By domain
Mathematics
Everything in Nalanda that sits under mathematics.
30 entries
Angles, rotation, and radians
A radian measures rotation as the arc length it sweeps out divided by the radius, a definition that makes the same amount of turning read identically on any size circle.
6 min
ConceptArithmetic operations as transformations
Each arithmetic operation is a transformation performed on a quantity, addition and multiplication combine two quantities into one, subtraction and division reverse a combination to recover a missing part.
7 min
ConceptCircles, radius, and pi
Every circle's circumference divided by its diameter equals the same fixed number, pi, a ratio fixed by the geometry of flatness rather than by any circle's particular size.
6 min
ConceptCommutative, associative, and distributive structure
The commutative, associative, and distributive laws state precisely which rearrangements of an arithmetic expression are guaranteed to leave its value unchanged, and part of what makes an operation interesting is which of these guarantees it fails.
7 min
ConceptCongruence and similarity
Two figures are congruent when one can be placed exactly on the other by moving it without stretching, and similar when one is an exact scaled copy of the other, same shape, possibly different size.
7 min
ConceptCoordinates and reference frames
A location is meaningless until you fix an origin, a set of directions, and a scale, and describing the same point from a different origin or set of directions gives different numbers for an identical place.
7 min
ConceptCounting and cardinality
Counting is a procedure carried out one tag at a time; cardinality is the fact that procedure reveals, that the last tag names the size of the whole group regardless of order.
7 min
ConceptCross product as oriented area and rotation
The cross product of two vectors is a third vector, perpendicular to both, whose length equals the area of the parallelogram they span and whose direction, fixed by a right hand convention, records the sense of rotation from the first vector to the second.
7 min
ConceptDot product as alignment
The dot product of two vectors is a single number, equal to the product of their lengths and the cosine of the angle between them, that measures how much one points along the other.
7 min
ConceptEquality as a balance relation
Equality is a claim that two expressions name the same quantity, checkable and preservable under matched changes to both sides, not a command to compute.
7 min
ConceptEquations as constraints
An equation is not a puzzle with one ritual answer; it is a filter that sorts every candidate value into allowed or disallowed, and the solution set can be empty, one value, several, or infinite.
7 min
Mental modelEstimation and orders of magnitude
An order-of-magnitude estimate breaks an unknown quantity into a chain of roughly known factors, so that the errors in each factor partially cancel and the product lands within a defensible range of the truth.
7 min
ConceptFractions as numbers, ratios, and operators
A fraction is a single number, the answer to a division that whole numbers alone cannot finish, and its many everyday faces, a part of a whole, a ratio, a scaling operator, a point on a line, all name that same number.
7 min
ConceptFunctions as machines and relationships
A function is a rule that assigns to every element of an allowed set of inputs exactly one element of an output set, regardless of whether the rule is a formula, a table, or a picture.
7 min
ConceptGraphs as pictures of relationships
A graph is a picture in which position stands for a fact, so shape, slope, and turning points are frozen statements about how one quantity depends on another.
7 min
ConceptLength, area, and volume
Length, area, and volume are all the same operation, counting how many unit pieces tile a shape, applied in one, two, and three dimensions respectively, and every area or volume formula is a shortcut for that count.
7 min
ConceptNegative numbers as direction and debt
A negative number is the answer that must exist if subtraction is to always make sense, and it names an opposite, in direction or in owed amount, rather than a smaller kind of nothing.
7 min
ConceptPatterns, sequences, and rules
A pattern is not proven by noticing repetition in a few terms; it is established by stating a rule that generates every term, and a short list is compatible with more than one such rule.
8 min
ConceptPlace value and number bases
Place value lets a fixed, small set of symbols represent any quantity, however large, by letting a digit's position multiply its meaning.
7 min
ConceptPoints, lines, angles, and planes
Geometry builds everything, triangles, circles, solids, from a small set of undefined primitives, point, line, and plane, plus a short list of relations, incidence, betweenness, and angle, that constrain how those primitives can meet.
7 min
ConceptQuantity before number
Before any numeral exists, a quantity is just how much of something there is, and you can compare, match, and preserve it without ever naming a number.
5 min
ConceptRatios, proportions, and similarity
A ratio is a relationship between two quantities, of the form how many times one contains the other, that can stay exactly fixed while the quantities themselves grow or shrink together.
6 min
ConceptSine, cosine, and tangent as geometric ratios
Sine, cosine, and tangent are ratios between the sides of a right triangle that depend only on one angle, never on the triangle's size, because similar triangles always share the same side ratios.
7 min
ConceptSymmetry and invariance
A symmetry is a transformation that leaves an object or a situation looking exactly the same, and whatever stays unchanged under that transformation is called invariant, a fact that can be used to shortcut a calculation rather than merely decorate its answer.
7 min
ConceptThe Pythagorean relationship
In any right triangle, the area of the square on the hypotenuse always equals the combined area of the squares on the other two sides, a fact provable by rearranging area rather than by measuring it.
7 min
ConceptTriangles and rigidity
A triangle with three fixed side lengths has exactly one possible shape, which is why triangulated frames resist deformation that four sided frames cannot.
7 min
ConceptUnits and dimensional reasoning
A measured number is incomplete without its unit, because the unit names the standard the number is a multiple of, and mismatched units make an equation meaningless regardless of whether the digits agree.
6 min
ConceptVariables and unknowns
A variable is a symbol paired with a domain of allowed values, standing for an unknown, a varying quantity, or a placeholder for generality depending on how it is used.
7 min
ConceptVectors as directed quantities
A vector is a quantity that has both a magnitude and a direction, and that combines with others by placing them tip to tail rather than by ordinary addition of numbers.
7 min
ConceptZero as number, placeholder, and boundary
Zero is not one idea but four: a placeholder that fixes position, a count of an empty collection, the boundary between positive and negative, and the number that leaves every quantity unchanged under addition.
7 min