Multiplication
the multiplication (v. lat.: multiplicare = multiply, also mark taking mentioned) are one of the four basic operations of arithmetic in arithmetic. The multiplication of natural numbers results to same addends from repeated adding (add):
- <math>
\ begin {matrix}
\ underbrace {b+b+ \ cdots+b} \ \ {A} \ \ [- 4ex]
\ end {to matrix} = \ sum_ {i=1} ^ {A} b= A \ b /math <cdot>
one calls A and b factors or multiplicands. The result, spoken “A times b”, is called product.
For example one writes 3 · 4 for 4 + 4 + 4, and speaks this term as“three times four”.
In place of 3 · 4 one writes sometimes also 3 × to 4. In computer programs one often uses the indication *, in other texts should one it however avoid. With the multiplication with variables the point is often omitted(5 x, XY). To the correct way of writing see mark characters.
With the multiplication of several or many numbers one knows the product symbol <math> \ prod< /math> (Pi)
use:
- <math> 3 \ 5 \ cdot 7 \ cdot 9 \ cdot 11 = \ prod_ {i=1 cdot} ^5 (2i+1) =10,395< /math>
or also
- <math> \ frac {3} {1} \ cdot \ frac {4} {2} \ cdot \ frac {5} {3} \ cdot \; \ dots \; \ \ frac {n+2 cdot} {n} = \ prod_ {i=1} ^n \ frac {i+2} {i} = \ frac {(n+1) (n+2)}{2}< /math>
Those among other things in the stochastics frequently verwendetete faculty is a special multiplication of natural numbers:
- <math> 1 \ cdot 2 \ cdot 3 \ cdot \ dots\ n = \ cdot prod_ {i=1} ^n i = n! </math>
Repeated multiplying by the same factor leads to strengthening, e.g. math
- <2> is \ cdot 2 \ cdot 2 \ cdot 2 \ cdot 2 \ cdot 2 = 2^6 = 64< /math>
The descriptive Verallgemeinerung of the multiplication andtheir arithmetic rules on the rational and real numbers one reaches A and b by regarding a rectangle with the side lengths (in a given unit of length). The area of this rectangle (in the appropriate unit area) is defined as product A·b .
The multiplicationrational numbers can be also formally defined by breaks. Likewise one can define the multiplication during the construction procedure of the real from the rational numbers.
The reverse operation for multiplying is dividing, also as multiplying alsothe reciprocal value to be understood knows.
Table of contents |
computing laws
in a body <math> K \,< /math> (thus esp. <math> K= \ Bbb {Q}, \ R, \ Bbb {C}< /math>) apply <math> \ forall for A, b, C \ in K< /math> (see mathematics)
Associative law | <math> A \ cdot (b \ cdot C) = (A \ cdot b) \ cdot C = A \ cdot b \ cdot C< /math> |
Commutative law | <math>A \ cdot b = b \ cdot A< /math> |
Distributive law | <math> A \ cdot (b + C) = A \ cdot b + A \ cdot C< /math> |
neutral element | <math> A \ cdot 1 = A </math> |
inverse element | <math> A \ cdot a^ {- 1} = 1 \ quad\ forall A \ neq 0< /math> |
absorbing element | <math> A \ cdot 0 = 0 </math> |
Gauss sum factor rule
the Gauss sum factor rule means that a multiplication with any number of factors then the largest product, with continuous sum of the factors achieves, ifthe total difference between the factors is as small as possible. The total difference is calculated, by adding the differences between all factors.
Example:
10 * 10 * 10 = 1000 Gesamtdiff.: 0 (0 + 0 + 0) 9 * 11* 10 = 990 Gesamtdiff.: 4 (2 + 1 + 1) 8 * 11 * 11 = 968 Gesamtdiff.: 6 (3 + 3 + 0) 8 * 12 * 10 = 960 Gesamtdiff.: 8 ( 4+ 2 + 2) 7 * 12 * 11 = 924 Gesamtdiff.: 10 (5 + 4 + 1) 7 * 13 * 10 = 910 Gesamtdiff.: 12 (6 + 3 + 3) [...]
like one, becomes smaller the product sees with rising total difference, although the sum of all factors is with each task of multiplication of 30.
as two factors
the product of more than two factors is more or less defined in such a way that one ofleft beginning multiplies ever two factors and continues in such a way, until only one number remains. The associative law mentioned now that the sequence is actually all the same, one can begin thus also from right, or begin (due to the commutative law) with two arbitrary factors.
Also the product of only one or of no factors is defined, although one does not have to multiply in addition no more: The product of a number is this number, and the product of zero factors is 1 (generally the neutral element the multiplication).
It is also possible to form an infinite product. The sequence of the factors plays however a role, one can the factors thus no longer at will exchange, and also arbitrary summaries to partial products are not possible always. (Similarlyas is the case for infinite sums.)
multiplication with the fingers
not only adding, but also multiplying, can be managed to limited extent with the fingers. For this both factors must be appropriate for the same decade half in, thus eitherboth on numbers between 1 to 5 or on numbers between 6 to 0 end.
In the first case one nummeriert the fingers beginning with the small finger with (d-1) 1 to (d-1) 5 for the thumb through, whereby D for thoseDecade of the appropriate number stands (thus bspw. 11 to 15 for the 2. Decade). Afterwards one holds the two fingers, whose product one wants to calculate, together. One receives the appropriate product, by counting lower fingers and with (d-1) *10multiplied, to it the product of the fingers of the left hand by the fingers of the right hand and finally additives a constant (d-1) *2*100 adds.
