (x+1)^3)

Simplify Expression

Simplify (e.g. 2x + 3x - x)

Symbolic Derivative

Expression, variable (e.g. x^3 + 2x, x)

Symbolic Integral

Expression, variable (e.g. x^2, x)

Solve Equation

Enter equation (e.g. x^2 - 4 = 0)

Quadratic Formula

Enter a, b, c for ax² + bx + c = 0
a
b
c

System of Linear Equations (2x2)

a₁x + b₁y = c₁   a₂x + b₂y = c₂
a1,b1,c1
a2,b2,c2

📐 Complex Residue Calculator

Residue theorem, contour integrals, pole analysis

Residue at a Pole

f(z) expression, pole Re, pole Im, order
Pole Re
Pole Im
Order

Residue Theorem: Sum of Residues

Sum residues to get contour integral: ∮ f(z) dz = 2πi × Σ Res(f, z_k)
Enter residues (comma-separated):

Laurent Series Coefficients

f(z) centered at z₀ (e.g. 1/(z*(z-1)))
Center z₀ (real)
Terms

Quick Reference

Residue Theorem: If f is analytic inside and on a simple closed contour C except for isolated singularities z₁,...,z_n inside C, then:
∮C f(z) dz = 2πi × Σk=1n Res(f, z_k)

Simple pole: Res(f, z₀) = limz→z₀ (z - z₀) f(z)
Double pole: Res(f, z₀) = limz→z₀ d/dz [(z - z₀)² f(z)]

Common residues:
• Res(1/(z-a), a) = 1
• Res(1/(z-a)^n, a) = 0 for n > 1
• Res(e^z/z, 0) = 1
• Res(sin(z)/z, 0) = 0

📐 Vector Operations

Vector arithmetic, dot/cross products, projections, angles

Vector Arithmetic

Vector A (comma-separated)
Vector B

Dot & Cross Product

Uses vectors A and B from above

Magnitude & Angle

Uses vectors A and B from above

Projection

Project A onto B

Unit Vector & Normalization

Normalize vector A

2D Vector Visualization

📊 Series Explorer

Taylor series, power series, convergence, and visualization

Taylor / Maclaurin Series

f(x) expression (use x as variable)
Center a
Number of terms
View range [-R, R]

Quick Series

Convergence Test

Enter terms a_n as expression in n (e.g. 1/n^2)

Series Visualization

Series Terms

🔀 Function Composer

Compose, transform, and visualize function combinations

Function Composition f ∘ g

f(x) - outer function
g(x) - inner function

Evaluate Composition at x

x value

Function Transformations

f(x) =
a (vertical)
b (horizontal)
h (shift x)
k (shift y)
Transform: a · f(b(x - h)) + k

Quick Compositions

Graph

↘ Limit Lab

Evaluate limits numerically, graphically, and via L'Hôpital's rule

Limit Evaluator

Function f(x)
Approaches x →
Side

Numerical Approach Table

L'Hôpital's Rule

Detects 0/0 or ∞/∞ forms and applies L'Hôpital iteratively

Quick Examples

Function Graph (zoom near limit point)

Zoom (±) 1

⫽ Partial Fractions Decomposer

Decompose P(x)/Q(x) into partial fractions step-by-step

Rational Function Input

Numerator P(x)
Denominator Q(x)

Quick Examples

Step-by-Step Solution

⋚ Inequality Studio

Solve and graph one-variable inequalities on a number line

Inequality Input

Enter inequality in x
Supported: >, <, >=, <=, = (equation)

Step-by-Step Solution

Quick Examples

Number Line Visualization

∭ Multivariable Calculus

Partial derivatives, gradient, Hessian, critical points for f(x, y)

Function f(x, y)

Expression
Evaluate at x₀
Evaluate at y₀
Direction u₁
Direction u₂

Quick Examples

Critical Points

Partial Derivatives

Gradient ∇f at (x₀, y₀)

Hessian Matrix H(f)

Directional Derivative D_u f

⌒ Newton's Method

Iterative root-finding with tangent-line visualization

Newton Iteration Setup

Function f(x)
Initial guess x₀
Max iterations

Convergence Table

Quick Examples

Visualization (function + tangent lines)

⨋ Improper Integrals

Evaluate improper integrals (infinite limits, unbounded integrands) with convergence tests

