Our number systems are now most generally described as sets. But here, they will be referred to as sets only if their set properties are being discussed. Else they will be called systems when their specific number notion is at the core of the discussion.
Before mathematics became about sets, it was about the numbers. The study of numbers arose from the need to count, and, more generally, to measure, and more generally, to solve. Their measurement utility is intuitive. Their ability to solve became evident with creation of the number abstractions provided by classical algebra.
What are Numbers?
Being mathematical objects, numbers of any kind owe their existence to the human mind and its ingenuity. Mathematically-minded humans built them for our own utility.
Our main number systems exist as a hierarchy of infinite sets, each set’s elements corresponding to a specific number notion, each more general set containing the prior set as a proper subset: Natural (), Integer (), Rational (), Real (), Complex (), Quaternion (), Octonion (). The letter symbolic of the integers is from the German word for number, Zahlen. The letter symbolic of the quaternions is from the name of the mathematician who first shone light on their qualities and utility, William Rowan Hamilton.
There is a great bifurcation in the number systems. On one side are the discrete sets of numbers. On the other side are the reals and their power sets, all derived from the notion of continuum. An n-dimensional Euclidean space is denoted by , where , aka , is another way of visualizing .
forms the heart of our number sets. The lower layer sets are countable subsets of . The higher layers are extensions of to higher dimensions. Classical mathematics is supported by the first extension of , to ; Gauss’ proof of the Fundamental Theorem of Classical Algebra ensures this.
The natural numbers are embedded in each number set in this hierarchy, so it is not unexpected that the Peano Arithmetic (PA) forms the axiomatic basis of all our systems. Commutivity and associativity are fundamental properties of this arithmetic up to . With , we lose commutivity. With , we further lose associativity.
Beyond the main hierarchy of number sets above, mathematics offers other well-defined number notions: numbers forming countable subsets of (algebraic, Euclidean), transfinite numbers (ordinal, cardinal), alternative number notions (surreal, hyperreal, p-Adic).
The Search for Solutions to Polynomial Equations Spawned our Number Systems
Number theorists are concerned with solutions to equations involving polynomials, expressions of the form , where the coefficients are integers.
Simplest linear polynomials of form have solutions when also. But the solutions of have no solutions in when . Thus we need to provide all solutions of such linear polynomial equations.
In the more general linear polynomials of form where , the integers do not contain all the solutions; we need the rationals to provide all solutions.
Simple polynomials of degree 2 (quadratics) force us to jump to to express solutions, as with . The similar looking quadratic, , forces us to for a solution.
All solutions to polynomial equations exist in a countable subset of the complex numbers, called the algebraic numbers, .
Concepts Useful For Characterizing Number Systems
Following are attributes that distinguish our principal number sets and their underlying number notions:
- cardinality (set size; how many numbers)
- dimension (how many parts [basis elements] to each number)
- algebraic structure (arithmetic, ordering)
- topological structure (connectedness, closure, completeness)
The terms connected and disconnected characterize the great bifurcation of our number sets, the divide between separable and inseparable number notions. Topologically, we can intuit a disconnected set as one we can cut without losing any elements. Or a bit more mathematically (but not yet concisely), any disconnected set can be defined by some union of disjoint open subsets.
Cardinality explicitly characterizes this bifurcation. While great variation in cardinality exists for various abstract set notions, our number sets are ‘regular’ sets that are characterized by only the two most fundamental transfinite cardinality notions, the notion of countable sets of disconnected numbers (for which there exists a 1-1 correspondence with the natural numbers), and the notion of uncountable sets of connected numbers (very much larger sets than the countable sets).
The connectedness distinction is confirmed by our geometrical intuition, our notions of separability of natural numbers and integers and smoothness of curves and surfaces in space. The countable number systems are associated with disconnected points in our spatial geometries. The connected numbers pack together inseparably to define continuous curves and surfaces. (We intuit such packing as ‘weight’, or in mathematical terms, measure. Conceptually, the measure of the real numbers is the same as the measure of the real numbers without its countable subsets. Thus countable subsets are said to have measure-.
