Tag
Foundations
Entries tagged foundations.
11 entries
Counting 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
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
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
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
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
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
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