Principle of permanence

The principle of permanence is a term from the didactics of number area extensions . It says that when building a complex mathematical theory, the mathematical structures of the underlying theory should be preserved as much as possible.

This working principle was established by Hermann Hankel in 1867 for the axiomatic construction of mathematical theories . The principle of permanence is a development of the scientific principle of economy, which is also known under the name of " Occam 's razor " and which can be reduced to the formula "simple is best".

Application in the axiomatic definition of the number system

Equivalence classes: fields of
the same color belong to the same equivalence class

A typical example of the application of the principle of permanence is the axiomatic definition of the number system. This is based on a simple number range - e.g. B. the natural numbers - and constructs a more complex number space on this basis. The motivation for building a more complex theory is the attempt that all calculation rules should apply as universally as possible. ${\ displaystyle \ mathbb {N}}$

Addition and subtraction

The whole numbers are defined as equivalence classes of pairs of natural numbers, and the previous natural numbers can be canonically embedded in the whole numbers: ${\ displaystyle \ mathbb {Z}}$

{\ displaystyle 0 \ implies \ lbrace (a, b) \ mid b = a \ rbrace}

{\ displaystyle 1 \ implies \ lbrace (a, b) \ mid b = a + 1 \ rbrace}

{\ displaystyle 2 \ implies \ lbrace (a, b) \ mid b = a + 2 \ rbrace}

and so on (including natural numbers)

{\ displaystyle a, b}

This embedding should be compatible with the arithmetic operations according to the principle of permanence.

example: With that pertains to a natural integer number is called. " " Denotes the addition in the natural numbers, " " denotes the addition in the whole numbers. Then the principle of permanence demands: ${\ displaystyle G (n)}$ ${\ displaystyle +}$ ${\ displaystyle \ oplus}$

{\ displaystyle G (n + m) = G (n) \ oplus G (m)}

The negative numbers are then exactly the equivalence classes in which is: ${\ displaystyle a> b}$

{\ displaystyle -1 \ implies \ lbrace (a, b) \ mid a = b + 1 \ rbrace}

{\ displaystyle -2 \ implies \ lbrace (a, b) \ mid a = b + 2 \ rbrace}

and so on.

In the graphic opposite, the equivalence classes can be seen as fields of the same color.

The notation is nothing more than an abbreviation for an equivalence class where is. ${\ displaystyle -1}$ ${\ displaystyle a = b + 1}$

The calculation rules known from the natural numbers must now be defined on these equivalence classes. In principle, there are many ways to do this.

The principle of permanence demands that the rules be defined in such a way that the laws that apply in basic theory, e.g. B. Commutative and associative law as well as the order relations - should also apply in the newly constructed theory.

One can show that there is then essentially exactly one possibility to define the calculation rules in this way (the "new" addition is marked with to distinguish it from the "old" addition within the natural numbers): ${\ displaystyle \ oplus}$

{\ displaystyle \ lbrace (a, b) \ mid b = a + x \ rbrace \ oplus \ lbrace (c, d) \ mid c = d + y \ rbrace: = \ lbrace (e, f) \ mid e = f + (x + y) \ rbrace}

Multiplication and division

At the next stage - construction of the rational numbers - this principle is applied again. Rational numbers are initially again defined as equivalence classes of pairs of integers. The calculation rules are defined according to the principle of permanence in such a way that all laws and rules in integers also apply to rational numbers. ${\ displaystyle \ mathbb {Q}}$

The addition and the multiplication in the natural numbers can be carried out without restriction, the subtraction, however, only if the minuend is greater than the subtrahend . The division can only be carried out if the dividend is a multiple of the divisor.

With the introduction of the rational numbers and the real numbers, the four basic arithmetic operations can now be carried out universally (apart from division by zero); the exponentiation, on the other hand, is only possible to a limited extent: can only be carried out if the base is a square number , etc. ${\ displaystyle a ^ {1/2}}$ ${\ displaystyle a}$

With the introduction of real numbers , exponentiation with a positive base can also be carried out universally; in the case of a negative basis again only to a limited extent. ${\ displaystyle \ mathbb {R}}$

This last limitation is finally removed with the introduction of complex numbers . However, you lose the order relation: The complex numbers cannot be arranged. The principle of permanence cannot be fully implemented here. ${\ displaystyle \ mathbb {C}}$

Used when dividing by zero

Division by Zero:
Fiction or Reality?

