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
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- Can 7 Drinks with 7 Flavors Make Infinite Varieties?With continuously variable proportions, seven flavors produce uncountably many mathematical mixtures; counting only flavor subsets gives 127 nonempty choices.
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- Does a Group Generated by Two Involutions Have to Be Infinite?A group generated by two elements of order 2 need not be infinite or non-abelian; its structure is determined by the order of their product.
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- How to Prove (∀x P(x) → A) → ∃x(P(x) → A)A classical proof uses cases on A and requires a nonempty domain; the statement is not generally valid in intuitionistic logic.
- What Is the Laplace Transform of $t e^{at}\sin(bt)$?The Laplace transform of t e^{at} sin(bt) is 2b(s−a)/([(s−a)^2+b^2]^2), for real a, nonzero b, and Re(s)>a.