Multiplication
The multiplication (by Latin multiplicare, multiply ', also called Multiplication ) is one of the four basic operations in arithmetic. Its inverse operation is the division. The arithmetic operator for multiplication is the mark "·" and " × ".
- 6.1 Statement
Naming
The multiplication of natural numbers is caused by the repeated adding ( summing ) of the same summands:
A and b are referred to as factors, also referred to as a multiplier and multiplicand as well as b.
The bill, pronounced " a times b" means multiplication. The result product.
Hint:
For example, to write 3 x 4 for 4 4 4, and talks this term as " three by four ". Also 3 × 4 or 3 * 4 is sometimes written instead of 3 x 4.
For multiplication with variables, the point is often omitted (5x, xy). To see correct spelling mark.
For multiplication, several or many numbers you can use (derived from the great Greek Pi ) is the product of characters:
Are integers running variable is called. In case the product is defined as 1.
Examples:
Or
The faculty frequently used among others in combinatorics is a special multiplication of natural numbers:
Repeated multiplication by the same factor results in potentiating such as is
The descriptive generalization of multiplication and their calculation rules on the rational and real numbers can be reached by considering a rectangle with sides a and b ( in a given unit of length ). The surface area of the rectangle ( in the corresponding unit area) is defined as the product of a × b.
The multiplication of rational numbers can also formally define by means of fractures. Similarly, one can define the multiplication during the construction process of the real from the rational numbers.
The reverse operation for multiplying the division, which may also be regarded as a multiplication with the reciprocal value.
Computational rules
In a body ( ie in particular ) apply to all (see Mathematics )
Algorithm
The multiplication of can be calculated according to the following algorithm, where r represents the radix (base) and n, m each place:
The algorithm is based on the individual digits of a number is multiplied by the other number and pushed. At the end of the ausmultiplizierten and pushed numbers are added.
Gaussian sum factor regulating
Wherein a number of the multiplication factors of any then the maximum product is attained, when at a constant sum of the factors of the overall difference between the factors is minimized. The total difference is calculated by adding all the differences between the factors.
Product of three factors. The sum of the factors is respectively 30 As the total difference between the factors, the product is less than (usually ).
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) ... 0 ∙ 29 ∙ 1 = 0 Gesamtdiff. 58 ( 29 1 28) 0 ∙ 30 ∙ 0 = 0 Gesamtdiff. 60 (30 0 30) Gaussian summation factor control is equivalent to the statement that the content of the body is at a maximum when the sides are of equal length. Thus, for example, The square at the same scale the rectangle with the largest area.
More or less than two factors
The product of more than two factors is defined so that multiplies starting from the left two factors and thus continue to remain still just a number. The associative law now states that you can start at any point; So also from the right. Due to the commutative and the order is irrelevant, so that with any two factors (which therefore do not have to stand together directly ) can be started.
Also, the product of a single or of no factors defined, although this does not need to multiply: The product of a number is that number itself, and the product of any factor is 1 ( generally the neutral element of multiplication).
It is also possible to form an endless product. In this case, however, the order of the factors playing a role, the factors can not therefore any exchange, and also any desired summaries partial products are not always possible. ( Similar to infinite sums. )
Multiplication with the fingers
Not only adding but also multiplying can be done with the fingers to a limited extent. To this end, both factors must be in the same half of the Decade, so either both terminate on digits from 1 to 5 or digits between 6-0.
In the first case numbered fingers starting with the little finger with 10 · (d-1 ) 1 to 10 · (d-1 ) 5 for the thumb through, where d is the decade of the corresponding number ( ie for example 11 to 15 for the second decade). Then you hold the two fingers whose product you want to work out together. The corresponding product is obtained by counting the lower finger (the two together held to count fingers to ) and with ( d-1) × 10 multiplied, the product of the lower finger of the left hand with the lower fingers of the right hand (each with the fingers held together ), and finally an additive constant ( d-1) ² x 100 added.
In the second case numbered the fingers of 10 · (d-1 ) 6 to 10 · d by ( say for example 16 to 20 ). Then maintained similarly to the first case, the two fingers of the desired factors together, counts the lower finger, but multiplied now gives this d · 10 and is one of this the product of the upper finger (again, without the held together fingers ) are added and the additive constant as ( d-1) · d · 100
- To calculate, for example, 7 times 8, you count the lower finger - there are 5 - and multiply it by 10 (d = 1). This gives 50 Now you multiply the upper fingers of one hand - here 3 - with the other - here 2 - and comes on 3 x 2 = 6 Now add the two intermediate results, ie 50 6 = 56, and we obtain the final result. The additive constant ( d-1) · d · 100 is here 0 · 1 · 100 = 0
- Multiplying the counts 24 and 22 to the lower finger 6, multiplying it by 20 ( ( d-1) x 10 = 2 × 10) 120 is added to the product of the lower finger 4 x 2 = 8 and the additive constant ( d-1) ² x 100 = 400 and 528 thereby receives
Particularly, this method is suitable for fast computing of square numbers without a calculator. For factors of different decades and decade halves can still apply this method by splitting the factors into sums.