The introduction of zero in compliance with the principle of permanence was demonstrated in the previous section - as a by-product of the introduction of negative numbers. However, it is a consequence of the calculation rules (basic arithmetic operations, ring axioms ), above all of the distributive law , that zero is inevitably an absorbing element of multiplication, i.e. H.

{\ displaystyle \ forall a \; \, \ colon \; a \ cdot 0 = 0 \ cdot a = 0}

.

From this it follows immediately that any introduction of a result of a division by zero must violate the usual calculation rules. But the question remains:

Can one use the number range "meaningful", i. H. under the "smallest possible" violation of the calculation rules, expand so that division by zero is also possible?

Since there is no such thing as "mathematics" as an unchangeable discipline, only various mathematical theories, the concept of "number" in mathematics is open and expandable. So you can z. B. on the space of continuous functions of calculation rules and even define an order relation . So these functions are also something like “numbers”. Conversely, one can also restrict existing theories and examine which laws still apply in the restricted theory. An example of this is intuitionist mathematics , which excludes not just a number but a law of logic , the law of the excluded third party . Such studies can be extremely fruitful and provide deep insights into the nature of the underlying axioms . ${\ displaystyle [0,1] \ to [0,1]}$

The analysis extends the calculation areas of real and complex numbers by limit formations . This increases the number ranges are compacted . The Einpunktkompaktifizierung is possible in both areas.

In many cases, de l'Hospital's rule allows a careful calculation of the limit of so-called indefinite expressions .

One point compactification

After the number space has already been expanded several times in order to be able to carry out special arithmetic operations on the entire number space, the question arises: Can the number space (meaningfully) be expanded in such a way that division by zero is possible?

In the article Division (mathematics) it is shown that an extension of a number space with two links addition and multiplication by a solution of an equation has to go beyond the known rules of calculation (ring axioms). However, there are definitions that meet at least some of the requirements and are therefore of practical importance. ${\ displaystyle R}$ ${\ displaystyle x}$ ${\ displaystyle 0 \ cdot x = a \ in R}$

When trying to define division by zero, the simplest case according to the principle of permanence results:

{\ displaystyle {0 \ over 0} = 1}

(Requirement 1, because for a ≠ 0 applies ).

{\ displaystyle {a \ over a} = 1}

However, on the other hand:

{\ displaystyle {0 \ over 0} = {0 \ cdot 0 \ over 0} = 0 \ cdot {0 \ over 0} = 0}

because for a ≠ 0 holds .

{\ displaystyle 0 \ cdot a = 0}

With consistent application of the principle of permanence, there is a violation of well-definedness. Conversely, every “unambiguous” definition leads to B. the division 0/0 automatically leads to a violation of the principle of permanence.

Since a violation of the uniqueness outweighs a violation of the principle of permanence, one usually makes a determination of the following kind. To do this, one extends the number range by another number, which one could call Θ, and which is determined as the result of any division by 0 becomes:

Definition Θ: Θ: = a / 0 a∈ for all R .

The following arithmetic operations are defined:

Θ 1: a + Θ: = Θ and Θ + a = Θ

Θ 2: a - Θ: = Θ and Θ - a = Θ

Θ 3: a * Θ: = Θ and Θ * a = Θ

Θ 4: a / Θ: = Θ and Θ / a = Θ

Θ 5: a ^ Θ: = Θ and Θ ^ a = Θ

each for a ∈ R ∪ {Θ}. The result of any expression in which Θ appears anywhere is determined to be Θ.

With these definitions, many previous calculation rules continue to apply, such as B. a + b = b + a, a + (b + c) = (a + b) + c, a * (b + c) = a * b + a * c.

On the other hand

a / a = 1 no longer applies if a = Θ: Θ / Θ = Θ
a - a = 0 no longer applies if a = Θ: Θ - Θ = Θ
0 * a = 0 no longer applies if a = Θ: 0 * Θ = Θ

The order relation can also be defined. There are several possibilities: either Θ is larger than all other numbers or Θ is smaller than all other numbers. However, this order relation is no longer compatible with the calculation rules mentioned above.

