Core Math & Algebra

Adjacency List Graph Adjacency List Generator Paste your edges, choose whether the graph is directed/undirected and weighted/unweighted, and we’ll build the adjacency list for you. We also show a JSON version and an adjacency matrix so you can compare different graph representations. 1. Graph input Accepted edge formats (one per line): A B , A,B , A B 5 (weighted), 1 2 , node_1 node_2 1.25 . We auto-detect separators. A B A C B D C D 2 D E Directed graph Weighted edges Sort nodes Generate Adjacency list JSON Adjacency matrix No graph generated yet. {"A": ["B","C"], "B":["D"], "C":["D"], "D":["E"]} Adjacency list vs adjacency matrix For a graph \( G = (V, E) \): Adjacency list: store, for every vertex \(v\), the list of neighbors it connects to. Space is \(O(|V| + |E|)\). Adjacency matrix: store a full \( |V|\times|V| \) table. Space is \(O(|V|^2)\), but lookups are \(O(1)\). When to use which Use an adjacency list for sparse graphs, typical interview problems, and traversals like BFS/DFS. Use an adjacency matrix when you need instant edge existence checks or when the graph is dense. Weighted edges If you tick “weighted”, neighbors will be shown as node(weight) , e.g. C(2) , so you can directly feed it to Dijkstra or Prim implementations. Related Graph Tools Graphing & Visualization Core Math & Algebra Adjacency List (this page) Logistic Regression Tips Keep node names consistent (use either letters or numbers). For directed graphs, remember that edge A→B does not imply B→A.

Atbash Cipher Atbash Cipher Encoder/Decoder Type or paste your text to encode or decode using the classic Atbash monoalphabetic substitution cipher. The tool works both ways because Atbash is its own inverse. 1. Input We support A–Z and a–z. Non-letters (numbers, spaces, punctuation) are kept unchanged. HELLO WORLD Preserve original case Encode

Baconian Cipher Encoder/Decoder Baconian Cipher Encoder/Decoder (BAC) Turn text into Bacon’s 5-letter A/B code or decode existing A/B groups back to normal letters. Supports classic 24-letter Bacon, 26-letter extension, and custom symbols like 0/1 or X/Y. You can even map it onto a “cover text” using uppercase/lowercase steganography. Encode (text ➜ A/B) Decode (A/B ➜ text) Alphabet variant Original 24-letter (I=J, U=V) Modern 26-letter (all letters) Choose the same variant your source used. Symbol for “A” Symbol for “B” Group in 5s Keep non-alphabet chars Input text Francis Bacon Output Convert Clear Optional: hide A/B in a cover text Paste a cover text. We will replace characters with uppercase (=A) and lowercase (=B) according to your encoded sequence. Extra cover characters will stay the same. Cover text Stego output (A=UPPER, B=lower) Build stego text How the Baconian cipher works The cipher turns every letter into a block of 5 characters, each either “A” or “B”. For example, in the original 24-letter alphabet: A = AAAAA B = AAAAB C = AAABA ... I/J = ABAAA U/V = BAABB Once you have the A/B sequence, you can hide it in many ways: two fonts, two typefaces, two colors, or simply upper/lowercase. A recipient who knows the scheme can map it back to As and Bs and then to letters. 24 vs 26 letters 24-letter (classic): I=J and U=V are merged to fit 24 patterns. 26-letter (extended): every letter A–Z is unique. Pick this if you don’t want merged letters. Common decoding problems Make sure the symbol you use for A and B is the same throughout. If the encoded message length is not a multiple of 5, the last letter may be incomplete. If the original text used the 24-letter alphabet, decoding with 26-letter mode will produce odd results. Related Crypto & Encoding Tools Caesar Cipher Encoder/Decoder Atbash Cipher Substitution Cipher Solver RSA Encryption/Decryption Baconian Cipher (alt) Tips for stronger stego Use a cover text of similar length to your encoded text. Mix in punctuation and numbers. Don’t repeat very obvious upper/lowercase patterns.

