Number system

This article is about different sets of numbers. For different methods of expressing numbers with symbols, see numeral system.

In mathematics, a 'number system' is a set of numbers, (in the broadest sense of the word), together with one or more operations, such as addition or multiplication.

Examples of number systems include: natural numbers, integers, rational numbers, algebraic numbers, real numbers, complex numbers, p-adic numbers, surreal numbers, and hyperreal numbers.

For a history of number systems, see number. For a history of the symbols used to represent numbers, see numeral system.

Logical construction of number systems

Natural numbers

Main article: Set-theoretic definition of natural numbers

Simply put, the natural numbers consist of the set of all whole numbers greater than zero. The set is denoted with a bold face capital N or with the special symbol $\mathbb{N}$ . (In some books, the natural numbers begin with 0. There is no general agreement on this subject.)^[1][2]

Giuseppe Peano developed axioms for the natural numbers, and is considered the founder of axiomatic number theory.

Axiom one: There is a natural number 0.

Axiom two: Every natural number a has a successor, denoted by S(a).

Axiom three: There is no natural number whose successor is 0.

Axiom four: Distinct natural numbers have distinct successors: a = b if and only if S(a) = S(b).

Axiom of induction: If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers.

From these five axioms, all of the properties of the natural numbers can be deduced.

The number one is defined as 1 = S(0).

Most number systems include the idea of equality. In mathematics, equality is an equivalence relation, meaning it obeys the three axioms of equality:

Reflexive axiom: a = a.

Symmetric axiom: a = b implies b = a.

Transitive axiom: a = b and b = c implies a = c.

(Taken together, the symmetric and transitive axioms imply Euclid's Common Notion One: "Things equal to the same thing are equal to each other.")

Operations can be defined on the natural numbers. Addition is essentially repeated application of the successor function, defined by

a + 0 = a

a + S(b) = S(a + b).

(This implies that S(a) = S(a + 0) = a + S(0) = a + 1, so S(x) is written x + 1 from now on.) The axiom of induction allows us to conclude that this defines a + b for all natural numbers b. We proceed to define multiplication of natural numbers as repeated addition, formally

a · 0 = 0

a · (b+1) = a · b + a,

which inductively defines multiplication for all natural numbers b. Multiplication a · b is also written a × b, a * b, or simply ab.

Exponentiation can then be defined similarly using repeated multiplication,

a⁰ = 1

a^b+1 = a^b · a.

The exponentiation a ^b is often written a ^ b or a ** b, especially in media where superscripting is impossible or undesirable.

The process of defining further operators in this way is covered in the Tetration article.

Operations may have inverses. Subtraction is defined as the inverse of addition. By definition, a − b = c means b + c = a. Division is similarly the inverse of multiplication. By definition a / b = c means b · c = a. Note that under these definitions, subtraction and division are undefined for many pairs of natural numbers. In the natural number system, no meaning is assigned to 3 − 5 or to 5/3. Exponentiation has two inverses, extraction of roots and logarithms. To say that the bth root of a is c, ^b√a = c, means that c ^b = a. To say that the logarithm to base b of a is c, log_ba = c, means that b^c = a. In other words, if x^y = z, x is the root ^y√z, and y is the logarithm log_xz. As with division, ^b√a and log_ba are not defined for all values of b and a.

An alternative way to approach the operations is using axiomatics. In these axioms, the usual order of operations is assumed.

Axioms of operations:

Commutative axioms: a + b = b + a; a · b = b · a.

Associative axioms: a + (b + c) = (a + b) + c; a · (b · c) = (a · b) · c.

Distributive axioms: a · (b + c) = a · b + a · c; (b · c) ⁿ = b ⁿ · c ⁿ.

Identity axioms: a + 0 = a; a · 1 = a; a ⁰ = 1 .

Inverse axioms: − a exists and has the property that a − a = 0; if a is non-zero, then 1/a exists and has the property that a · (1/a) = 1.

All of these axioms can be proven as theorems, starting with inductive definitions and the Peano Axioms, except the inverse axioms.

Integers

The natural numbers can be extended to the number system called the integers (denoted with Z) as follows. For every non-zero natural number a, there exists an integer denoted −a, which is not a natural number. As a special case −0 is defined as the natural number 0. The successor function can be extended to the integers by the rule S(−a) = −(a − 1).