Improper Integral Setup

Integrand f(x)
Type
Lower bound a
Upper bound b
Truncation L (large)
N subintervals

Convergence Test

Quick Examples

Visualization

Convergence Diagnostic Table

🎯 Eigenvalue Visualizer

Eigenvalues, eigenvectors, diagonalization, and spectral decomposition

2x2 Matrix Eigenanalysis

Enter matrix [[a,b],[c,d]]

3x3 Matrix Eigenanalysis

Enter 3x3 matrix (row by row)

Quick Matrices

Eigenspace Visualization

Transformation Visualization

♪ Music Math Tools

Frequencies, scales, chords, tuning systems, and acoustics

Note Frequency Calculator

MIDI note number (A4 = 69, C4 = 60)

Scale Generator

Root note (0=C, 1=C#, ..., 11=B) and scale type
Root (0-11)
Scale

Chord Builder

Root MIDI note + chord type
Root MIDI
Chord

Interval Calculator

Two frequencies (Hz)
f1 (Hz)
f2 (Hz)

Equal Temperament Reference

Harmonic Series

Fundamental frequency (Hz)
Number of harmonics

⚡ Circuit Calculator

Ohm's law, series/parallel, voltage dividers, RC/RL/RLC analysis, op-amps

Ohm's Law Triangle

V (Volts)
I (Amps)
R (Ohms)

Power Calculator

Voltage (V)
Current (A)

Series Resistors

Resistances (comma-separated, Ohms)

Parallel Resistors

Resistances (comma-separated, Ohms)

Voltage Divider

Vin (V)
R1 (Ω)
R2 (Ω)

RC Time Constant

R (Ω)
C (F)
Vin (V)

LED Resistor Calculator

Vsource (V)
VLED (V)
ILED (mA)

Capacitor Energy & Charge

Capacitance (F)
Voltage (V)

⭐ Astronomy Tools

Celestial mechanics, orbital mechanics, stellar physics, cosmology

Kepler's Third Law

For orbits around a central body
Semi-major axis a (AU)
Central mass (M☉)

Orbital Velocity

Altitude (km)
Body radius (km)
GM (km³/s²)

Escape Velocity

Mass (kg)
Radius (m)

Stellar Luminosity

L = 4πσR²T&sup4; (Stefan-Boltzmann)
Radius (R☉)
Temperature (K)

Distance Modulus

m - M = 5 log10(d/10pc)
Apparent mag m
Absolute mag M

Hubble's Law

v = H₀ × d
Distance (Mpc)
H₀ (km/s/Mpc)

Solar System Quick Reference

♥ Medical Calculators

Clinical calculations, dosing, unit conversions, vital statistics

BMI Calculator

Weight (kg)
Height (cm)

Body Surface Area (BSA)

Weight (kg)
Height (cm)

Drug Dosage Calculator

Dose (mg/kg)
Weight (kg)
Concentration (mg/mL)
Frequency (per day)

Ideal Body Weight (Devine)

Height (cm)
Sex

Mean Arterial Pressure (MAP)

Systolic BP (mmHg)
Diastolic BP (mmHg)

Unit Conversions (Medical)

Value

☰ Menu

All tools, views, and features
Create or import a custom function

⬭ Non-Euclidean Geometry

Hyperbolic · Spherical · Taxicab · Curvature comparison

Poincaré Disk (Hyperbolic)

Click inside the disk to place points. Hyperbolic geodesics (arcs ⊥ boundary) connect consecutive points. Angle sum < π.

Click inside the disk to add points.

Hyperbolic Triangle Solver

Enter side lengths a, b, c (in hyperbolic units). Computes angles via hyperbolic law of cosines. Angle sum < π, area = π − (α+β+γ).

a
b
c

Taxicab (Manhattan) Geometry

d = |Δx| + |Δy|. Circles are diamonds. Compare with Euclidean distance.

Click "Show taxicab circle" to begin.

Spherical Triangle Solver

Enter side lengths a, b, c (angular, in degrees, 0–180). Computes angles via spherical law of cosines. Angle sum > π, area = R²·E where E = α+β+γ−π.

a (°)
b (°)
c (°)

Curvature Comparison

Visualize the same triangle (α=β=γ=60°) under three geometries. Adjust the triangle scale to see how angle sum deviates.