Big Number Big Number Calculator Do arithmetic on huge integers (hundreds or thousands of digits), format them nicely, and get a human name when possible. 1. Enter your numbers Digits only. Hyphens, commas, and spaces will be stripped. You can paste very long numbers. Number A 123456789012345678901234567890 Number B 98765432109876543210 2. Choose operation A + B A − B A × B Format A Multiplication is okay for a few thousand digits combined. For extremely large inputs, we’ll warn you. 3. Result Raw big integer — With thousands separators — Scientific notation (approx) — English name (short scale) — How big numbers are handled here JavaScript’s normal numbers (double precision) break at 9,007,199,254,740,991 (2 53 −1). To work with numbers bigger than that, our calculator keeps them as strings of digits and performs arithmetic the way you would do it by hand. Supported operations Addition : arbitrary length. Subtraction : if A ≥ B. Multiplication : grade-school style; very large inputs may slow down. Formatting : comma grouping, scientific notation, and name lookup (thousand, million, billion, …). Naming large numbers We use the US/short scale: thousand (10³), million (10⁶), billion (10⁹), trillion (10¹²), quadrillion, …, centillion (10³⁰³) . Above that, we’ll tell you it’s out of our name table. Related tools Number to Words Converter Words to Numbers Converter Roman Numeral Converter Core Math & Algebra When to use this Good for cryptography exercises (non-secure), factorial growth examples, combinatorics, educational “how big is it?” demos, and handling IDs too large for normal math.

Big-O Complexity Big-O Complexity Explorer Pick a complexity class, plug in n, and see how fast it grows. Then compare with real algorithms like sorting, search, or graph traversals. Quick Explorer Compare Classes Reference 1. Choose complexity & input size Complexity O(1) O(log n) O(n) O(n log n) O(n²) O(2ⁿ) O(n!) n (input size) Try 10, 100, 1000… Estimate growth 2. Output Complexity — Estimated steps — Growth feeling — Example — Compare common complexities for the same n n Compare Class Formula used Steps @ n Comment Click “Compare” to fill the table. Common algorithms and Big-O Algorithm Time Notes Linear search O(n) Scan one by one. Binary search O(log n) Requires sorted array. Merge sort O(n log n) Optimal general sort. Bubble sort (worst) O(n²) Educational, not optimal. DFS

Checksum Calculator Checksum Calculator Compute CRCs and simple checksums directly in your browser. Paste binary/hex/text payloads from embedded or SCADA systems and compare the result with your device. 1. Input data Select input type and paste your data. Hex can be space/comma separated or continuous. Input type Text

Correlation Coefficient Correlation Coefficient Calculator (Pearson’s r) Paste two columns of numbers and we’ll compute the Pearson correlation coefficient r, r², sample size, and the best-fit line. This is the classic statistic to measure linear association. Enter your data Separate values with comma, space, semicolon, or line break. X and Y must have the same length. X values variable X 10, 12, 15, 18, 20 Y values variable Y 8, 11, 14, 17, 19 Decimal places 2 3 4 Calculate correlation Results Pearson r – r² (explained) – n (pairs) – Direction – Least-squares regression line (Y on X) – Interpretation: for every 1-unit increase in X, Y changes by the slope. Correlation coefficient formula This calculator uses the sample Pearson correlation coefficient: r = Σ[(xᵢ − x̄)(yᵢ − ȳ)]

Depth-First Search (DFS) Depth-First Search (DFS) Interactive Tool Model your graph with an adjacency list, pick a start node, and see the DFS visit order, timestamps, and edge classification. Ideal for studying algorithms, preparing technical interviews, or debugging graph code. 1. Graph input Format: one node per line, use colon then space-separated neighbors. Example: A: B C B: D C: D E D: F E: F: A: B C B: D C: D E D: F E: F: Start node Directed graph Run DFS Clear 2. DFS results Visit order — Node Discovery time (d) Finish time (f) Parent Run DFS to populate Edge classification (directed) We classify edges as tree, back, forward, or cross (for directed graphs). For undirected graphs, non-tree edges are typically back edges. — How DFS works Depth-first search explores a graph “deeply”: from the current node, it visits one unvisited neighbor, then one of its neighbors, and so on, backtracking when no unvisited neighbors remain. This tool follows the classic CLRS-style DFS with discovery and finish timestamps. DFS(G): for each vertex u in G.V: color[u] = WHITE, parent[u] = NIL time = 0 for each vertex u in G.V: if color[u] == WHITE: DFS-Visit(u) DFS is fundamental in algorithms for: Topological sort (on DAGs) Detecting cycles Finding connected components Classifying edges (tree, back, forward, cross) Time complexity With an adjacency list, DFS runs in O(V + E) , which is optimal for a full traversal. Directed vs undirected In directed graphs, edge classification is meaningful (tree/back/forward/cross). In undirected graphs, every non-tree edge is effectively a back edge to an ancestor. Related graph tools Adjacency List Builder Graphing & Visualization Bayesian Network Tips Keep node labels simple (A, B, C or 1, 2, 3). Our parser ignores blank lines and trims spaces. If the start node is missing, DFS will still run on all nodes in alphabetical order.

