Inequality (mathematics): notation, properties and applications

The four comparisons listed are not independent relations: Each of them can be expressed by any other, so it is justified, despite the different comparisons, to speak of the order of natural, real etc. numbers. numbers despite the various comparisons. For example, the other comparisons can be expressed by the relation as follows:

• holds exactly when hold.
• exactly when y < x
• holds exactly when hold.

Equality and inequality are also clearly defined by each of the four comparisons, but the comparisons cannot be expressed by equality or inequality alone. For example:

• is valid exactly when neither nor < x
• applies exactly when or y < x

On the natural numbers, the comparison ≤ by means of the successor function defined as the minimal relation satisfying the following properties:

• If then .
• If is the successor of and then .

Or in other words: is exactly not greater than if is reachable from by means of the successor function.

In von Neumann's model of the natural numbers, is defined as a ∈ (i.e., the set is an element of ) and by (i.e., is subset of ).

For integers, the following definition of possible:

• If and are both non-negative, then holds exactly if holds for and taken as natural numbers.
• If is negative and is not, then .
• If is negative and is not, then not .
• If and are both negative, then holds exactly when holds.

Rational numbers can be represented as fractions. Let two rational numbers be given by fractions and (where integers and positive natural numbers). Then defined by .

In order theory, the real numbers can be defined as Dedekind cuts of rational numbers. If α , subsets of the rational numbers, subsets to the real numbers (that is, α or β the set of all rational numbers is smaller than or ), then if α {\displaystyle} is a subset of β

Real numbers can also be represented as Cauchy sequences of rational numbers. Let and be rational Cauchy sequences representing the real numbers respectively. Then let true exactly if (i.e. the equivalence of the two Cauchy sequences) or for all except for finitely many .

For numbers with < z x < z {\displaystyle well. This property is called transitivity of Furthermore, either < x This property is called trichotomy. Starting from these two properties of order on said number ranges, one abstracts in mathematics and calls any relation of mathematical objects that satisfies these two properties a strict total order. In this sense, is also a strict total order. Irreflexivity also follows from these properties: For no number from the respective number range is valid. Similarly, the asymmetry follows: If then apply

Transitivity also applies to ≤ : If and then always holds as well. Another property is reflexivity: For any number from the respective number range . The relation ≤ is antisymmetric: For and cannot hold at the same time. The property that for every two numbers at least or hold is called totality. Again abstracting, any relation satisfying these properties is called a total order. These properties apply analogously to ≥ which thus also forms a total order.

The order of natural, real etc. numbers is compatible with addition. numbers is compatible with addition: If applies a {\displaystyle any such number, then x + a also applies. Conversely, from follows a < y {\displaystyle If the subtraction is defined (which is not the case for the natural numbers, but for instance for the integers, the rational and the real numbers), holds y - Similarly, y - comparison between and is thus determined by whether the difference is positive or negative.

The formation of the additive inverse, i.e. the mapping that assigns the number each number (geometrically speaking, a mirroring), on the other hand, is not compatible with order. Rather, applies - x

In the case of multiplication, a distinction is necessary: Compatibility applies completely analogously to multiplication by a positive number. A two-sided multiplication with zero, on the other hand, always leads to equality: for any numbers . Therefore, for ≤ and ≥ compatibility with multiplication of non-negative numbers at least still holds: If and non-negative, then also holds. Conversely, however, it does not necessarily follow from this inequality that . Multiplication by a negative number, on the other hand, can be expressed as the above reflection followed by multiplication by a positive number (e.g. ). Thus, for two numbers with negative a {\displaystyle a} x ⋅ a

For the mathematical abstraction of these compatibility properties, see ordered body.

The orders presented here on the natural, the integers, the rational and the real numbers have certain properties that are independent of the arithmetic structure, for example, but do not apply to arbitrary total orders.

The order on the natural numbers has a minimum, the number (in some definitions also , here for simplicity always contained in the natural numbers). Every natural number has a successor, i.e. a minimal number which is greater than . This is just the number .

• The order of is a discrete one.
• The natural numbers are unlimited (upwards) - there is no maximum natural number.
• The is the only natural number that has no predecessor.
• The natural numbers are well-ordered, i.e. every non-empty subset of the natural numbers has a minimum.

The integers form a discrete order. In them, each element a predecessor and a successor . There is also no maximum element, but also no minimum element. They are therefore no longer well-ordered.

The rational numbers not form a discrete order: in the rational numbers no number has a predecessor or a successor, much more, between every two rational numbers (at least) a third rational number, e.g. with Thus the rational numbers form a dense order.

The real numbers also form a dense order. An additional important property is the supremum property or completeness of order: every limited subset has a supremum and an infimum, i.e. a smallest upper bound and a largest lower bound, respectively. The natural numbers lie confinally in the rational and even the real numbers, i.e. for every real number there is a natural number that is at least as large. The order on the real numbers thus has countable confinality. The orders each induce an order topology. With regard to the order topology of the real numbers, the rational numbers lie densely in the real numbers.