Triangle scale (side length)
Euclidean (K=0)
Spherical (K>0)
Hyperbolic (K<0)

Reference: Key Formulas

Hyperbolic distance (Poincaré disk):
d(z₁,z₂) = arcosh(1 + 2|z₁−z₂|² / ((1−|z₁|²)(1−|z₂|²)))
Hyperbolic law of cosines:
cosh(c) = cosh(a)·cosh(b) − sinh(a)·sinh(b)·cos(γ)
Hyperbolic area = π − (α+β+γ) (angle defect)
Spherical law of cosines:
cos(c) = cos(a)·cos(b) + sin(a)·sin(b)·cos(γ)
Spherical area = R²·(α+β+γ−π) (spherical excess)
Taxicab distance: d = |Δx| + |Δy|

▦ Tessellation Lab

Regular, Archimedean & Penrose tilings · wallpaper groups

Tiling Type

Family
Tile size (px)
Iterations (self-similar)

Wallpaper Group Classification

Select a tiling family to see its symmetry group.

Schläfli & Vertex Configuration

—

Tiling Visualization

⊠ Projective Geometry

Homogeneous coords · cross-ratio · perspective · collineations

Cartesian ↔ Homogeneous

x
y
w

Cross-Ratio (4 collinear points)

[A,B;C,D] = (A−C)(B−D) / ((A−D)(B−C)). Projectively invariant.

A
B
C
D

Perspective Projection (Pinhole Camera)

X (world)
Y (world)
Z (depth)
Focal length f

Projective Transformation (3×3 Collineation)

Enter matrix H rows (3 comma-separated numbers each) and a point (x, y) to transform.

H row 1 (a,b,c)
H row 2 (d,e,f)
H row 3 (g,h,i)
Point x
Point y

Visualization (grid + H-transform)

⌢ Differential Geometry

Frenet-Serret frame · curvature κ · torsion τ · osculating circle

Parametric Curve r(t) = (x(t), y(t), z(t))

x(t)
y(t)
z(t)
t min
t max
Eval at t₀

Frenet-Serret Frame at t₀

Compute to see T, N, B vectors.

Curvature, Torsion & Arc Length

—

Curve & Osculating Circle Visualization

⟳ Symmetry Group Finder

Point groups · frieze groups · symmetry elements

Pattern Source

Pick a preset pattern

Detected Symmetry Elements

Choose a preset and click Identify.

Group Classification

—

Pattern with Symmetry Axes Overlay

Reference: Common Groups

Point groups (2D): Cₙ (n-fold rotation only) | Dₙ (n-fold + n mirrors) | Cₛ (single mirror) | C₁ (trivial)
Frieze groups (7): p111, p1a1, p1m1, p112, pm11, pma2, pmm2
Wallpaper groups (17): p1, p2, pm, pg, cm, pmm, pmg, pgg, cmm, p4, p4m, p4g, p3, p3m1, p31m, p6, p6m

🗺 Map Projections & Geodesy

Great-circle distance · coordinate conversion · world map projections

Great-Circle Distance (Haversine)

Point A latitude (°)
Point A longitude (°)
Point B latitude (°)
Point B longitude (°)
Sphere radius R (km)

City Pair Presets

Map Projection

Projection type
Center longitude (°)
Show graticule?

World Map (selected projection)

Click 'Render Map' to draw.

⫧ Boundary Value Problem Solver

Shooting method & finite-difference for y'' = f(x,y,y') on [a,b]

BVP Equation

y'' = f(x, y, y') (use y, yp for y')
x₀ (a)
x₁ (b)
y(a) = α
y(b) = β
N (steps)
Method

Solution Table (sample)

—

Reference

Shooting: converts BVP → IVP by guessing y'(a), uses secant to match y(b)=β
FD: discretizes y'' ≈ (yᵢ₊₁−2yᵢ+yᵢ₋₁)/h², solves tridiagonal linear system
Stability: FD is 2nd-order accurate, O(h²); shooting inherits IVP stability

Solution y(x)

🌀 Nonlinear Dynamical Systems

Lorenz, Rössler, logistic map & bifurcation diagrams

System

System
σ / a
ρ / b
β / c
Steps
Δt (continuous)
x₀,y₀,z₀ (comma-sep)

Bifurcation Diagram (logistic map)

r min
r max
Iterations
Plot last

Lyapunov Exponent (logistic)

r
Iterations
—

Phase Portrait (X–Y projection)

