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
- Superposition of Wavefunctions: Meaning and ExamplesLearn when wavefunctions can be superposed, how to normalize the result, and why probability amplitudes produce quantum interference.
- What Is the Support of a Regular Borel Measure?For a regular Borel measure, its support is the closed set of points whose every open neighborhood has positive measure; regularity makes its complement null.
- How Many Whole Numbers Are Between 100 and 999?There are 900 whole numbers from 100 through 999. For a uniform random choice, count the numbers meeting the event and divide by 900.
- Prove That A Is a Subset of C If A × B ⊆ B × CUse nonemptiness to compare the coordinates in the Cartesian-product inclusion: it gives A ⊆ B and B ⊆ C, hence A ⊆ C.
- How to Find Suprema and Infima Without ProofFind a set’s supremum and infimum by checking its bounds and endpoints, then distinguish an attained maximum or minimum from a limiting value.
- Supremum and Infimum of x/sin x on [0, π/2]The quotient x/sin x is undefined at zero. On (0, π/2] its infimum is 1 and its maximum, hence supremum, is π/2.
- What Is the Supremum of the Negative of a Function?For a nonempty function range bounded below, the supremum of its negative is the negative of its infimum; attainment is a separate issue.
- Surface Area When y = √(x + 1) Rotates About the x-AxisThe curve y = √(x + 1) has no single surface area without bounds. On [a, b], its area of revolution is π/6[(4b + 5)^(3/2) − (4a + 5)^(3/2)].
- How to Solve Homogeneous Equations with the SVDUse the singular value decomposition to solve Ax = 0: the null space is spanned by right singular vectors with zero singular values.
- Swapping a Limit and an Integral: A CounterexampleSee why limits and integrals cannot always be interchanged, using a spike counterexample and standard conditions that make the swap valid.
- How to Run Sweave from a Windows Batch FileUse a Windows batch file to run Sweave with Rscript, then compile its generated .tex file with MiKTeX; includes paths and error checks.
- Can a TI-84 Perform Symbolic Algebra?A standard TI-84 cannot simplify expressions or solve equations symbolically, but it can evaluate numbers, graph functions, and find numerical roots.