Frequently Asked Questions

Start on the Topics page, choose an area and level, open a topic, play with the interactive visual, take the mini-quiz and mark it complete. Then test yourself in Quiz or Tests and track results in Progress.

Open Topics → Arithmetic & Number Sense and filter by Beginner. Work through counting, place value and the four operations, then move to fractions, decimals and percentages before starting Algebra.

Beginner (arithmetic and early geometry), Intermediate (school algebra, geometry, trigonometry basics), Advanced (calculus, linear algebra, statistics, proofs) and Expert (university-level analysis, abstract algebra, topology, differential equations).

Yes, completed topics, quiz history and your streak are stored in your browser’s local storage. Clearing site data or using a different browser starts fresh. You can reset on the Progress page.

Press / or Ctrl+K anywhere, then type. Results cover topic names, summaries, formulas and FAQ entries; arrow keys move through results and Enter opens one.

Quiz gives instant feedback and explanations after every question. Tests are timed, let you move between questions, and only reveal answers at the end with a detailed review.

No. Numeric questions are generated randomly and topic questions draw random distractors, so every attempt is different.

Division asks “what number times 0 gives a?”. If a ≠ 0 no number works, and if a = 0 every number works, so the result is either impossible or ambiguous. Hence it is left undefined.

Yes. Let x = 0.999…; then 10x = 9.999…, so 9x = 9 and x = 1. Equivalently, the geometric series 9/10 + 9/100 + … sums to exactly 1.

To keep the distributive law valid: 0 = (−1)(1 + (−1)) = −1 + (−1)(−1), so (−1)(−1) must equal 1. It also means “reversing a reversal”.

Because aⁿ ÷ aⁿ = a⁰ by the exponent rule, and anything (non-zero) divided by itself is 1. It keeps the pattern aⁿ⁺ᵐ = aⁿ·aᵐ consistent.

No. A prime has exactly two distinct positive divisors; 1 has only one. Excluding 1 keeps the Fundamental Theorem of Arithmetic (unique prime factorisation) true.

Permutations count arrangements where order matters (P(n,r) = n!/(n−r)!). Combinations count selections where order does not matter (C(n,r) = n!/(r!(n−r)!)).

A derivative measures instantaneous rate of change (slope); an integral accumulates quantities (area). The Fundamental Theorem of Calculus shows they are inverse operations.

Radian measure makes formulas simple: d/dx sin x = cos x only holds in radians, and arc length is just s = rθ. Degrees are an arbitrary convention based on 360.

i is defined by i² = −1. Complex numbers a + bi extend the reals so every polynomial has roots, and they describe rotations and waves naturally.

The value a function gets arbitrarily close to as the input approaches a point, regardless of the value at the point itself.

Mean is the sum divided by the count, median is the middle value and mode is the most frequent value. Median resists outliers; the mean does not.

It is the typical distance of data points from the mean. Small values mean data clusters tightly; large values mean it is spread out.

Correlation means two variables move together; causation means one produces the other. Ice-cream sales and drownings correlate because both rise in summer, not because one causes the other.

Matrices represent linear transformations and systems of equations. They power computer graphics, machine learning, networks, physics and economics.

A proof is a logical argument showing a statement must be true in all cases, not just examples. Patterns can fail; proofs cannot.

φ = (1+√5)/2 ≈ 1.618, the positive solution of x² = x + 1. Ratios of consecutive Fibonacci numbers approach it.

π cannot be written as a fraction of integers; Lambert proved this in 1761. Its decimals never end or repeat, and π is also transcendental (not a root of any integer polynomial).

Seven famous unsolved problems chosen by the Clay Mathematics Institute (such as the Riemann Hypothesis and P vs NP), each with a $1 million prize. Only the Poincaré conjecture has been solved.

Consistency beats cramming: 30–45 focused minutes daily with active practice (solving problems) beats hours of passive reading. Use the streak counter to stay motivated.

Read twice, draw a picture, define variables, write an equation, solve, and check that the answer makes sense in context. Practise translating phrases into algebra.

Read the explanation, open the linked topic, try to explain the idea in your own words, and retry a similar question later. Mistakes are the fastest way to learn.

Memorise a small core (like the quadratic formula and Pythagorean theorem) and understand how the rest are derived. The topic pages show each formula with an interactive visual to build intuition.

Arithmetic → Algebra → Geometry → Trigonometry → Statistics → Calculus → Linear Algebra → Discrete Math → Differential Equations → Real/Complex Analysis → Abstract Algebra and Topology.

Take the timed Tests regularly, review mistakes, keep a formula sheet in your own words, and practise under exam conditions.