Time series

Bifurcation diagram

⚖ Stability Analysis

Jacobian eigenvalues · Routh–Hurwitz · Lyapunov

Jacobian Eigenvalue Stability

Jacobian J (rows semicolon-sep)
—

Routh–Hurwitz (polynomial)

Coefficients (highest → lowest power, comma-sep)
—

Lyapunov Equation

A matrix
Q matrix (identity if blank)
—

Reference: 2D Classifications

Stable node: both eigenvalues real < 0
Unstable node: both real > 0
Saddle: real eigenvalues, opposite signs
Stable spiral: complex with Re(λ) < 0
Unstable spiral: complex with Re(λ) > 0
Center: purely imaginary (Re=0)
Routh–Hurwitz: all 1st-column entries > 0 ⇒ stable
Lyapunov: P > 0 and Q > 0 ⇒ A is Hurwitz

Complex eigenvalue plot

∮ Integral Transforms

Laplace, Mellin, Hankel & Z-transforms — symbolic & numeric

Laplace Transform

f(t)
t min
t max
—

Mellin Transform

f(x)
s (real)
x max
—

Z-Transform

x[n] — formula in n (e.g. 0.5^n, sin(n), n^2)
N terms
|z| (eval radius)
—

Hankel Transform

f(r)
Order ν
k max
—

Magnitude spectrum

≈ Asymptotic & Perturbation Methods

WKB · regular perturbation · dominant balance

WKB Approximation

y'' + k²(x)·y = 0 — enter k(x)
x min
x max
—

Regular Perturbation (y'' + ε·g(y) = 0)

g(y) — small nonlinearity
ε
Order
—

Dominant Balance

f(x) — examine as x → ∞
—

Asymptotic vs Exact

Compares exact (numerical) solution against asymptotic approximation over the requested range.

∥ Method of Characteristics

1st-order PDE a·uₓ + b·u_y = c

PDE a(x,y)·uₓ + b(x,y)·u_y = c(x,y,u)

a(x, y)
b(x, y)
c(x, y, u)
Initial curve: y = g(x₀) — enter g(x₀)
x₀ min
x₀ max
Number of chars
t end

Output

Compute to view characteristic summary.

Characteristic curves in (x, y) plane

◐ Symplectic & Hamiltonian Integrators

Verlet, leapfrog, Stormer–Verlet, RK4 vs symplectic comparison

Hamiltonian System

H = T(p) + V(q) — enter V(q)
q₀
p₀
Δt
Steps
Method

Energy drift (conservation check)

—

Reference

Symplectic: preserves phase-space volume & bounded energy drift
Verlet: 2nd-order, time-reversible
Leapfrog: 2nd-order, popular for N-body & MD
Yoshida s4: 4th-order composition of Verlet steps
RK4: non-symplectic, energy drifts slowly even for short simulations

Phase space (q, p)

Energy H(t) over time

⟘ QR Decomposition

Gram–Schmidt, Householder & Givens rotations

Matrix A (rows semicolon-sep)

Q matrix

—

R matrix

—

Diagnostics

—

Reference

Gram–Schmidt: classical (CGS) — orthogonalize columns sequentially
Householder: reflects columns to upper-triangular via reflections (more stable)
Givens: rotates pairs to zero out entries below diagonal
Use cases: least squares, eigenvalue algorithms (QR), linear system solving

Visualization: A and Q basis vectors

⊥ SVD Applications

Rank-k approximation · PCA · image compression · pseudoinverse

SVD of A (rows semicolon-sep)

—

Rank-k Approximation (Eckart–Young)

k (rank)
—

PCA on data matrix

2D points (x,y per line, semicolon-sep)
—

Reference

SVD: A = UΣVᵀ, where U,V orthogonal, Σ diagonal with σ₁ ≥ σ₂ ≥ ... ≥ 0
Eckart–Young: best rank-k approximation is Aₖ = Σᵢ₌₁ᵏ σᵢuᵢvᵢᵀ
Pseudoinverse: A⁺ = VΣ⁻¹Uᵀ (used in least squares)
PCA: right singular vectors of centered data = principal directions

Singular values spectrum

ᵉ Matrix Functions

exp, log, sqrt, polar decomposition

Matrix A (rows semicolon-sep, square)

Result

—

Method info

—

Reference

exp(A): Σₙ Aⁿ/n! — solves linear ODE ẋ = Ax
Padé approximant: efficient rational approximation used internally
log(A): inverse of exp, requires A nonsingular with no eigenvalues on negative real axis
sqrt(A): X² = A (principal branch via Schur decomposition)
Polar: A = UP, U orthogonal, P positive-semidefinite Hermitian
cos(A), sin(A): Σ (-1)ⁿA²ⁿ/(2n)!, (-1)ⁿA²ⁿ⁺¹/(2n+1)!

