Knowledge object
Points, lines, and planes
Points, lines, and planes provide foundational geometric language for position, direction, collinearity, coplanarity, and incidence.
OCME · Bilingual mathematics & derivations
Knowledge objects
Understand concepts through definitions, assumptions and formulas, then follow guides, practice and evidence.
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Angle
In elementary Euclidean geometry, an angle is formed by two rays with a common endpoint, and its opening can be described in degrees.
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Triangle
In the plane, three distinct noncollinear points together with the three segments joining them form a nondegenerate triangle.
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Right triangle
A triangle having one interior angle that is a right angle.
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Euclidean length
The distance between two points in the plane is defined as the nonnegative square root of the sum of squared coordinate differences.
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Pythagorean theorem
In a right triangle, the sum of the squares of the two leg lengths equals the square of the hypotenuse length.
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Natural number
Natural numbers are basic numbers for counting and discrete order; this entry adopts the convention that they include 0.
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Integer
The integers consist of zero, the positive integers, and the negative integers, extending natural numbers to represent opposite directions and differences.
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Rational number
A rational number can be represented as a ratio of two integers with a nonzero denominator.
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Real number
The real numbers contain the rational numbers and complete the rational system by adding its missing limit points.
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Arithmetic operations
Addition, subtraction, multiplication, and division are a family of basic operations on two quantities within a specified number system and domain.
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Variables and algebraic expressions
An algebraic expression uses variables, constants, and defined operations to represent a mathematical quantity depending on its inputs.
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Equation
An equation requires two expressions to take equal values over a specified variable domain; a solution is an allowed value that makes the equality true.
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Linear equation in one variable
A linear equation in one variable can be written as ax+b=0 with a nonzero linear coefficient a.
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Set
A set treats selected objects within a specified context as a whole and describes its contents through element membership.
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Proposition
A proposition is a meaningful statement that can be asserted or denied and discussed as true or false within a specified logical context.
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Logical connectives
Negation, conjunction, disjunction, implication, and biconditional form compound propositions from existing propositions.
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Mathematical proof
A mathematical proof is a checkable argument establishing a specified conclusion from explicit premises and rules of inference.
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Set membership
The notation x∈A states that the element x belongs to the set A.
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Subset
A⊆B means that every element of A is also an element of B.
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Set operations
Union, intersection, difference, and relative complement construct new sets from existing sets through element-membership conditions.
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Sample space
A sample space lists all possible elementary outcomes for one specified random experiment.
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Event
An event describes a set of outcomes of interest in a random experiment.
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Function mapping
The notation f:X→Y states that the function f maps the domain X into the codomain Y.