Again using our geometric intuition, we find utility in a notion of dimensionality. Dimension 1, the points on a straight line, is inherent in and its subsets. The other connected systems come with higher dimensions that are powers of 2: the plane (2); and more general space (n=4, 8). Geometry and number theory are closely related fields in mathematics and in our intuition.
Our number sets come with a variety of algebraic structure. Based on the imbedded natural numbers, their PA arithmetic comes with two binary operations: addition, multiplication (and the inverse operations, subtraction and division). The various system-imposed limitations on the general rules governing such operations determines the kind of algebraic structure associated with each number set. All our number sets are closed with respect to + and × operators. Some are not closed with respect to the ÷ operation, and some are not algebraically closed (closed with respect to roots of polynomials with coefficients from the system).
The 1-dimension number sets are ordered by a natural ordering that is inherent in PA, via the relation ≤ (less than or equal). With this relation, we identify systems that are ordered and well-ordered. In a well-ordered system, there is a least element in every subset. Number systems with dimension greater than 1 have no defined order (in our current arithmetically-supported sense).
Some number systems do not contain their topological closure. This means the limit point of a sequence of converging numbers is not necessarily a number within that system. Such a system will find itself embedded in a larger system that does contain the subset limit points. A system that contains all its limit points is called complete. The reals are complete. The rationals are not complete because limit points of rational sequences are real. (There are always an infinitude of real numbers between two rationals, no matter how close together the two rationals become).
The algebraic structure properties of a number system serve to axiomatize the system beyond the Peano Arithmetic. Algebra and Number Theory thus have a close working relationship.
Counting With Natural Numbers ( or )
Our counting or natural numbers are the basic number layer of our mathematics. They enter our domain of mathematical discourse as a primitive notion. Through PA, we define the primitive notion of consecutive numbers generated by a successor relation. PA further defines the primitive notion of mathematical induction, based on the more primitive notion of an infinite sequence of numbers. The natural numbers comprise the smallest successor set, aka induction set. Through the ZFC axioms, the natural numbers are constructed as an infinite set of finite sets, in conjunction with the ZFC Axiom of Infinity.
The natural numbers have a mathematical utility in counting set elements and thus calculating set size. There are two notions of characterizing set size, via the cardinal numbers that represent 1-1 mappings (isomorphisms) between sets of equivalent size, and via the ordinal numbers, representing order-isomorphisms between sets.
For finite sets, the notions of size and order are both expressed via the natural numbers. It was Cantor that extended these notions to infinite sets, where the natural numbers no longer suffice to order or to count. The transfinite ordinal and cardinal numbers provide this extended utility, as will be discussed later.
The counting number symbols we assign to natural numbers are our Hindu/Arabic decimal numerals.
Some sense of the natural numbers does exist outside of consciousness, wired into the brains of various non-sentient creatures (as well as man). Such primitive sensory capabilities are artifacts of the underlying biology of the mind, and possibly even constitute a precursor of consciousness. The ability of brains to instantaneously enumerate a few items has evolved as a life-saving and hence evolutionary life skill. The natural numbers of mathematics exist as an abstraction of this counting process, just as circles and lines exist as abstractions of natural forms.
Algebraically, the natural numbers together with zero () constitute a commutative monoid under either addition or multiplication. This means there is both an additive and multiplicative identity, and addition and multiplication are commutative. The positive natural numbers are a commutative monoid only under multiplication.
The natural numbers are well-ordered by ≥, meaning every subset has a least element.
The Integers ()
The integers extend the natural numbers by including their additive inverses (the negative numbers). They are ordered by the usual ≤, and have dimension 1. They are not well-ordered under ≤; not every subset has a least element.
Algebraically, the additive inverses make the integers an non-zero commutative ring (the product of two non-zero factors is non-zero). This is also called an integral domain; lacking a multiplicative inverse, is not a field.
The Commensurable (Rational) Numbers ()
Beyond the natural (whole) numbers, five millennia ago the ancient Egyptians refined measurement by devising the unit fractions, reciprocals of integers of form 1/n. Two and a half millennia ago, the ancient Greeks generalized fractions to ratios, devising the rational numbers.
