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
- What Are the Group Tables for a Group of Four Elements?A group with four elements is isomorphic to either the cyclic group C4 or the Klein four-group V4; see both multiplication tables and why.
- Does the Set of Real Numbers Form a Group?The real numbers form a group under addition, but not under multiplication; the nonzero real numbers form a multiplicative group.
- What Are Groupification and Perfection of a Commutative Monoid?Groupification embeds a commutative monoid into a universal group when cancellation holds; p-perfection adjoins compatible p-th roots via Frobenius.
- Hamilton’s Principal and Characteristic Functions: What’s the Difference?Hamilton’s principal function S depends on position and time; for a time-independent system, its characteristic function W satisfies a fixed-energy equation.
- How Does Hamilton’s Principle Work with Semiholonomic Constraints?For nonintegrable velocity constraints, distinguish Hamilton’s principle on constrained paths from the d’Alembert variation used for ideal constraints.
- Has the Speed of Light Changed Over Time?The vacuum speed of light is exactly 299,792,458 m/s by SI definition; whether it varied cosmologically is a separate, dimensionless, unconfirmed question.
- How to Put a Table of Contents in a Beamer HeadlineUse Beamer’s built-in section navigation for a compact headline, and put a multicol table of contents on a regular frame for reliable columns.
- What Can You Use Instead of the Time-Averaged Poynting Vector?For periodic fields, use the cycle-averaged Poynting vector; for pulses, integrate instantaneous flux over time. The right choice depends on what you need.
- How to Change Comment Layout in algorithmicx PseudocodeRedefine algorithmicx’s algorithmiccomment command to move comments, change their marker or color, or place them on a separate pseudocode line.
- How Can You Imagine Curled-Up Dimensions?Picture a curled-up dimension as a tiny circular direction at every point in space, like the cross-section of a hose seen from far away.
- Are Sulfate Bonds Nonpolar Covalent or Ionic?Sulfate is a 2− polyatomic ion with polar covalent sulfur–oxygen bonds; ionic attraction occurs between sulfate and counterions in salts.
- Why Does a Polygon Have the Same Number of Vertices and Sides?A polygon has the same number of vertices and sides because each side joins consecutive vertices in one closed cycle.