Visualization

∇ Vector Calculus

grad · div · curl · Laplacian · Jacobian & Hessian

Scalar field f(x, y, z)

f
x₀
y₀
z₀
—

Vector field F = (P, Q, R)

P(x,y,z)
Q(x,y,z)
R(x,y,z)
—

Identities

grad(f): (∂f/∂x, ∂f/∂y, ∂f/∂z)
div(F): ∂P/∂x + ∂Q/∂y + ∂R/∂z
curl(F): (∂R/∂y − ∂Q/∂z, ∂P/∂z − ∂R/∂x, ∂Q/∂x − ∂P/∂y)
Laplacian: ∇²f = ∂²f/∂x² + ∂²f/∂y² + ∂²f/∂z²
curl(grad f) = 0 · div(curl F) = 0
div(∇f) = ∇²f · curl(curl F) = grad(div F) − ∇²F

Field visualization

H Hypothesis Testing

z, t, χ², F & ANOVA

One-sample z-test

Sample mean x̄
Population μ₀
σ (population sd)
n (sample size)
α (significance)
—

Two-sample t-test

Sample 1 (comma-sep)
Sample 2 (comma-sep)
α
Type
—

Chi-square goodness-of-fit

Observed (comma-sep)
Expected (comma-sep, blank for uniform)
—

One-way ANOVA

Groups (semicolon-sep; values comma-sep within each)
—

F-test (variances)

σ₁² (variance 1)
σ₂² (variance 2)
n₁
n₂
—

λ Reliability & Survival Analysis

Kaplan–Meier · Weibull · hazard & MTBF

Kaplan–Meier Estimator

Times (comma-sep, with + for censored)
—

Weibull Fit

Shape k
Scale λ
t (eval at)
—

System Reliability

Component reliabilities (comma-sep)
Configuration
k (if k-of-n)
—

Reference

Kaplan–Meier: Ŝ(t) = Πᵢ (nᵢ − dᵢ)/nᵢ over events with tᵢ ≤ t
Weibull R(t): exp(−(t/λ)ᵏ)
Weibull h(t): (k/λ)(t/λ)ᵏ⁻¹
MTBF: λ·Γ(1 + 1/k)
Series: R = Π Rᵢ (worst-case)
Parallel: R = 1 − Π(1−Rᵢ)
k-of-n: Σ C(n,i) Rⁱ (1−R)ⁿ⁻ⁱ for i ≥ k

Survival & hazard curves

⏱ Timer & Stopwatch

Countdown timer and stopwatch with lap tracking
00:00.00

🚀 Relativity Lab

Special & general relativity — Lorentz transforms, time dilation, length contraction, relativistic energy-momentum, Schwarzschild metric, gravitational redshift

Lorentz Factor γ

v / c (β)
Or v (m/s)

Time Dilation

Proper time Δt₀ (s)
v / c

Length Contraction

Proper length L₀ (m)
v / c

Relativistic Velocity Addition

u / c (object velocity)
v / c (frame velocity)

Relativistic Doppler Shift

Source freq f₀ (Hz)
v / c
Angle θ (deg)

Relativistic Energy & Momentum

Rest mass m₀ (kg)
v / c
Or KE (J)

Schwarzschild Metric

Mass M (M☉)
Radius r (km)
θ (rad)

Gravitational Redshift

M (M☉)
r_emit (km)
f₀ (Hz)

Twin Paradox Calculator

Trip duration (yr)
v / c
Turnarounds

Reference

γ = 1/√(1−β²) · Δt = γΔt₀ · L = L₀/γ
w = (u+v)/(1+uv/c²) · E = γm₀c² · p = γm₀v
E² = (pc)² + (m₀c²)² · Rs = 2GM/c²
Schwarzschild: ds² = (1−Rs/r)c²dt² − (1−Rs/r)⁻¹dr² − r²dΩ²
Grav. redshift: f_obs = f_emit √(1−Rs/r)
Doppler: f_obs = f₀γ(1−βcosθ)⁻¹