Our rational numbers are assigned symbols suggesting their definition as ratios of integers, aka fractions, the bane of 3rd graders (4th graders in my time, but the world is speeding up). The integers are contained among the rationals as those rationals of form n/1.
The rationals have countable cardinality, dimension 1, are topologically disconnected. They are imbedded in , which is their topological closure. Algebraically, is an ordered field via ≤. Their algebraic closure is via the algebraic numbers, .
The Incommensurable (Real) Numbers ()
The ancient Greeks thought commensurable numbers were the be all and end all of number systems until they discovered no commensurable number could measure the diagonal of a square. They then needed the concept of incommensurables, a number concept more powerful than fractions.
represents the 1-dimensional continuum of real numbers. Topologically speaking, the disconnected rationals are densely imbedded among the reals, which are connected. The rationals are characterized by finite continued fractions, the reals by infinite continued fractions. All the real numbers that are not rational are called irrational.
Algebraically, is a complete ordered field, its completeness distinguishing it from . Any complete ordered field is isomorphic to . is constructively derivable from as equivalence classes of Cauchy sequences, or alternatively as Dedekind Cuts.
is imbedded in , which provides its algebraic closure.
The Complex Numbers ()
The real numbers were subsequently found inadequate for describing certain solutions to non-linear equations. This was first noticed by Cardano in the mid-16th century, while investigating solutions to cubic equations.
Because non-linear curves exist in more than one dimension, we can intuit that a number system spanning space would be necessary to describe them. For functions of two variables, generating plane curves, the complex numbers were defined to extend the number line reals to the entire plane, and the Fundamental Theorem of Algebra was conceived.
may be understood as the set of all pairs of real numbers (x,y). A complex number is written x+iy, where i=√-1. The real numbers are complex numbers with . Complex numbers where are called pure imaginary numbers.
Algebraically, is a field, but not an ordered field; there are no arithmetically-useful orderings. is the algebraic closure of .
contains a subset of disconnected spatial points, the imbedded countable subset of algebraic numbers.
The shortest path between two truths in the real domain passes through the complex domain.
The Quaternions ()
The quaternions are an extension of the complex numbers to four dimensions. As well as being algebraically interesting, they find utility in today’s space/computer age for applications involving spatial mechanics. They were devised by Hamilton, and his rules for multiplication are not commutative.
Algebraically, is one of the two finite dimension division rings containing as a proper subring. It is also the largest associative finite dimension algebra over .
The Octonions ()
The octonions are an extension of the complex numbers to eight dimensions. Also called the Cayley numbers, they are neither commutative nor associative in their multiplication.
The numbers in are most easily visualized as pairs of numbers in (Cayley-Dickson construction), just as can be visualized as the union of two complex planes.
are the four identified normed, finite dimension division algebras over the real numbers.
finds use in theoretical physics that deals with space in more than 3 dimensions.
Algebraic Numbers ()
The algebraic numbers comprise all roots of non-zero polynomials in one variable with integer coefficients. These comprise a countable subset of the complex numbers. comprises the algebraic closure of and is a subfield of .
Euclidean (Constructible) Numbers ()
The Euclidean numbers can be constructed geometrically, given an unmarked straightedge, a compass, and a unit interval line segment. Speaking abstractly, one generates circle arcs and straight lines, recording their intersections (solutions to the corresponding pairs of linear and quadratic equations). Certain solutions (among the algebraic numbers) can be found to generate a finite tower of field extensions of . Eventually, all solutions will reside in the nth tower extension, which we call the set of Euclidean Numbers.
Algebraically, the Euclidean numbers are the quadratic closure of the rationals and hence are a subfield of the algebraic numbers. It is the smallest extension of that is closed under the operations of square root and complex conjugation.
The Euclidean numbers help us understand and prove the three great unsolved geometric problems that confounded the ancient Greeks: Can one, with ruler and straightedge construction, double the cube, square the circle, or trisect an angle? We now know all three problems involve solving polynomial equations of degree higher than two, and hence require non-constructible numbers for their construction.
On an seeming unrelated note, we now also know that any constructible n-sided regular polygon must have n restricted to those numbers whose Euler totent .