Science and math questions, solved step by step.
Worked solutions in mathematics, statistics, physics, chemistry, biology, computer science and LaTeX, each checked before publication, plus free calculators that explain every step.
Browse by topic
All topics →Mathematics
Algebra, calculus, analysis, topology, linear algebra and number theory, worked out step by step.
Statistics & Probability
Distributions, estimators, expected values and hypothesis tests with clear derivations.
Physics
Mechanics, electromagnetism, thermodynamics, relativity and quantum physics explained.
Chemistry
Reactions, bonding, equilibrium, thermochemistry and lab calculations.
Biology
Evolution, biochemistry, physiology and ecology, answered with evidence.
Computer Science
Algorithms, complexity, automata, number representation and digital logic.
LaTeX & TikZ
Working solutions for tables, figures, bibliographies, TikZ and pgfplots.
Academia
Publishing, peer review, PhD life and research careers.
Calculators
All calculators →Quartiles
Find Q1, median, Q3 and IQR from a list of values or from histogram bins (grouped data), with every step shown.
Standard deviation
Sample and population standard deviation, variance and mean, with the full sum-of-squares working.
Normal probability
Given the mean and standard deviation, find P(X < b), P(X > a) or P(a < X < b) with z-scores.
Dice probability
Exact distribution, mean and standard deviation for the sum of any number of fair dice.
Matrix
Determinant with cofactor expansion, inverse, symmetric eigen-decomposition and singular values.
Boolean simplifier
Turn minterms or a truth table into a minimal sum-of-products expression using Quine–McCluskey.
Recently updated
- Asymptotics of the Central Binomial CoefficientUsing Stirling’s formula, the central binomial coefficient satisfies C(2n,n) ~ 4^n divided by √(πn), with useful correction terms.
- Average or Individual Molar Heat Capacity?Learn when to use individual or average molar heat capacities, with formulas for pure substances, mixtures, temperature ranges, and reactions.
- Bead on a Rotating Hoop: Equilibrium and StabilityDerive the motion, equilibrium positions, and stability of a frictionless bead on a hoop rotating about its vertical diameter.
- Best Practices for Publishing Computer Code with LaTeXLearn when to use verbatim, listings, or minted, with complete LaTeX examples for readable, maintainable, and safe code publication.
- How to Calculate Probabilities in a Markov ChainCalculate Markov-chain probabilities using transition matrices, initial distributions, matrix powers, path products, and hitting-probability equations.
- Can We Prove the Irrationals Are Uncountable Without Cantor’s Diagonal Argument?Yes. A nested-interval argument proves the real numbers, and therefore the irrationals, are uncountable without decimal diagonalization.
- Christoffel Symbols in Terms of $\sqrt{|g|}$The contracted Christoffel symbols satisfy Γᵘᵢᵤ = ∂ᵢ ln√|g| = (1/√|g|)∂ᵢ√|g|, with g the metric determinant.
- How to Combine Transformation MatricesCombine transformation matrices by multiplying them in reverse application order for column vectors, including translations with homogeneous coordinates.
- How Does Paraffin Combustion Work?Learn how paraffin wax burns, how to balance its complete-combustion equation, and why limited oxygen produces soot and carbon monoxide.
- How Are Computer Science Conferences Ranked?Computer science has no universal conference ranking: evaluate venues by subfield reputation, review standards, selectivity, program quality, and indexing.
- When Is the Absolute Value of a Function Continuous?If f is continuous, then |f| is continuous. The converse is generally false, with a simple sign-changing counterexample.
- When Is a Proper Injective Map a Homeomorphism onto Its Image?A continuous proper injective map into a first-countable Hausdorff space is an embedding; without Hausdorffness, the statement can fail.