F-Test Calculator F-Test Calculator (Two-Sample Variance Test) Compare two sample variances and test whether the underlying population variances are equal. We automatically put the larger variance on top. 1. Sample data You can enter either the sample variance (s²) or the sample standard deviation (s). If you fill both for a sample, we will use the variance. Sample 1 n₁ (sample size) s₁² (variance) s₁ (std dev) Sample 2 n₂ (sample size) s₂² (variance) s₂ (std dev) 2. Test setup Significance (α) Tail Two-tailed Right-tailed (var1 > var2) Left-tailed (var1 < var2) Calculate F-test Two-tailed is the default for testing equality of variances. 3. Results F statistic — df₁ — df₂ — p-value — How the F-test works The classical F-test for two variances tests: H₀: σ₁² = σ₂² H₁: σ₁² ≠ σ₂² (two-tailed) or σ₁² > σ₂²

Graphing & Visualization Graphing & Visualization Playground Three mini-visualization tools in one: plot math functions, make instant charts from data, or draw simple graphs. Designed to be lighter than Tableau/RAWGraphs but faster for quick checks. 1) Function Plotter 2) Data Chart 3) Node/Edge Graph Enter a function of x, e.g. sin(x) , x*x - 2*x , exp(-x)*sin(3*x) . Uses radians. f(x) Color (CSS) x min x max samples Draw Paste numbers separated by commas or line breaks. Optionally add labels as first line: Jan,Feb,Mar then second line 10,20,15 . Data Chart type Line Bar Color Title Draw chart Enter edges as A-B per line or comma-separated. We'll auto-place nodes in a circle. Node color Edge color Draw graph Canvas Tip: bigger ranges or very large data will auto-scale to fit. About graphing & visualization Visualization tools all do the same 3 things: map data ➜ to visual variables (x, y, color, size), choose a layout, and render to a surface. Here we’re doing it in the simplest way possible—directly in canvas. Supported mini-workflows Function plotter: convert f(x) into points and draw a polyline. Data chart: parse rows & columns; scale to canvas; draw axes and shapes. Node/edge graph: parse edges; find unique nodes; place them on a circle; connect. When to use a heavier tool If you need interactions, large datasets, or specific chart types (Sankey, violin, force-directed with physics), tools like RAWGraphs, Tableau, or d3-based apps will be more flexible. Related Math & Data Core Math & Algebra Correlation Coefficient Confidence Interval Geometry Visualization tips Label your axes. Use consistent colors. Don’t show more precision than the data supports. And always tell viewers what the chart is about.

Hill Cipher Hill Cipher Encoder/Decoder Enter a 2×2 key matrix (mod 26), check invertibility, then encrypt or decrypt text. We strip non-letters and pad with X if needed. 1. Key matrix (2×2) Values are integers mod 26 (0–25). A common demo key is [[3, 3], [2, 5]]. Check key invertibility 2. Text Input text (we keep letters A–Z, uppercase, pad with X) ATTACKATDAWN For decryption, paste ciphertext here and click “Decrypt”. 3. Actions Encrypt Decrypt 4. Output Result — Numeric vector (A=0 … Z=25) — How the Hill cipher works (2×2) 1. Choose a 2×2 key matrix K with entries mod 26. 2. Convert text to numbers using A=0, B=1, …, Z=25, and group into 2-element column vectors. 3. For each vector P , compute C = K · P mod 26 . Convert back to letters. K = [a b; c d], det(K) = ad − bc. K is invertible mod 26 if gcd(det(K), 26) = 1. To decrypt, we need the inverse matrix K⁻¹ mod 26 so that P = K⁻¹ · C mod 26 . This page computes that for you. Related cryptography tools RSA Encryption/Decryption Baconian Cipher Encoder/Decoder Checksum Calculator Big Number Calculator Tips If your key is not invertible mod 26, tweak one entry or pick a different key. A determinant with gcd(det, 26)=1 is required.

