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A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number , other examples being square numbers and cube numbers . The n th triangular number is the number of dots in the triangular arrangement with n dots on each side, and is equal to the sum of the n natural ...
In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, n th, etc.) aimed to extend enumeration to infinite sets. [ 1 ] A finite set can be enumerated by successively labeling each element with the least natural number that has not been previously used.
A Gödel numbering is a function mapping the set of well-formed formulae (a finite sequence of symbols on which the ordinal notation function is defined) of some formal language to the natural numbers. This associates each well-formed formula with a unique natural number, called its Gödel number.
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, = = =, where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles (see figure 2), while R is the radius of the triangle's circumcircle.
Any ordinal number can be turned into a topological space by using the order topology. When viewed as a topological space, ω 1 {\displaystyle \omega _{1}} is often written as [ 0 , ω 1 ) {\displaystyle [0,\omega _{1})} , to emphasize that it is the space consisting of all ordinals smaller than ω 1 {\displaystyle \omega _{1}} .
In a somewhat obsolete variant usage, the abscissa of a point may also refer to any number that describes the point's location along some path, e.g. the parameter of a parametric equation. [1] Used in this way, the abscissa can be thought of as a coordinate-geometry analog to the independent variable in a mathematical model or experiment (with ...
Any finite natural number can be used in at least two ways: as an ordinal and as a cardinal. Cardinal numbers specify the size of sets (e.g., a bag of five marbles), whereas ordinal numbers specify the order of a member within an ordered set [9] (e.g., "the third man from the left" or "the twenty-seventh day of January").
Representation of the ordinal numbers up to . Each turn of the spiral represents one power of ω {\displaystyle \omega } . Transfinite induction requires proving a base case (used for 0), a successor case (used for those ordinals which have a predecessor), and a limit case (used for ordinals which don't have a predecessor).
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