Multiplication table

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In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Before Calculators were cheap and plentiful people would use mathematical tables &mdashlists of numbers showing the results of calculation with varying arguments&mdash to simplify In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two

The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the world, as it lays the foundation for arithmetic operations with our base-ten numbers. The decimal ( base ten or occasionally denary) Numeral system has ten as its base. It is necessary to memorize the table up to 9 × 9, and often helpful up to 12 × 12 to be proficient in traditional mathematics. Traditional mathematics (sometimes classical math education) is a term used to describe the predominant methods of Mathematics education in the United States As noted below many schools in the United States adopted standards-based mathematics texts which completely omitted use or presentation of the multiplication table, though this practice is being increasingly abandoned in the face of protests that proficiency in elementary arithmetic is still important.

1 In basic arithmetic
2 Traditional use
3 In abstract algebra
4 Standards-based mathematics reform in the USA
5 See also
6 External links

In basic arithmetic

A multiplication table ("times table", as used to teach schoolchildren multiplication) is a grid where rows and columns are headed by the numbers to multiply, and the entry in each cell is the product of the column and row headings. Elementary arithmetic is the most basic kind of Mathematics: it concerns the operations of Addition, Subtraction, Multiplication, and division Traditionally, the heading for the first row and first column contains the symbole multiplication operator. In Operator theory, a multiplication operator is a Linear operator T defined on some vector space of functions and whose value at a function

×	1	2	3	4	5	6	7	8	9	10	11	12	13	14
1	1	2	3	4	5	6	7	8	9	10	11	12	13	14
2	2	4	6	8	10	12	14	16	18	20	22	24	26	28
3	3	6	9	12	15	18	21	24	27	30	33	36	39	42
4	4	8	12	16	20	24	28	32	36	40	44	48	52	56
5	5	10	15	20	25	30	35	40	45	50	55	60	65	70
6	6	12	18	24	30	36	42	48	54	60	66	72	78	84
7	7	14	21	28	35	42	49	56	63	70	77	84	91	98
8	8	16	24	32	40	48	56	64	72	80	88	96	104	112
9	9	18	27	36	45	54	63	72	81	90	99	108	117	126
10	10	20	30	40	50	60	70	80	90	100	110	120	130	140
11	11	22	33	44	55	66	77	88	99	110	121	132	143	154
12	12	24	36	48	60	72	84	96	108	120	132	144	156	168
13	13	26	39	52	65	78	91	104	117	130	143	156	169	182
14	14	28	42	56	70	84	98	112	126	140	154	168	182	196

So, for example, 3×6=18 by looking up where 3 and 6 intersect.

This table does not give the zeros. That is because any real number times zero is zero. In Mathematics, the real numbers may be described informally in several different ways

Multiplication tables vary from country to country. They may have ranges from 1×1 to 10×10, from 2×1 to 9×9, or from 1×1 to 12×12 to quote a few examples. 10 x 10 is essential for use in long multiplication, but knowledge to 12 x 12 and higher can be used as shortcuts in other calculation methods. The most common example of such a table in the 1960s and 1970s was inside the reference section of the Pee Chee folder commonly used in United States schools and in many other places. The yellow Pee Chee Folder was a very common American item in the second half of the 20th century

Traditional use

The traditional rote learning of multiplication was based on memorization of columns in the table, in a form like

1 × 7 = 7

2 × 7 = 14

3 × 7 = 21

4 × 7 = 28

5 × 7 = 35

6 × 7 = 42

7 × 7 = 49

8 × 7 = 56

9 × 7 = 63

10 x 7 = 70

11 x 7 = 77

12 x 7 = 84

13 x 7 = 91

14 x 7 = 98

15 x 7 = 105

Learning the content of the (10x10) table is much less work than it superficially seems to be. Rote learning is a Learning technique which avoids understanding of a subject and instead focuses on memorization. (It should not be learnt as the table itself, but rather as connections between any two single-digit factors and the resulting product, until the connection becomes intuitive, much like vocabulary in a foreign language. ) Because of the symmetry of the table 45 entries are in fact duplicates (55 entries left). The connection between 1 and any number as well as 10 and any number are trivial (36 entries left), the connections between 5 and any number can easily be derived from the multiplication by 10 and adding the occasional 5 for odd numbers(28 entries left). Multiplication by 2 is generally considered easy as well (21 entries left) and finally multiplication by 9 has an easily memorized pattern as well. Taking all those entries out of the table leaves all of 15 entries to be learnt by rote.

Bold text==Patterns in the tables==

For example, for multiplication by 6 a pattern emerges:

  2 × 6 = 12
  4 × 6 = 24
  6 × 6 = 36
  8 × 6 = 48
 10 × 6 = 60

 number × 6 = half_of_number_times_10  + number

The rule is convenient for even numbers, but also true for odd ones:

 1 × 6 = 05 +  1 =  6
 2 × 6 = 10 +  2 = 12
 3 × 6 = 15 +  3 = 18
 4 × 6 = 20 +  4 = 24
 5 × 6 = 25 +  5 = 30
 6 × 6 = 30 +  6 = 36
 7 × 6 = 35 +  7 = 42
 8 × 6 = 40 +  8 = 48
 9 × 6 = 45 +  9 = 54
10 × 6 = 50 + 10 = 60

In abstract algebra

Multiplication tables can also define binary operations on groups, fields, rings, and other algebraic systems. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In such contexts they can be called Cayley tables. A Cayley table, after the 19th century British Mathematician Arthur Cayley, describes the structure of a For an example, see octonion. In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real

Standards-based mathematics reform in the USA

In 1989, the NCTM developed new standards which were based on the belief that all students should learn higher-order thinking skills, and which played down the teaching of traditional methods that relied on rote memorization, such as multiplication tables. The National Council of Teachers of Mathematics (NCTM was founded in 1920. Widely adopted texts such as TERC omit aids such as multiplication tables, instead guiding students to invent their own methods, including skip counting and coloring in multiples on 100s charts. Skip counting is a mathematics technique taught as a kind of multiplication in Standards-based mathematics textbooks such as TERC. It is thought by many that electronic calculators have made it unnecessary or counter-productive to invest time in memorizing the multiplication table. Standards organizations such as the NCTM had originally called for "de-emphasis" on basic skills in the late 1980s, but they have since refined their statements to explicitly include learning mathematics facts. The National Council of Teachers of Mathematics (NCTM was founded in 1920. Though later versions of texts such as TERC have been rewritten, the use of earlier versions of such texts has been heavily criticized by groups such as Where's the Math and Mathematically Correct as being inadequate for producing students proficient in elementary arithmetic. Mathematically Correct is a website created by educators parents mathematicians and scientists who were concerned about the direction of reform mathematics curricula based

External links

In ancient Indian mathematics, a Vedic square is a special type of 9 × 9 Multiplication table.

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