Hypergeometric Distribution Hypergeometric Distribution Calculator Model “successes without replacement” — for example, drawing red balls from an urn. Enter N, K, n, k and get the probability and distribution table. 1. Input parameters Population size (N) Total items in the population Number of success states (K) How many of the N are “success” Sample size (n) Drawn without replacement Observed successes (k) Exact number you want the probability for Calculate 2. Results P(X = k) — Cumulative P(X ≤ k): — P(X ≥ k): — Mean — Variance — 3. Distribution table All valid k from 0 to n (subject to K and N). Handy for homework

IQR Interquartile Range (IQR) Calculator Paste your dataset, and this tool will sort it, calculate Q1, median, Q3, the IQR, and show the usual Tukey outlier fences. 1. Enter your data You can separate values with commas, spaces, or line breaks. Example: 12, 15, 15, 18, 21, 22, 26 12, 15, 15, 18, 21, 22, 26 Calculate IQR Reset 2. Results Q1 (25%) — Median (Q2) — Q3 (75%) — IQR (Q3 − Q1) — Count (n) — Outlier fences (Tukey) Lower fence: — Upper fence: — Any value < lower fence or > upper fence is a potential outlier. Detected outliers — 3. Sorted data — What is the IQR? The interquartile range (IQR) is a robust measure of variability. It tells you how spread out the middle 50% of your values are. Because it ignores the lowest 25% and highest 25% of data, it is much less sensitive to outliers than the full range. How we computed your quartiles This tool uses the classic “median-of-halves” method (Tukey style): Sort the data. Find the median (Q2). If n is odd, exclude the median from both halves. Q1 = median of the lower half, Q3 = median of the upper half. IQR = Q3 − Q1 Lower fence = Q1 − 1.5 × IQR Upper fence = Q3 + 1.5 × IQR Why IQR is useful It’s used in boxplots . It’s a good default for outlier detection . It’s robust and easy to explain. Related statistics tools Covariance Confidence Interval p-value Tip If you have fewer than 4 data points, the IQR won’t tell you much. Try to collect more data for a meaningful spread.

ISBN Validator ISBN Validator Validate ISBN-10 and ISBN-13, strip hyphens and spaces, see checksum steps, and convert ISBN-10 → ISBN-13. 1. Enter ISBN You can paste: 978-0-306-40615-7 or 0-306-40615-2 or even 0306406152 . Validate ISBN 2. Validation Result Type detected — Is valid? — Normalized ISBN — 3. Conversion ISBN-10 → ISBN-13 only when valid. ISBN-13 → ISBN-10 only when it starts with 978. ISBN-10 ➜ ISBN-13 — ISBN-13 ➜ ISBN-10 — 4. Checksum Steps Validate an ISBN to see step-by-step checksum here. How ISBN validation works ISBN-10 For ISBN-10, after stripping hyphens, you must have 10 characters. The checksum is: sum(i × d i ) for i = 1..10 must be divisible by 11 The last digit can be X which means 10. ISBN-13 For ISBN-13, after stripping hyphens, you must have 13 digits. Multiply digits alternately by 1 and 3, sum them, and the total must be divisible by 10. checksum = (d1×1 + d2×3 + d3×1 + d4×3 + ... + d12×3 + d13×1) mod 10 = 0 This tool implements exactly these algorithms so you can trust the result. Related tools Checksum Calculator Number to Words Converter Words to Numbers Converter Core Math & Algebra Why validate ISBNs? To catch typos in book databases, library records, ecommerce listings, or form inputs before saving them.

