Mathematical division)
In mathematics, division is a partially defined operation on the set of natural numbers and integers; On the other hand, in the case of rational, real and complex numbers it is always possible to carry out the division, requiring that the divisor be different from zero, whatever the nature of the numbers to be divided. In the event that it is possible to carry out division, it consists of finding out how many times a number () is "contained" in another number ('dividend). The result of a division is called a quotient.
The «exact» division (main subject of this article) must be distinguished from the «division with remainder» or residue (the Euclidean division). Unlike addition, subtraction, or multiplication, division by whole numbers is not always defined; in effect: 4 divided by 2 equals 2 (a whole number), but 2 divided by 4 equals ½ (a half), which is no longer a whole number. The formal definition of "division" "divisibility" and "commensurability" will then depend on the definition set.
Like any operation, in the result of a division has to be unique, so there is a definition for quotient and remainder
Definition
Conceptually, the division describes one or two related notions, albeit different, that of "separating" and that of "sharing". In a formal way, the division is a binary operation that two numbers associate the product of the first by the reverse of the second. For a non-neutral number, the "division for that number" function is the reciprocal "multiplication for that number". Thus, the quotient a{displaystyle a } divided b{displaystyle b } is interpreted as the product a{displaystyle a} for 1b{displaystyle {tfrac {1}{b}}}}.
If the division is not exact, that is, the divisor is not contained an exact number of times in the dividend, the operation will have a remainder or remainder, where:
- dividendor=divisorr× × corciente+restor.{displaystyle {rm {dividendo=divisortimes quotient+rest}}}
Etymology: the word derives from the Latin dividere: to split, to separate.
Notation
In algebra and science, the division is usually denoted as a fraction, with the written dividend on the divider. For example 34{displaystyle {tfrac {3}{4}}}} read: Three divided four. An oblique bar may also be used: 3/4{displaystyle 3/4,}; this is the most common mode in computer or computer programming languages, since it can be easily registered as a simple ASCII code sequence.
Another way of indicating a division is through the symbol of the bone (♪ ♪ {displaystyle div }) (also called the “sign of division”). This symbol is also used to represent the division operation itself, as is frequently used in calculators. Other variants are the two points (:) or the point and comma (;).
Properties
Division is not properly speaking an «operation» (that is, a law of internal composition defined everywhere), its «properties» have no structural implications on the set of numbers, and must be understood within the context of the fractional numbers.
- non-conmutative, counterpart: 5♪ ♪ 3I was. I was. 3♪ ♪ 5{displaystyle 5div 3neq 3div 5};
- non-associative, against
- neutral pseudoelement to the right: 1
- a1=a{displaystyle {dfrac {a}{1}}=a};
- absorbent pseudoelement to the left: 0
- Yeah.bI was. I was. 0,0b=0{displaystyle {mbox{mbox}}bneq 0,{dfrac {0}{b}=0};
- equivalent fractions:
- ab=cd ad=bc{displaystyle {dfrac {a}{b}}}={dfrac {c}{d}{d}}iff ad=bc }.
Algorithms for division
Until the 16th century the algorithm of division by galley was very common, very similar to long division and to dessert (replaced by this as the preferred method of division). The usual process of division (long division) is usually represented under the diagram:
| Corciente{displaystyle {rm {quotient}} | |||
| Divisorr{displaystyle {rm {Divisor,}}} | Dividendor{displaystyle {rm {Dividend}}} | ||
| Restor{displaystyle {rm {Resto,}}} |
An equivalent diagram is also used with the line below the dividend
| Divisorr{displaystyle {rm {Divisor,}}} | Dividendor{displaystyle {rm {Dividend}}} | |
| (orperaciornes){displaystyle {rm {,_{(operations)}{,}}} | Corciente{displaystyle {rm {quotient}} | |
| Restor{displaystyle {rm {Resto,}}} |
And another equivalent diagram is also used
| Dividendor{displaystyle {rm {Dividend}}} | Divisorr{displaystyle {rm {Divisor,}}} | |
| (orperaciornes){displaystyle {rm {,_{(operations)}{,}}} | Corciente{displaystyle {rm {quotient}} | |
| Restor{displaystyle {rm {Resto,}}} |
Another method consists of using an "elementary table", similar to multiplication tables, with pre-established results.
Division of numbers
Division of natural numbers
Let us consider the set ℕ = {0, 1, 2,...n,...} of natural numbers and let a,b not null, c natural numbers, we will say that
- a♪ ♪ b=c{displaystyle adiv b=c}
yes
- a=b⋅ ⋅ c{displaystyle a=bcdot c}
If so, a is said to be the dividend; b, the divisor; and c, the quotient if it exists.
However, given two natural numbers a and b 0, there are two unique natural numbers q and r such that relationships are fulfilled <math alttext="{displaystyle a=bcdot q+r,0leq ra=b⋅ ⋅ q+r,0≤ ≤ r.b{displaystyle a=bcdot q+r,0leq rafterb}<img alt="{displaystyle a=bcdot q+r,0leq r.
The algorithm that finds q and r, given a and b, is called integer division, among other names.
Division of integers
Division is not a closed operation, which means that, in general, the result of dividing two integers will not be another integer, unless the dividend is an integer multiple of the divisor.
There are divisibility criteria for integers (for example, any number ending in 0,2,4,6 or 8 will be divisible by 2), used particularly to decompose integers into prime factors, what is used in calculations such as the least common multiple or the greatest common factor.
Division of rational numbers
The division in Q is always possible, since the divider is not zero. Well, the quotient x♪ ♪ and{displaystyle xdiv y}, it is but the product x⋅ ⋅ and− − 1{displaystyle xcdot y^{-1}}
In rationals, the result of dividing two rational numbers (provided that the divisor is not 0) can be calculated with any of the representative fractions. It can be defined as follows: given p/q and r/s,
- pq♪ ♪ rs=pq⋅ ⋅ sr=p⋅ ⋅ sq⋅ ⋅ r{displaystyle {p over q}div {r over s}={p over q}cdot {s over r}={pcdot s over qcdot r}}}}}}
This definition shows that division works like the inverse operation of multiplication.
Division of real numbers
The result of dividing two real numbers is another real number (as long as the divisor is not 0). It is defined as a/b = c if and only if a = cb and b ≠ 0.
Division of quadratic binomial forms
- (a+b2)♪ ♪ (c+d2)=(a+b2)× × (c+d2)− − 1{displaystyle (a+b{sqrt {2}})div (c+d{sqrt {2}})=(a+b{sqrt {2})times (c+d{sqrt {2})^{-1}}
Division by zero
Dividing any number by zero is an "undefined". This results from the fact that zero multiplied by any finite quantity is again zero, that is to say that zero does not have a multiplicative inverse.
Division of complex numbers
The result of dividing two complex numbers is another complex number (as long as the divisor is not 0). is defined as
- p+iqr+is=pr+qsr2+s2+iqr− − psr2+s2{displaystyle {p+iq over r+is}={pr+qs over r^{2+}s^{2}}+i{qr-ps over r^{2}+s^{2}}}}}
where r and s are not both equal to 0.
In the trigonometric form r(# a+iwithout a)♪ ♪ s(# b+iwithout b)=(r♪ ♪ s)(# (a− − b)+iwithout (a− − b)){displaystyle r(cos a+isin a)div s(cos b+isin b)=(rdiv s)(cos(a-b)+isin(a-b))}}}
In exponential form:
- peiq♪ ♪ reis=(p♪ ♪ r)ei(q− − s).{displaystyle {pe^{iq}div re^{is}}=(pdiv r)e^{i(q-s)}.}