In the second case one nummiert the fingers of (d-1) 6 to (D) 0 through (thus bspw. 16 to20). Afterwards one holds similarly to the first case the two fingers of the desired factors together, counts the lower fingers, but multiplies these now by d*10 and adds to this the product of the upper fingers and the additives constant results initself as (d-1) *d*100.
- around for example 7 times 8 to count, one counts the lower fingers - here there is 5 - and multiplies it by 10 (d=1). One receives 50. Now one multiplies thoseupper finger hand - here 3 - with that the others - here 2 - and comes on 3*2= 6. Now the two intermediate results add, thus 50 + 6 = 56, and one receives the final result. The additives constant (d-1) *d *100 is here 0*1 *100 = 0.
- when multiplying 24 and 22 one counts the lower fingers on 6, multiplies this by 20 ((d-1) *10 = 2*10) to 120, added in additionthe product of the upper fingers 4*2= 8 and the additives constant (d-1) *2*100= 400 and receive thereby 528.
This procedure for fast calculating of Quadratzahlen without pocket calculators is particularly suitable. For factors of different decades and decade halves one can do thisProcedures still use, by splitting the factors up into sums.
Background for this procedure is the fact that one can write such products as:
(a+x) * (a+y) =< math> a^2< /math> + (x+y) *a+x*y
and products of the second decade half to calculate can, by one the complements of the last numberconcerning. 10 forms. The last number is then the product of the complements, the decimals the complement of the sum of the complements.
- Ex. 9*7
of complements: | 1 (= 10 - 9), 3 (= 10 - 7) |
decimals: | 6 (= 10 - ( 1 + 3 )) |
One: | 3 (= 1 * 3) |
product: | 63 |
Vedi multiplication
this calculation method “lightning " multiplication of also large factors comes from India and is suitable to whenever these are scarcely under the same power of ten (toovedisch: S.A.Veda, Vedi language).
The calculation method is the basis the following relationship: <math> A< /math> and <math> b< /math> two numbers are close below a power of ten <math> 10^n< /math> and <math> \ without A< /math> and/or. <math> \ without b< /math> the differences for this. Then math
- < A> is cdot \ b = (10^n- \ without A) \ cdot (10^n - \ without b) = (10^n - \ without A \ without b) \ cdot 10^n + \ without A \ without b = (A \ without b) \ cdot 10^n + \ without A \ without b </math>
If now <math> \ without A\ without b < 10^n< /math> is, one knows the two number sequences of< math> (A \ without b) </math> and <math> \ without A \ without b </math> write simply next to each other, in order to arrive in such a way at the solution of the multiplication. (Note: Prominent zeros of the second term must .) examples
are written:
95 * 97 = 9215 992 * 988 = 980096 Fact. Diff. Fact. Diff. A, b to 100 A, b to 1000 -------------- ----------------- 95 5 992 8 \ * \ * 97 3 988 12 -------------- ----------------- 9215,980,096 (95-3) (5*3) (992-12) (8*12)
natural results in a permutation of the factors the same result, there: <math> (A \ without b) = (10^n - \ without A \ without b) = (b - \ without A) </math> is.
strange kindthe multiplication (Russian farmer multiplication)
A and B are integral factors. The product P = A · B can be determined also in the following - apparently strange - kind:
- Step: Divide A and the results so long by 2,until 1 adjusts itself as result. A not integral result is rounded off on the next whole number and afterwards the division by 2 is continued.
- Step: Double to B sequentially
- step: Cross out all lines, in which in the column A onestraight number stands.
- Step: Add all not painted numbers of the column B. The received sum is the looked for product P.
Example: 11 · 3 =?
Split A Split B 11 · 3 5 6 2 12 painted because of (2= straight) in column A 1 24 _______________________ Sum 33 =======================
the apparently strange at this method is that the calculation is always correct, although in the column A generally roundnesses are made.
explanation
in the columnA are made cancellations, where with the decimal number 11 in the binary representation zeros stand: 11 (decimally) = 1011 (binary). The column A is to be read from bottom to top. This method is also the simplest kind, decimal numbers into transform binary. The sequential duplications in the column B correspond to the power-of-two numbers of the binary number system, multiplied by the second factor. Where in column A a zero are located, the appropriate number in B is multiplied by 0, painted therefore. Everythingremaining numbers of the column B belong to the product and are summed up.
One can formulate this also easily differently.
- <math> 11 \ 3 = 3 + 6 + 24 \ Leftrightarrow 11 \ cdot 3 = 3 \ cdot (1 + 2 + 8) \ Leftrightarrow cdot 11 = 1 +2 + 8 \ Leftrightarrow 11 = 2^0 + 2^1 + 2^3< /math>
The last equation equals the binary representation 1011 of 11.
Verallgemeinerungen
the well-known multiplication of real numbers can be generalized for the multiplication of complex numbers, by one one imaginary unit i introduces and the factors to the form A + bi formally out-multiplied.
By demand of some of the computing laws indicated above one arrives at algebraic structures with two linkages, an addition and a multiplication. In a ring it givesan addition, with which the quantity forms a Abel group, and a multiplication, which are associative and distributiv. If the multiplication has a neutral element, one calls the ring unitarily. Additionally if the division is always possible, one receives an inclined body. Additionally if the multiplication is commutative, one receives a body.
With this multiplication to confound other linkages, which are designated generally also than products, are not e.g. the dot product in Euclidean vector spaces, the scalar multiplication in vector spaces and that Cross product in the three-dimensional area <math> \ R^3< /math>. Of multiplication one speaks also at size values of physical dimension.
see also
- linear factor, prime factorization
- Russian farmer multiplication, multiplicator, mark character
- mathematics of the school
- Schönhage road algorithm, Karatsuba algorithm, Toom Cook algorithm.