Divide check

In Mathematical, a divide check is a division in which the divider would be Zero. Thus, a divide check would be written $\ frac {X} {0}$ , where $x$ would be the dividend.

In Algebra, the divide check is not defined. In analyzes, under certain conditions, it is possible to calculate the limit of a quotient whose denominator is a continuation or a function of null limit.

In data processing, an attempt at divide check cause an error which can, if the result is not tested by it, distort the result of the program or sometimes cause the premature stop of it. In the modern machines functioning with the standard IEEE, the result is simply a particular configuration of bits which one indicates under the term of NaN (" not has number"). NaN combined by an operation with any other number gives NaN.

Explanation of the impossibility of division by 0

The explanation which follows is valid in a unspecified ring commutative, i.e. a unit provided with two laws (addition and multiplication) checking the usual algebraic properties. In particular, the whole of the Integers (relative), that of the Real numbers, or that of the Complex numbers return within this framework. One notes $0$ the neutral element of the addition (the “zero” in question), and $1$ the neutral element of the multiplication. It is a question of showing that zero cannot be an invertible element of the ring.

If it were the case, by noting $a$ the reverse of zero, one would have, by definition of a reverse, $0 \ times a=1$ (what one could also note $a= \ frac {1} {0}$ ). But, in any ring, one easily shows that one has, for all $a$ , the equality $0 \ times a=0$ (it is said that $0$ is element absorbing for the multiplication). Indeed, by Distributivité of the multiplication on the addition, one can write $(0+0) \ times a=0 \ times a+0 \ times a$ , and as one has $0+0=0$ , that involves, after having withdrawn $0 \ times a$ from each member, the equality $0=0 \ times a$ . The equality $0=1$ would then be obtained, from which one proves easily that any element of the ring is equal to $0$ . In short, the only ring where the concept of divide check would have a direction would be tiny room to only one element, which one in general excludes from the definition of a ring (which requires thus that $0$ and $1$ is distinct).

This is why the divide check does not have not only a direction in the whole of usual numbers (whole, real or complex), but more generally in any whole of numbers checking the usual algebraic properties with respect to the addition and of the multiplication (what is called a ring). There is thus no hope to build a new whole of numbers which would give a direction contrary to zero (as that of the complex numbers a direction to the square Racine of $-1$ gives), except if one agrees to lose essential properties of the usual calculus (in particular distributivity of the multiplication on the addition).

Concept of limit

One cannot thus give direction to the quotient $\ frac {1} {0}$ . But in analyzes, one can try to give a direction to the Limite of the quotient $\ frac {1} {X}$ when $x$ tends towards zero. Thus, if $x$ indicates a Real number, the quotient $\ frac {1} {X}$ can have a absolute Value as large as one wants, on the condition of taking $x$ sufficiently near to zero. That is written

$\ lim_ {X \ to 0} \ left|\ frac {1} {X} \ right|=+ \ infty,$

and it is said that the absolute value of this quotient admits for limit “plus the infinite one” when $x$ tends towards zéro.
If one wants to do without the absolute value, two cases should be distinguished, according to whether $x$ tends towards zero by negative values (in this case, the quotient tends towards $- \ infty$ ) or positive (the quotient tends then towards $+ \ infty$ ).

One writes: $\ lim_ {X \ to 0^+} \ frac {1} {X} = + \ infty,$
and: $\ lim_ {X \ to 0^-} \ frac {1} {X} = - \ infty,$

See too

Indetermination of form 0/0

Simple: Division by zero

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