🔥 Thermodynamics Studio

Ideal gas, Carnot cycle, entropy, Gibbs/Helmholtz free energy, Maxwell relations, phase transitions, heat engines, partition functions

Ideal Gas Law (PV = nRT)

P (Pa)
V (m³)
n (mol)
T (K)
Solve for

Carnot Cycle Efficiency

T_hot (K)
T_cold (K)

Entropy Change

Q (J)
T (K)
Type

Thermodynamic Potentials

U (J)
T (K)
S (J/K)
P (Pa)
V (m³)
N (mol)

Heat Engine Analysis

Q_h (J)
Q_c (J)
W_out (J)

Van der Waals Equation

n (mol)
T (K)
V (L)
a (L²·atm/mol²)
b (L/mol)

Partition Function (Ideal Gas)

N molecules
T (K)
V (m³)
Mass (amu)
DoF

Maxwell Relations & Reference

Ideal Gas: PV = nRT, R = 8.314 J/(mol·K)
Carnot: η = 1 − T_c/T_h
ΔS: ∫dQ/T (reversible)
Helmholtz: F = U − TS
Gibbs: G = H − TS = U + PV − TS
Enthalpy: H = U + PV
Maxwell: (∂T/∂V)_S = −(∂P/∂S)_V
Van der Waals: (P+a·n²/V²)(V−nb) = nRT
Boltzmann: S = k_B ln Ω
Partition: Z = Σ exp(−ε_i/k_BT)

⚡ Electromagnetics Lab

Coulomb, Biot-Savart, Maxwell's equations, wave propagation, impedance, skin depth, Poynting vector, transmission lines, antenna basics

Coulomb's Law

q₁ (μC)
q₂ (μC)
r (m)

Biot-Savart (straight wire)

I (A)
r (m)

EM Wave Parameters

Frequency (Hz)
Or wavelength (m)

Skin Depth

f (Hz)
μ_r
σ (S/m)

Poynting Vector & Power

E (V/m)
H (A/m)
Area (m²)

Impedance & Admittance

R (Ω)
X_L (Ω)
X_C (Ω)

Transmission Line

Z₀ (Ω)
Z_L (Ω)
βl (rad)

Maxwell's Equations (Reference)

Gauss (E): ∇·E = ρ/ε₀
Gauss (B): ∇·B = 0
Faraday: ∇×E = −∂B/∂t
Ampère: ∇×B = μ₀J + μ₀ε₀ ∂E/∂t
Wave speed: c = 1/√(μ₀ε₀)
Impedance: Z₀ = √(μ₀/ε₀) ≈ 377 Ω
Skin depth: δ = √(2/(ωμσ))
Poynting: S = E × H (W/m²)

🤖 Robotics & Kinematics

Denavit-Hartenberg, forward/inverse kinematics, Jacobian, PID control, trajectory planning, workspace analysis, dynamics

2R Planar Forward Kinematics

L₁ (m)
L₂ (m)
θ₁ (°)
θ₂ (°)

2R Inverse Kinematics

L₁ (m)
L₂ (m)
x (m)
y (m)

Jacobian (2R Planar)

L₁
L₂
θ₁ (°)
θ₂ (°)

PID Controller Simulator

Kp
Ki
Kd
Setpoint
Disturbance
Steps

DH Parameter Table

Denavit-Hartenberg convention:
Tᵢ = Rot_z(θᵢ) · Trans_z(dᵢ) · Trans_x(aᵢ) · Rot_x(αᵢ)

iθᵢdᵢaᵢαᵢ
1θ₁0L₁0
2θ₂0L₂0

Reference

FK: x = L₁cosθ₁ + L₂cos(θ₁+θ₂), y = L₁sinθ₁ + L₂sin(θ₁+θ₂)
IK: cosθ₂ = (x²+y²−L₁²−L₂²)/(2L₁L₂)
J: 2×2 Jacobian maps joint velocities to end-effector velocity
Singularity: det(J) = 0 when arm is fully extended or folded
PID: u(t) = Kp·e + Ki·∫e dt + Kd·de/dt

