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.
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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
- Why Every Real Symmetric Matrix Is DiagonalizableA real symmetric matrix has an orthonormal basis of eigenvectors, so an orthogonal change of basis diagonalizes it.
- Prove That If x + 5 Is Odd, Then x² Is EvenFor integer x, x+5 odd forces x even, so x² is even. See the direct proof, a numerical example, and why the integer assumption matters.
- Why Is the Shortest Tangent Intercept of an Ellipse $a+b$?For an ellipse with positive semiaxes a and b, the shortest segment cut off by the coordinate axes on a tangent has length a+b.
- Why Every Square Matrix Splits into Symmetric and Skew-Symmetric PartsEvery real square matrix is uniquely the sum of a symmetric and a skew-symmetric matrix; use transpose to construct both parts and prove uniqueness.
- Is (A ∩ B) ∪ C = A ∩ (B ∪ C)?The proposed identity is false for arbitrary sets. See a counterexample, the correct distributive law, and the exact condition for equality.
- How to Prove That a Formal Grammar Is CorrectProve grammar correctness by showing soundness and completeness: the grammar generates every intended string and no unintended strings.
- How to Prove an Expression Is Not a Well-Formed FormulaTo prove an expression is not a well-formed formula, state the grammar and show that no formation rule can generate or parse it.
- How to Prove a Closed Form for n = k + 1 by InductionTo prove a closed-form formula at n=k+1, assume it holds at n=k, substitute into the next case, and simplify to the target expression.
- How to Prove the Inequality \(\frac{ab}{c^3}+\frac{bc}{a^3}+\frac{ca}{b^3}\geq\frac1a+\frac1b+\frac1c\)For positive a, b, and c, multiply by abc and apply the sum-of-squares inequality twice; equality holds exactly when a = b = c.
- How to Prove That Free Modules Are Flat Without Projective ModulesFor any ring, identify a free module with a direct sum of copies of the ring; tensoring turns an injection into a coordinatewise injection.
- How to Prove Minkowski Metric Invariance Under Lorentz TransformationsThe Minkowski metric is invariant because every Lorentz matrix satisfies ΛᵀηΛ = η, preserving the spacetime interval between events.
- How to Prove the Alternate Interior Angles TheoremAIP is an ambiguous geometry abbreviation. Under the common interpretation of alternate interior angles.