Logistic Regression Logistic Regression Calculator Train a binary logistic regression model directly in your browser. Paste your data, click train, and get the intercept, weights, predicted probabilities, and log-odds interpretation. 1. Paste your dataset Format: each row = 1 observation. Last column = target (0 or 1). Columns separated by comma, semicolon, tab, or space. First row can be a header. age,income,clicked 25,35000,0 32,48000,1 41,62000,1 23,27000,0 36,54000,1 2. Training parameters Learning rate Epochs Decimals 3 4 6 Train 3. Model output No model trained yet. 4. Predict with the trained model After training, we’ll show the feature list. Enter values in order and get the probability \( P(y=1 \mid x) \). Predict Logistic regression essentials Logistic regression models the log-odds (logit) of the positive class as a linear combination of predictors: logit(p) = ln( p

P-value P-value Calculator Enter your test statistic, pick the distribution, and instantly see the p-value. Works for z, t, chi-square, and F. 1. Test input Test

Q-Q Plot Generator Q-Q Plot Generator Check normality (and more) in seconds. Paste your data, pick the distribution, and see the Q-Q plot with a reference line. 1. Paste your data Use comma, space or new line separated values. Example: 12, 10.5, 9, 13, 14.2 ... 12 10.5 9 13 14.2 11.7 10.9 12.5 9.5 15.1 2. Choose distribution & options Theoretical distribution Normal (estimate μ, σ) Uniform(0,1) Exponential(λ=1) Normal is the most common for normality tests. Reference line Least-squares fit line Line through quartiles None Generate Q-Q plot 3. Q-Q plot — 4. Sample vs Theoretical Quantiles # Ordered sample Theoretical quantile Difference How this Q-Q plot generator works This tool sorts your sample into order statistics \(x_{(1)}, \dots, x_{(n)}\) and pairs each one with a corresponding theoretical quantile from the chosen distribution. For the normal distribution, we first estimate the sample mean \(\hat{\mu}\) and standard deviation \(\hat{\sigma}\), then transform standard normal quantiles to match. Formula for the hyperbolic plotting positions We use the popular rule: p_i = (i - 0.5)

Roman Numeral Converter Roman Numeral Converter Convert between Arabic numbers and Roman numerals, validate your Roman strings, and even convert dates like 2025-11-07 → VII • XI • MMXXV. 1. Number ➜ Roman Enter a number (1–3999) Convert to Roman Roman numeral — 2. Roman ➜ Number Enter a Roman numeral (I–MMMCMXCIX) Convert to Number Arabic number — 3. Date ➜ Roman Useful for inscriptions or fancy headings. We convert day, month, year separately. Select a date Convert date Date in Roman numerals — Roman numeral rules Roman numerals are built from these symbols: Symbol Value Notes I 1 V 5 X 10 L 50 C 100 D 500 M 1000 IV, IX 4, 9 Subtractive notation XL, XC 40, 90 Subtractive notation CD, CM 400, 900 Subtractive notation This tool enforces standard modern form (no overlines, no 5000+, no non-standard repetitions). Related Math & Conversions Number to Words Converter Words to Numbers Converter Measurement Unit Conversions Core Math & Algebra Common conversions 2024 → MMXXIV, 1999 → MCMXCIX, 44 → XLIV If you get “invalid Roman numeral”, check for too many repetitions (e.g. IIII), or illegal subtractive pairs (like IL).

RSA RSA Encryption

Runge-Kutta Runge-Kutta (RK4) Method Calculator Numerically solve a first-order ODE \( \frac{dy}{dx} = f(x, y) \) using the classical Runge–Kutta method of order 4 (RK4). We show every intermediate slope (k1–k4) and the resulting y values. 1. Problem setup Use JavaScript/Math syntax: x + y , x*y , Math.sin(x) , y - x*x . We already expose common Math functions via with(Math){...} . dy/dx = f(x, y) Number of steps x₀ y(x₀) Step size h Run RK4 2. Solution steps No computation yet. Fill the form and click “Run RK4”. Runge–Kutta 4th order formula Given \(\frac{dy}{dx} = f(x, y)\) and a current point \((x_n, y_n)\) with step \(h\): k₁ = f(xₙ, yₙ) k₂ = f(xₙ + h/2, yₙ + h·k₁/2) k₃ = f(xₙ + h/2, yₙ + h·k₂/2) k₄ = f(xₙ + h, yₙ + h·k₃) yₙ₊₁ = yₙ + (h/6)(k₁ + 2k₂ + 2k₃ + k₄) This method is popular because it strikes an excellent balance between accuracy and computational cost. Tips If the solution diverges, try a smaller step size \(h\). For stiff ODEs, RK4 might not be the best choice. Always check your function syntax if you get NaN results. Related Math Tools Core Math & Algebra Graphing & Visualization Correlation Coefficient Confidence Interval When RK4 is overkill If you just need a quick rough value, Euler’s method is simpler. RK4 is ideal when you want good accuracy but don’t want to implement adaptive step-size methods.