Background for this process is the fact that you can write such products as:
Can calculate and products of the second half of the Decade, by regarding the complements of the last digit of 10 forms. The last digit is then the product of the complement, the complement of the numeric sum of complements.
Vedic multiplication
This type of calculation is from India and is part of the so-called Vedic Mathematics. In this computing system first the numbers are analyzed and then selected a suitable method for their calculation. So there is, for example, a process which always lends itself to " flash" multiplication of large factors if these are close to or above the same order of magnitude ( to " Vedic " see also: Veda, Vedic language ).
The calculation method is based on the following relationship: and be two numbers just below a power of ten and or differences at the moment. Then
If now, you can see the two digit consequences of and just write next to each other in order to reach the solution of multiplication. (Note: Leading zeros of the second term must also be recorded. )
95 ∙ 97 ∙ 992 = 9215 988 = 980 096 = 156 98 12 ∙ 13 ∙ 102 Fact. Diff. Fact. Diff. Fact. Diff fact. Diff a, b, 100 a, b, 1000 a, b to 10a, b to 100 -------------------------------------------------- ----------- 95 -5992 to 8 2 12 98-2 \ ∙ \ ∙ \ ∙ \ ∙ 97 -12 -3988 13 3102 2 -------------------------------------------------- ----------- 92 15 980 096 15 6 99 96 ( 95-3 ) (-5 ∙ -3) ( 992-12 ) (-8 ∙ -12) ( 12 3 ) (3 ∙ 2 ) (98 2-1 ) (100 ( -2) ∙ 2 ) ( 97-5 ) (5 ∙ 3 ) ( 988-8 ) ( 8 ∙ 12) (13 2 ) (3 ∙ 2) (102-2-1) (100-2 ∙ 2 ) In the latter case a number higher than 100 and a lower, since in this case the product is left on the even number, a carry must be obtained, that is left, right.
Of course, a permutation of the factors gives the same result as: is, see the last line of the example. Since equal sign when multiplying two numbers are always , you can leave them out for these cases also indicated in the last line.
As the base can also be used and. Calculated as in here, just right, or is formed as the difference between left and multiplied by 2 ( base 20 ) and divided by 2 ( base 50 ). To the base 50 only the integer part after division in the case that the left-hand sum is odd, used by 2 and added as a carry to the right. Use of evidence according to through and reshape.
Russian peasant multiplication
A and B are integer factors. The product P = A · B is also on the following - apparently curious - are determined type:
Example: 11 x 3 =?
Column A Column B 11 · 3 5 6 2 12 canceled due to (2 = even ) in column A 1 24 _______________________ sum 33 ======================= The seemingly curious thing about this method is that the bill is always correct, although be made in column A generally rounding.
Explanation
In column A shall be made to where to stand at the decimal number 11 in binary representation zeros: 11 (decimal) = 1011 ( binary). The column A should be read from bottom to top. This method is also the easiest way to transform decimal numbers into binary. The continuous doublings in the B column are the powers of the binary number system, multiplied by the second factor. Where there is a zero in column A, the respective number in B is multiplied by 0, therefore deleted. All other numbers in column B are part of the product and are summed.
One can also easily rephrase this.
The last equation is the binary representation of 11 is equal to 1011.
Multiplication by ruler and compass
For a graphical multiplication by ruler and compass, you can use the tendons sentence: Through a point O to draw a line and transmits from O to be multiplied lengths and down in opposite directions. This results in two points A and B. By O to draw a second line. On this one carries on a line of length one, making another point E is formed. The second straight line is cut by the circle through the points A, B and E in a point C. The distance from O and C after the tendon has set the desired length
The required circuit can be constructed as a perimeter around the plane defined by A, B and E triangle.
In addition to the tendons and the Sekantensatz set for the construction of the product of two numbers is useful. Using the Sekantensatzes the starting point O is located outside the circle, and the variables A and B are seen from O removed in the same direction. Accordingly, then C of O of view lies in the same direction in which the fuel was removed.
Generalizations
The well-known multiplication of real numbers can be generalized to the multiplication of complex numbers by introducing an imaginary unit i and the factors in the form a b * i formally multiplied out.
By request of some of the above computational rules leads to algebraic structures with two links, an addition and a multiplication. In a ring, there is an addition with which the set forms an abelian group, and a multiplication is associative and distributive. Has the multiplication a neutral element, called the ring unitary. If, in addition, the division is always possible to obtain a skew field. If, in addition, the multiplication is commutative, we obtain a body.
With this multiplication, not to be confused are other shortcuts that are commonly referred to as products, eg, the scalar product in Euclidean vector spaces, the scalar multiplication in vector spaces, matrix multiplication and the cross product in three dimensions. From multiplication is called even if the size values of physical quantities.
Definition and understanding of the term in the Economic and Social Sciences
Multiplication of social innovation is deemed to exist when a ( standing that is under multiple conditions and interactions) through the complex process of dissemination of practices, skills and attitudes by means of multiplier (s) takes place induced learning process at different levels, the new practices, skills and attitudes among prospective innovators / adopters and new structures of the system results.