Addition can be defined on the integers inductively as follows. If a and b are natural numbers, then −a + −b = −(a + b). If a is any integer, then a + 0 = a. If b is a non-zero integer, then a + b = (a − 1) + S(b). It is then necessary to show that addition is well-defined in the case where b is a natural number. The definition of subtraction extends to the integers unchanged, and now it can be proven that a − b is defined for all integers a and b. To justify the use of − for both "minus" and "negative", one proves that a − b = a + −b. Multiplication can be defined as follows. For all natural numbers a and b, −a · b = a · −b = −(a · b). It follows as a theorem that −a · −b = a · b. The definition of division extends to the integers unchanged, but division is not defined in every case. At this point we can define natural number powers of integers in exactly the same way we defined natural number powers of natural numbers. However, we need a larger number system to define negative number powers of integers. It is also notable that our definition of roots becomes ambiguous in the integers: √4 can mean either 2 or −2. It is customary to consider the positive root when two roots exist.

From these definitions, it can be proven that all of the axioms of operations hold for integers except the multiplicative inverse. A number system with this property is called a commutative ring with identity.

Rational numbers

The rational numbers (denoted with Q) are the number system that extends the integers to include numbers which can be written as fractions. It allows division to be defined for all pairs of numbers except for division by zero.^[3] It also allows the definition of exponents to be extended to negative integer exponents, and to some, but not all, rational exponents.

We define a fraction a / b to be an ordered pair, where a is any integer and b is any non-zero natural number. We define equality of fractions by a / b = c / d if and only if a · d = b · c, and define a/1 = a, which embeds the integers in the set of all fractions.^[4] These definitions of equality partition the set of fractions and integers into equivalence classes. The canonical representative of an equivalence class is an element a / b where b is positive and relatively prime to a, or the integer a if b=1.^[5] Finding the canonical representative of an equivalence class of rational numbers is also called reducing to lowest terms. The set of rational numbers $\mathbb{Q}$ is defined to be either the set of equivalence classes or the set of canonical representatives.^[6]

We justify using the same symbol for fractions and division by proving that the fraction a / b = c = c/1 just when a = b · c. We define addition of fractions by (a / b) + (c / d) = (a · d + c · b)/(b · d). We define multiplication of fractions by (a / b) · (c / d) = (a · c)/(b · d). Since b and d are non-zero, b · d is also non-zero. We can then define subtraction as the inverse of addition and division as the inverse of multiplication, just as we did for integers, and prove that for any rational a / b and c / d, if c is non-zero, then (a / b)/(c / d) = (a · d)/(b · c). Thus division is now defined for any two rational numbers, provided the divisor is not zero.

We can now extend the definition of exponentiation to include negative exponents, by defining (for any natural number n and nonzero rational number a) a ⁻ⁿ = 1/(aⁿ). We can also define the use of rational exponents in some cases, by defining a^{m / n} = b to mean a^m = bⁿ. In other words, a^{m / n} = ⁿ√(a^m), provided the root exists. This is unambiguous if n is odd. If n is even and m is odd, this definition would be ambiguous, and taking the positive root is again customary.^[7]

With these definitions, all of the axioms of operations hold without exception. A number system in which addition and multiplication are defined for all pairs of numbers, and in which the axioms of operations hold, is called a field.

Polynomials

Polynomials are not usually called numbers, but they share many properties with numbers. All of the axioms of operations hold for polynomials except for the axiom of multiplicative inverses. Polynomials do not, in general, have multiplicative inverses. Thus the set of polynomials, like the integers, is a commutative ring (with identity).

Algebraic numbers

The algebraic numbers are a number system that includes all of the rational numbers, and is included in the set of real numbers. The construction of the algebraic numbers requires an understanding of the definition and properties of an extension field. Roughly speaking, one extends the rational numbers by appending all zeroes of polynomials with integer coefficents. This, however, would append complex numbers, which are usually excluded from the algebraic numbers, unless the set is called the complex algebraic numbers. It is, therefore, traditional to construct the real numbers first, and then define the algebraic numbers as a subset of the reals. The algebraic numbers form a field.

Real numbers

Main article: Construction of the real numbers

There are many ways to construct the real number (denoted R) system: equivalence classes of Cauchy sequences, transcendental extension fields, and Dedekind cuts, to mention just three. But the most elementary definition is that the real numbers are all numbers that can be written as decimals.

A decimal can only have finitely many digits to the left of the decimal point, but to the right of the decimal point there are three cases to consider. A decimal may terminate, repeat, or continue forever without ever becoming an infinite sequence of repeating strings of digits (in brief, a non-repeating decimal). In the first two cases, the decimal is rational, that is, it can be changed to a fraction. In the third case, the decimal is irrational.

