Division

• to determine the possible number of shares, of a size determined in advance. For example:

Vocabulary and notations - history

The current symbol of division is a horizontal feature separating the numerator ( dividend ) from the denominator ( dividing ). For example, has divided by B notes $\backslash \; frac\; ab$.

The denominator gives the denomination and the numerator enumerates: $\backslash \; frac\; 34$ indicates that it is about quarters , and that there is three of it → " three quarts"

Diophante and the Roman, at the 4th century wrote already fractions in a similar form, the Indiens also at the 12th century and the modern notation was adopted by the Arab .

The symbol : was used later by Leibniz.

The manufacturers of computers print the symbols ÷ or / on the key “operator division”. The use of these symbols is more ambiguous than the bar of fraction, since she asks to define priorities, but she is practical for the writing “in line” used in printing works or on a screen.

Today in France, class of 6th of college, the notations ÷ , : and / are used, because division has for the pupils a statute of operation. A nuance of direction is commonly allowed:

• has   ÷  B and has  :   B indicates an operation (not carried out), and the suitable vocabulary is dividend for has and dividing for B ;
• $\backslash \; frac\; \{has\}\; \{B\}$ and has   /  B indicates the fractional writing of the result of this operation, and the suitable vocabulary is numerator for has and denominator for B .

Definition

Being given a ring integrates (has, +, ×) , division on has is the law of composition : $A\; \backslash \; times\; has\; \backslash \; to\; A$, noted for example “÷”, such as $\backslash \; forall\; (has,\; B,\; c)\; \backslash \; in\; has\; \backslash \; times\; has\; \backslash \; times\; A$, has ÷ B = C if and only if B × C = has .

The integrity of the ring ensures that division has a single result well. On the other hand, it is defined only on $A\; \backslash \; times\; (\backslash \; \{0\; \backslash \}\; Has)$ if and only if has is a body, and to in no case definite for B = 0 .

If division is not everywhere defined, one can extend division jointly and the unit has while posing that $\backslash \; forall\; (has,\; b)\; \backslash \; in\; has\; \backslash \; times\; A$, has ÷ B is a number of this wide unit. One thus builds the body generated by the ring has . Thus one built $\backslash \; mathbb\; \{Q\}$ starting from $\backslash \; mathbb\; \{Z\}$.

A more rigorous construction of $\backslash \; mathbb\; \{Q\}$ defines it as the Ensemble quotient of $\backslash \; mathbb\; \{Z\}\; \backslash \; times\; (\backslash \; mathbb\; \{NR\}\; -\; \backslash \; \{0\; \backslash \})$ by the Relation of equivalence $R$ defined by $\backslash \; forall\; ((has,\; b),\; (a\text{'},\; b\text{'})),\; (has,\; B)\; R\; (a\text{'},\; b\text{'})\; \backslash \; yew\; ab\text{'}=a\text{'}\; b$.

This definition does not recover that of Euclidean Division, which is posed in a similar way but from which the direction is radically different.

In the idea, it is also used to reverse the multiplication (in , how much time B has). The problem of definition does not arise any more, since $\backslash \; forall\; (has,\; b)\; \backslash \; in\; \backslash \; mathbb\; \{NR\}\; \backslash \; times\; (\backslash \; mathbb\; \{NR\}\; -\; \backslash \; \{0\; \backslash \})$, is nonempty and raised part of $\backslash \; mathbb\; \{NR\}$, which thus admits a larger element.

This division, fundamental into arithmetic, introduced the concept of remains . Nevertheless, as for all divisions, the B of the definition cannot be Zero, indeed, a division by 0 would give an infinite result.

Properties

Division was not strictly speaking an operation (Law of composition interns, everywhere definite), its " propriétés" do not have implications structural on the units of numbers, and must be included/understood like properties of the numbers in fractional writing.

" Non-propriétés"

• not commutative because 5 ÷ 3 ≠ 3 ÷ 5
• not associative because 12 ÷ (4 ÷ 3) ≠ (12 ÷ 4) ÷ 3

Remarks

• pseudo neutral element on the right: 1
  $\backslash \; frac\; a1\; =\; a$
• pseudo element absorbing on the left: 0
  if B ≠ 0, $\backslash \; frac\; 0b\; =\; 0$
• equality of fractions
  • of the same dénominateur
    if B ≠ 0, $\backslash \; frac\; ab\; =\; \backslash \; frac\; cb\; \backslash \; yew\; a=c$
  • in general (which rises from the construction of $\backslash \; mathbb\; \{Q\}$)
    if B ≠ 0 and D ≠ 0, $\backslash \; frac\; ab\; =\; \backslash \; frac\; Cd\; \backslash \; yew\; ad=bc$
• ordre
  if B > 0, $\backslash \; frac\; ab$ and $\backslash \; frac\; cb$ are in the same order that has and C

Algorithm of division

This algorithm is used to determine a decimal writing of the quotient of two integers, which spreads with the quotient of two decimal numbers

In certain cases, division " pas" does not finish; , which means that the algorithm reiterates ad infinitum.

In this case, the quotient is rational nondecimal, and one can prove that its decimal development admits one period, of which the length is strictly lower than the divider.

In a nonexact division a$\backslash \; div$b ( has and B being two integers, B not no one), if one notes $q\_p$ and $r\_p$ respectively the quotient and the remainder obtained after p by pushing the iterations until obtaining p figures after the comma of the quotient, one obtains a framing or an equality:

An irrational number (real, without being rational) cannot be written in the form of fraction, by definition.

Mathematics and French language

One can divide an entity into a number of parts whose addition gives this entity, by an implicit or explicit means.

Thus, one can:

• to divide a cake into two shares, by a stab
• simply to divide a cake into two by an unspecified means
• to divide 1 into 2 halves, by the mathematical mental representation that one is made
• of it simply divide 1 into 36ème
• etc

One can also divide by dichotomy or by mischievousness , but to divide by 2 is a concept mathématique.
$\backslash \; frac\; ab\; =\; c$: “ has divided by B is equal to C ”.

See too

• Divisibility

• Euclidean Division
• Divide check
• Fraction
• Modulo
• Multiplication

Simple: Division