🦠 Epidemiology & Disease Models

SIR, SEIR, SIS, SIRD compartmental models, R₀, herd immunity, vaccination thresholds, age-structured models, stochastic epidemics, spatial & network spread, intervention modeling

SIR Model Parameters

β (transmission)
γ (recovery)
N (population)
I₀ (initial infected)
Days

SEIR Model (with Exposed)

β
σ (incubation rate)
γ

Spatial SIR — Cellular Automaton

Each cell is S/I/R; infection spreads to 8 neighbors stochastically. Click cells to infect them.
β (cell-to-cell)
γ (recovery)
Grid size

R₀ & Herd Immunity

R₀
Vaccine efficacy

SIRD Model (with Deaths)

β
γ (recovery)
μ (mortality)

Effective R with Vaccination

R₀
Vaccinated %
Efficacy %

Network Epidemic — Agent-Based

Epidemic spreads across a random social network. Watch the wave of infection percolate through contacts.
Nodes
Avg degree
β (per edge)

Vaccination Campaign Comparator

Compare final attack rates across 0%, 30%, 60%, 90% vaccination coverage with the same R₀.
R₀
γ (recovery)

Reference

SIR: dS/dt=−βSI/N, dI/dt=βSI/N−γI, dR/dt=γI
SEIR: adds dE/dt=βSI/N−σE, dI/dt=σE−γI
SIRD: dD/dt=μI, recovery uses (γ+μ) not γ
R₀ = β/γ (SIR) or β/γ+μ (SIRD)
Herd immunity: p_c = 1−1/R₀
R_eff = R₀(1−p·ε)
Final size: S_∞ = S₀·exp(−R₀(1−S_∞/N))
Spatial: 8-neighbor cellular automaton
Network: Erdős–Rényi graph, per-edge transmission

🧠 Machine Learning Core

Linear & logistic regression, kNN, naive Bayes, PCA, gradient descent, cross-validation, confusion matrix, ROC, regularization

Linear Regression (OLS)

Data points (x,y per line, semicolon-sep)

Logistic Regression

Data (x, label 0/1 per line, semicolon-sep)

k-Nearest Neighbors

k
Query x
Training (x, label per line, semicolon-sep)

Gradient Descent Visualizer

f(x) = ax² + bx
b
Learning rate
x₀ (start)
Steps

Confusion Matrix

TP
FP
FN
TN

Naive Bayes (Binary)

P(A)
P(B|A)
P(B|¬A)

PCA (2D Projection)

2D points (x,y per line, semicolon-sep)

Reference

OLS: β = (XᵀX)⁻¹Xᵀy
Logistic: P(y=1) = 1/(1+e⁻ᶻ)
kNN: majority vote of k nearest
Naive Bayes: P(A|B) = P(B|A)P(A)/P(B)
PCA: eigenvectors of covariance matrix
GD: xₙ₊₁ = xₙ − η∇f(xₙ)
Accuracy: (TP+TN)/(TP+FP+FN+TN)
F1: 2·Prec·Rec/(Prec+Rec)

🔗 Markov Chains

Transition matrices, steady-state distribution, absorbing chains, hitting times, Chapman-Kolmogorov, hidden Markov models, MCMC, PageRank

Transition Matrix & Steady State

P matrix (rows semicolon-sep, each comma-sep)
Power n

Chapman-Kolmogorov (Pⁿ)

Same P matrix as above
Steps n

n-Step Transition Probability

From state i
To state j
Steps n

Absorbing Chain Analysis

P matrix (with absorbing states on diagonal=1)

PageRank

Adjacency matrix (rows=from, cols=to)
Damping d

Simulate Markov Chain

Start state
Steps

HMM Forward Algorithm

Observations (comma-sep indices)
States
Symbols

Reference

Markov property: P(Xₙ₊₁|X₀...Xₙ) = P(Xₙ₊₁|Xₙ)
Steady state: π = πP (left eigenvector, eigenvalue 1)
Chapman-Kolmogorov: P(m+n) = P(m)·P(n)
Absorbing: N = (I−Q)⁻¹, absorption prob B = NR
PageRank: π = (1−d)/n + d·π·M
HMM: α_t(i) = Σ_j α_{t-1}(j)·a_ji·b_i(o_t)
Detailed balance: πᵢPᵢⱼ = πⱼPⱼᵢ (reversible)