By means of place value systems, natural numbers can be represented as sequences of digits. Using such representations, two natural numbers can be compared, i.e. it can be calculated whether the number represented by one sequence of digits is smaller than the other. If two natural numbers and each represented by their digit sequences without leading zeros in a place value system, then if

• the sequence of digits for is shorter than that for or
• both are of equal length and the digit sequence to is lexicographically smaller than that to

The lexicographic comparison is based on the comparison of one-digit numbers. By means of place value systems, natural numbers are also represented in modern digital calculators, on the basis of which comparisons are possible. The numbers that arithmetic-logic units of such computers can deal with directly have fixed magnitudes, i.e. they contain leading zeros, so that a comparison according to the lexicographic order is possible. Directly by means of the above definition of order, comparisons of arbitrary integers or rational numbers can also be calculated. When representing rational numbers in scientific notation, two numbers can be compared by first comparing the exponent and then, if the exponent is the same for both, the mantissa. This is especially true for floating point numbers representing dyadic fractions, which are often used on digital computers for calculations - especially approximate calculations. Many processors (such as those based on x86) provide their own instructions for comparing integer and floating point numbers.

Since the real numbers form a supra-countable set, there is no scheme according to which all real numbers can be represented. Thus, the question of a general calculation rule for comparison is also superfluous. An important basic approach is to represent certain real numbers by calculation rules that can calculate arbitrarily precise upper and lower bounds for the number in the form of rational numbers, for example by calculating further decimal places step by step. This leads to the concept of a calculable number. Two different computable numbers can be compared by calculating increasingly precise upper and lower bounds for both until the two intervals are separated (cf. interval arithmetic). On the other hand, the equality of two numbers represented in this way cannot be calculated, so the other comparisons cannot be calculated for possibly equal numbers either. For many applications it is sufficient, for example in numerical analysis, to allow a tolerance, i.e. the comparison is carried out correctly as long as the distance between the two numbers is greater than a fixed tolerance, which can be arbitrarily small, otherwise the numbers are regarded as equal. Such a comparison is computable for general computable numbers. In important special cases, on the other hand, an exact comparison is also possible: Algebraic numbers can be represented by polynomials with integer coefficients, whose zero they are, and an interval with rational minimum and maximum, which determines the respective zero. In an algebraic way, it is now possible to determine for two numbers represented in this way whether they are equal by determining common zeros. These are just determined by the greatest common divisor of the two polynomials. In the case of inequality, the comparison can then again be carried out via upper and lower bounds. These also make it possible to dispense with the algebraic calculation if the inequality has already been proven by calculated bounds. Assuming that Schanuel's hitherto unproven conjecture holds, an algorithm was also constructed that calculates comparisons for numbers that are given as zeros of systems of equations that may contain elementary functions. For algebraic numbers that are given as square root expressions or as zeros of low-degree polynomials, special procedures for comparison exist.

Such methods for exact comparisons are used in computer algebra systems and algorithmic geometry.

In the natural numbers, can be defined as the minimum element by means of the order. Accordingly, a definition of the successor function, i.e. the successor to each number, is possible. By means of the successor function, the arithmetic operations such as addition and multiplication can also be defined recursively. In the integers, rational and real numbers, however, no element is distinguished by the order, which is why the arithmetic operations (which would always distinguish the as a neutral element of addition) cannot be defined by means of the order.

Conversely, however, in all these cases the order can be defined via arithmetic. In the case of the natural numbers, an elementary definition is possible by means of addition alone (i.e. in Presburger arithmetic): It holds exactly then if an exists with . In the integers, the rational numbers and the real numbers, an unambiguous definition by means of addition alone is not possible, because the mapping in the respective number range is a group automorphism in the respective additive group, which, however, is not compatible with the order. With the addition of multiplication, i.e. in the respective ring structure, on the other hand, an elementary definition of the order is also possible. This is particularly simple in the real numbers and more generally in every Euclidean body: for there the non-negative numbers are characterised precisely by the fact that they have a square root. This provides the following definition:

In the integers, a corresponding definition is possible using the four-square theorem: an integer is non-negative exactly when it can be represented as the sum of four square numbers. This provides the definition

,

which can be transferred to the rational numbers (a rational number is exactly non-negative if it is the quotient of two sums of four square numbers).

• The real numbers can be extended to the hyperreal numbers, which also have an order compatible with addition and multiplication, but in turn have other order-theoretical properties.
• The natural numbers can be extended to the cardinal numbers and to the ordinal numbers, which are still well-ordered.
• The surreal numbers form another number range provided with an order.

• Comparison operator