Irrational numbers may be either algebraic or transcendental (non-algebraic). There is no easy way to tell whether a non-repeating decimal is algebraic or transcendental. In fact, there are many open questions on this subject.

The real numbers are (up to isomorphism) the only complete ordered field.

Examples

The number forty-two is a real number because it can be written as a decimal: 42.0. The number one half is both a rational number and a real number because it can be written 0.5. The number one third is both a rational number and a real number because it can be written 0.333... . The square root of two is an algebraic real number, which as a decimal is 1.4142135... . The ratio of the circumference of a circle to its diameter, π, is a transcendental real number, which as a decimal is 3.1415927... . The square root of negative one, i, is not a real number, and cannot be written as a decimal, because the square of any decimal is never negative.

Equivalence classes of decimal numerals

Just as the fractions 1/2 and 2/4 are equal, the real numbers 1.0 and 0.999... are equal. The easiest way to see this is to start with the equation 1/3 = 0.333... and multiply both sides by three. In general, any real number whose decimal form has an unending string of repeating nines is equal to the decimal obtained to removing all of the nines in the unending string that lie to the right of the decimal point, and increasing the rightmost non-nine digit by one. If all of the digits are nine, then it may be necessary to append a leading 0. Thus 123.789999999... = 123.79 and 999.999... = 0999.999... = 1000.0.

This fact is a consequence of the definition of the limit of an infinite series.

Complex numbers

The complex numbers are numbers which can be written in the form a + b · i, where a and b are real numbers and i the imaginary unit -- that is, a number whose square is minus one. The complex numbers can be viewed as abstract symbols, as representing points in the Argand plane, as an extension field of the real numbers, as a two-dimensional vector space with basis {1, i}, and in many other ways. Originally thought to be a pure abstraction, they have proved enormously useful in many practical applications, particularly in electrical engineering.

The complex numbers are a complete field, but cannot be ordered in any way that is consistent with the usual properties of inequalities. That is, there is no meaningful answer to the question, "Which is greater, 1 or i?"

More advanced number systems

The word number has no generally agreed upon mathematical meaning, nor does the word number system. Instead, we have many examples. Thus there is no rule to say what is a number and what is not. Some of the more interesting examples of abstractions that can be considered numbers include the quaternions, the octonions, ordinal numbers, and the transfinite numbers.

See also

• Abstract algebra
• Affinely extended real number system
• Large numbers
• Hypercomplex numbers
• development of number systems

Notes

1. ^ Billstein, Libeskind, and Lott, Mathematics for Elementary School Teachers 8th edition, Pearson, 2004, ISBN 0-321-15680-3
2. ^ Usage varies among mathematicians as to whether zero is to be included in the natural numbers. The Peano axioms include zero, but substituting "1" for "0" in rule one, rule three and the induction rule accurately describes the natural numbers without a zero.
3. ^ Division by zero cannot be defined in a way logically consistent with the axioms of operations.
4. ^ Since more than one fraction can represent a given rational number, a careful treatment must check every definition of an operation to insure that operation is well defined, i.e. that the definition depends on the number, and not on the fraction used to represent the number. Those definitions that may involve integers must also be consistent with the operations already defined on integers.
5. ^ That such a representative exists and is unique is a consequence of fundamental theorem of arithmetic.
6. ^ Opinions differ on whether the rational numbers are best defined as the set of equivalence classes or the set of canonical representatives. The former interpretation is somewhat simpler mathematically, while the latter ensures that the integers are technically a subset of the rationals.
7. ^ Exponents a^{m / n} where m and n are both even are defined by reducing the fraction m / n to lowest terms.

References

• Richard Dedekind, 1888. Was sind und was sollen die Zahlen? ("What are and what should the numbers be?"). Braunschweig.
• Edmund Landau, 2001, ISBN 082182693X, Foundations of Analysis, American Mathematical Society.
• Giuseppe Peano, 1889. Arithmetices principia, nova methodo exposita (The principles of arithmetic, presented by a new method). Bocca, Torino. Jean van Heijenoort, trans., 1967. A Source Book of Mathematical Logic: 1879–1931. Harvard Univ. Press: 83–97.
• B. A. Sethuraman (1996). Rings, Fields, Vector Spaces, and Group Theory: An Introduction to Abstract Algebra via Geometric Constructibility. Springer. ISBN 0-387-94848-1. 
• Solomon Feferman (1964). The Number Systems : Foundations of Algebra and Analysis. Addison–Wesley. 
• Stoll, Robert R., 1979 (1963). Set Theory and Logic. Dover.