🎲 Stochastic Processes

Brownian motion, Poisson processes, martingales, Itô calculus, Ornstein-Uhlenbeck, Wiener process, ARIMA, GARCH, Lévy processes, diffusion

Brownian Motion Simulator

Steps
Δt
Paths

Poisson Process

λ (rate)
T (time)

Ornstein-Uhlenbeck Process

θ (mean reversion)
μ (long-term mean)
σ (volatility)

GARCH(1,1) Model

ω
α
β
Steps

Itô Integral Approximation

f(x,t) = x^a
Partitions

AR(1) Process

φ
c
σ_ε

Reference

Brownian: W(t) ~ N(0,t), increments independent
Poisson: P(N(t)=k) = (λt)^k e^{-λt}/k!
O-U: dX = θ(μ−X)dt + σdW
GARCH: σ²_t = ω + αε²_{t-1} + βσ²_{t-1}
Itô: ∫₀ᵀ f dW (mean-zero martingale)
Itô lemma: df = (∂f/∂t + μ∂f/∂x + ½σ²∂²f/∂x²)dt + σ∂f/∂x dW
AR(1): X_t = c + φX_{t-1} + ε_t
Stationarity: |φ| < 1 for AR(1)

🌊 Wavelet Analysis

Haar wavelet, Daubechies, CWT, DWT, multi-resolution analysis, denoising, compression, mother wavelets, scaling functions, wavelet packets

Haar DWT

Signal (comma-sep, power-of-2 length)
Levels

Wavelet Denoising

Generate: sin(2π·f·t) + noise
Freq
Noise σ
N (power of 2)
Threshold

Continuous Wavelet Transform (Morlet)

Signal: sin(2π·f·t) + 0.5·sin(2π·2f·t)
f
Scales

Wavelet Compression

Keep % coefficients
Signal

Scaling Function φ & Wavelet ψ

Wavelet

Multi-Resolution Analysis

MRA: L²(ℝ) = ⊕_j W_j (orthogonal decomposition)
V_j ⊂ V_{j+1}: nested approximation spaces
φ(x): scaling function, φ_{j,k}(x) = 2^{j/2}φ(2^j x − k)
ψ(x): mother wavelet, ψ_{j,k}(x) = 2^{j/2}ψ(2^j x − k)
Two-scale: φ(x) = √2 Σ h_k φ(2x−k)
Reconstruction: f = Σ c_{J,k} φ_{J,k} + Σ_{j≥J} d_{j,k} ψ_{j,k}

Reference

Haar: ψ(x) = 1 on [0,½), −1 on [½,1)
DWT: c_k = Σ h_n x_{2k+n}, d_k = Σ g_n x_{2k+n}
CWT: W(a,b) = 1/√|a| ∫ f(t)ψ((t−b)/a) dt
Morlet: ψ(t) = π^{-1/4}(e^{iω₀t}−e^{-ω₀²/2})e^{-t²/2}
Parseval: ∫|f|² = Σ|c_{j,k}|² (energy conservation)
Heisenberg: Δt·Δω ≥ ½ (uncertainty principle)

🔷 Algebraic Geometry

Affine & projective varieties, Gröbner bases, ideal operations, Hilbert's Nullstellensatz, Bézout's theorem, elliptic curves, rational curves, divisors & line bundles

Polynomial Ideal & Varieties

Polynomials (one per line, e.g. x^2+y^2-1)

Elliptic Curve y² = x³ + ax + b

a
b
Point P: x
y

Bézout's Theorem

deg(f)
deg(g)

Gröbner Basis Info

Ideal generators (one per line)
Monomial order

Projective Space ℙⁿ

n (dimension)

Discriminant & Singularities

Curve: y²=f(x), deg f
f coefficients (x⁰,...)

Reference

V(I): {x ∈ kⁿ : f(x)=0 ∀f∈I} (variety)
I(V): {f : f|_V = 0} (vanishing ideal)
Nullstellensatz: I(V(I)) = √I (over algebraically closed k)
Bézout: |V(f)∩V(g)| ≤ deg(f)·deg(g) (ℙ²)
Elliptic: y² = x³+ax+b, Δ = −16(4a³+27b²)
Group law: chord-tangent construction
Gröbner: canonical generating set for ideal
dim ℙⁿ: n, [ℙⁿ] = n+1 homogeneous coords