Since some basic arithmetic rules no longer apply due to the introduction of the Θ, it is not an extension of the number range in the sense of the principle of permanence.

The number system can be expanded so that the result of division by zero is defined. However, this extension has some disadvantages:

The extension is not possible in a clear way. There are various options that are equal to one another.
The extension does not lead to a simplification of the rules - as with the extension of the natural numbers to the whole numbers - but to a greater complexity.
The extension is not compatible with the principle of permanence.

The IEEE 754 standard takes a different approach, that of two-point compactification with “PlusMinusUnendet” and NaN .

Two point compactification

For the real numbers, however, a two-point compactification is also of interest. The extension of the calculation rules is, for example, for everyone : ${\ displaystyle Z: = \ {- \ infty \} \ cup \ mathbb {R} \ cup \ {+ \ infty \}}$ ${\ displaystyle a \ in \ mathbb {R}, b \ in \ mathbb {R} _ {> 0}, x \ in Z}$

${\ displaystyle a + (\ pm \ infty) = \ pm \ infty}$
${\ displaystyle b \ cdot (\ pm \ infty) = \ pm \ infty}$
${\ displaystyle (-b) \ cdot (\ pm \ infty) = \ mp \ infty}$
${\ displaystyle \ pm \ infty + (\ pm \ infty) = \ pm \ infty}$
${\ displaystyle \ pm \ infty \, \ cdot \, \ (\ pm \ infty) = + \ infty}$
${\ displaystyle - \ infty <a <+ \ infty}$

As stated above, the ring axioms can not apply to the added elements . Limit value formations can give a "division by zero" an exact meaning: ${\ displaystyle \ pm \ infty}$

${\ displaystyle {\ tfrac {1} {+ 0}}: = \ lim _ {x \ to 0+} x ^ {- 1} = + \ infty}$
${\ displaystyle {\ tfrac {1} {- 0}}: = \ lim _ {x \ to 0-} x ^ {- 1} = - \ infty}$

However, such a limit value is different from division by zero.

The IEEE 754 standard essentially follows such a type of extension. There's a negative zero there. The result of an arithmetic expression with two limit values that run against each other, a so-called indefinite expression , such as

${\ displaystyle {\ tfrac {0} {0}} =}$ NaN
${\ displaystyle \ pm \ infty - (\ pm \ infty) =}$ NaN
${\ displaystyle 0 \ cdot (\ pm \ infty) =}$ NaN,

is still "excluded" via NaN (English for "Not a Number"). Obviously, calculating with the non-number gives NaN

${\ displaystyle x \, +}$ NaN NaN ${\ displaystyle =}$
${\ displaystyle x \; \ \ cdot}$ NaN NaN ${\ displaystyle =}$

always a non-number. So there are three types of objects:

the (finite) real numbers with which all four basic arithmetic operations can be carried out without restriction under the exclusion of division by zero and for which the ring axioms apply,
the two infinite "numbers" , with special, different calculation rules that allow division by zero (except ), ${\ displaystyle \ pm \ infty}$ ${\ displaystyle {\ tfrac {0} {0}}}$
the non-number NaN, which catches the “indefinite expressions” and which cannot be expected.

Conclusion

The proposed solutions are both unsatisfactory, both with regard to the rules of both arithmetic and total order. The extensions do not lead to a simplification of the rules, on the contrary: in the end, significantly more special cases have to be observed.

As a result, division by zero is not defined . However, the stronger statement applies that it cannot be defined within the framework of the rules mentioned . In this respect, a division by zero does not occur in the serious mathematical literature.

Nevertheless, it is helpful to offer options for further computing for the (especially spontaneous) use of computing devices. So the rules Θ 1 to Θ 4 z. B. has been implemented in Excel , which instead of Θ uses the notation # DIV0! used. The IEEE 754 standard is closer to the hardware.

Web links

Online version of Hankel's "Lectures on Complex Numbers and Their Functions"

Individual evidence

^ Hermann Hankel: Lectures on the complex numbers and their functions. 1867
↑ It is even stronger that an expansion structure that contains quotients with a divisor 0 is not a mathematical ring.