Statistical Power Statistical Power Calculator Estimate the power of a two-group experiment or A/B test from effect size, sample size, and alpha. Supports tests on means and on proportions. 1. Choose test type Difference in means (two-sample, equal n) Difference in proportions (two-sample, equal n) 2A. Inputs for means Example: control mean = 10, treatment mean = 12 → effect size = 2. SD is the common standard deviation. Mean (group A) Mean (group B) Standard deviation (common) Sample size per group 2B. Inputs for proportions Example: baseline conversion p1 = 0.10, new variant p2 = 0.13. Proportion p1 (group A) Proportion p2 (group B) Sample size per group 3. Test settings Alpha (significance) Tails Two-tailed One-tailed Calculate power 4. Results Estimated power — Effect size (absolute) — Indicative n per group for 80% power — What is statistical power? Statistical power is the probability that your test will detect an effect if the effect is really there. Low power → high chance of false negatives. Key determinants of power Effect size : bigger differences are easier to detect. Sample size : more data → narrower standard error → higher power. Alpha : a higher alpha (e.g. 0.1) makes it easier to reject H0, increasing power. Variability : lower standard deviation → higher power. Formula idea (z-approximation) For a two-sample test on means (equal n), the test statistic roughly follows z = (μ₂ − μ₁)

t-test Calculator t-test Calculator Choose the t-test type, enter your sample statistics, and get the t-value, degrees of freedom, and p-value (one- or two-tailed). This works for small samples and unknown population variance. 1) One-sample t-test 2) Independent t-test 3) Paired t-test Sample mean Sample SD Sample size (n) Hypothesized mean (μ₀) Assumes equal variances (pooled) by default. Group 1 mean Group 2 mean Group 1 SD Group 2 SD Group 1 size (n₁) Group 2 size (n₂) Assume equal variances (Student's t). Uncheck for Welch's correction. Enter summary stats of the paired differences (After - Before). Mean of differences SD of differences Pairs (n) Tail Two-tailed (≠) Right-tailed (>) Left-tailed (<) Significance α Used for interpretation only. Calculate t-test Results t statistic – Degrees of freedom – p-value – How this t-test works This calculator follows standard Student's t-test formulas. 1. One-sample t-test Used to test whether a sample mean differs from a hypothesized population mean μ₀. t = (x̄ − μ₀)

Words to Numbers Converter Words to Numbers Converter Turn spelled-out English numbers into digits. Works for “one hundred and twenty three”, “two thousand five”, “minus forty two”, and even decimals like “three point one four”. 1. Paste or type your number words Tip: you can paste an entire sentence — we’ll try to extract the numeric part. two thousand and fifty-six point seven five Convert to number 2. Result Numeric value — Examples you can try five hundred and twelve one million two hundred thirty four thousand five hundred sixty seven negative forty two three point one four one five nine How the English words → number logic works English number words follow a hierarchical pattern: small numbers (zero–nineteen), tens (twenty, thirty, ...), multipliers (hundred, thousand, million, billion). We scan your text token by token, sum groups, and apply multipliers when we find them. Supported words Base: zero, one, two, three, ... nineteen Tens: twenty, thirty, forty, fifty, sixty, seventy, eighty, ninety Multipliers: hundred, thousand, million, billion Connectors: and Sign: minus, negative Decimals: point (then digits as words: “point five six” → 0.56) Limitations This page is tuned for standard English financial/academic phrasing. Very free-form text or mixed languages may not convert cleanly. Related tools Number to Words Converter Core Math & Algebra Measurement Unit Conversions Why convert words to numbers? Useful for invoices, contracts, OCR text, exam answers, and datasets where numbers have been spelled out.