Multiplication table

Last updated
Multiplication table from 1 to 10 drawn to scale with the upper-right half labeled with prime factorisations Multiplication table to scale.svg
Multiplication table from 1 to 10 drawn to scale with the upper-right half labeled with prime factorisations

In mathematics, a multiplication table (sometimes, less formally, a times table) is a mathematical table used to define a multiplication operation for an algebraic system.

Contents

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 base-ten numbers. Many educators believe it is necessary to memorize the table up to 9 × 9. [1]

History

Pre-modern times

The Tsinghua Bamboo Slips, Chinese Warring States era decimal multiplication table of 305 BC Qinghuajian, Suan Biao.jpg
The Tsinghua Bamboo Slips, Chinese Warring States era decimal multiplication table of 305 BC

The oldest known multiplication tables were used by the Babylonians about 4000 years ago. [2] However, they used a base of 60. [2] The oldest known tables using a base of 10 are the Chinese decimal multiplication table on bamboo strips dating to about 305 BC, during China's Warring States period. [2]

"Table of Pythagoras" on Napier's bones PSM V26 D467 Table of pythagoras on slats.jpg
"Table of Pythagoras" on Napier's bones

The multiplication table is sometimes attributed to the ancient Greek mathematician Pythagoras (570–495 BC). It is also called the Table of Pythagoras in many languages (for example French, Italian and Russian), sometimes in English. [4] The Greco-Roman mathematician Nichomachus (60–120 AD), a follower of Neopythagoreanism, included a multiplication table in his Introduction to Arithmetic , whereas the oldest surviving Greek multiplication table is on a wax tablet dated to the 1st century AD and currently housed in the British Museum. [5]

In 493 AD, Victorius of Aquitaine wrote a 98-column multiplication table which gave (in Roman numerals) the product of every number from 2 to 50 times and the rows were "a list of numbers starting with one thousand, descending by hundreds to one hundred, then descending by tens to ten, then by ones to one, and then the fractions down to 1/144." [6]

Modern times

In his 1820 book The Philosophy of Arithmetic, [7] mathematician John Leslie published a multiplication table up to 99 × 99, which allows numbers to be multiplied in pairs of digits at a time. Leslie also recommended that young pupils memorize the multiplication table up to 50 × 50.

The illustration below shows a table up to 12 × 12, which is a size commonly used nowadays in English-world schools.

×0123456789101112
00000000000000
10123456789101112
2024681012141618202224
30369121518212427303336
404812162024283236404448
5051015202530354045505560
6061218243036424854606672
7071421283542495663707784
8081624324048566472808896
90918273645546372819099108
100102030405060708090100110120
110112233445566778899110121132
1201224364860728496108120132144

Because multiplication of integers is commutative, many schools use a smaller table as below. Some schools even remove the first column since 1 is the multiplicative identity.

11
224
3369
4481216
5510152025
661218243036
77142128354249
8816243240485664
991827364554637281
×123456789

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

  1 × 10 = 10
  2 × 10 = 20
  3 × 10 = 30
  4 × 10 = 40
  5 × 10 = 50
  6 × 10 = 60
  7 × 10 = 70
  8 × 10 = 80
  9 × 10 = 90

This form of writing the multiplication table in columns with complete number sentences is still used in some countries, such as Bosnia and Herzegovina,[ citation needed ] instead of the modern grids above.

Patterns in the tables

There is a pattern in the multiplication table that can help people to memorize the table more easily. It uses the figures below:

 
1232 4
456   
7896 8
 0 5  0 
Figure 1: OddFigure 2: Even
Cycles of the unit digit of multiples of integers ending in 1, 3, 7 and 9 (upper row), and 2, 4, 6 and 8 (lower row) on a telephone keypad Multiplication mnemonic.svg
Cycles of the unit digit of multiples of integers ending in 1, 3, 7 and 9 (upper row), and 2, 4, 6 and 8 (lower row) on a telephone keypad

Figure 1 is used for multiples of 1, 3, 7, and 9. Figure 2 is used for the multiples of 2, 4, 6, and 8. These patterns can be used to memorize the multiples of any number from 0 to 10, except 5. As you would start on the number you are multiplying, when you multiply by 0, you stay on 0 (0 is external and so the arrows have no effect on 0, otherwise 0 is used as a link to create a perpetual cycle). The pattern also works with multiples of 10, by starting at 1 and simply adding 0, giving you 10, then just apply every number in the pattern to the "tens" unit as you would normally do as usual to the "ones" unit.

For example, to recall all the multiples of 7:

  1. Look at the 7 in the first picture and follow the arrow.
  2. The next number in the direction of the arrow is 4. So think of the next number after 7 that ends with 4, which is 14.
  3. The next number in the direction of the arrow is 1. So think of the next number after 14 that ends with 1, which is 21.
  4. After coming to the top of this column, start with the bottom of the next column, and travel in the same direction. The number is 8. So think of the next number after 21 that ends with 8, which is 28.
  5. Proceed in the same way until the last number, 3, corresponding to 63.
  6. Next, use the 0 at the bottom. It corresponds to 70.
  7. Then, start again with the 7. This time it will correspond to 77.
  8. Continue like this.

In abstract algebra

Tables can also define binary operations on groups, fields, rings, and other algebraic systems. In such contexts they are called Cayley tables. Here are the addition and multiplication tables for the finite field Z5:

For other examples, see group, and octonion.

Chinese and Japanese multiplication tables

Mokkan discovered at Heijō Palace suggest that the multiplication table may have been introduced to Japan through Chinese mathematical treatises such as the Sunzi Suanjing, because their expression of the multiplication table share the character in products less than ten. [8] Chinese and Japanese share a similar system of eighty-one short, easily memorable sentences taught to students to help them learn the multiplication table up to 9 × 9. In current usage, the sentences that express products less than ten include an additional particle in both languages. In the case of modern Chinese, this is (); and in Japanese, this is (ga). This is useful for those who practice calculation with a suanpan or a soroban, because the sentences remind them to shift one column to the right when inputting a product that does not begin with a tens digit. In particular, the Japanese multiplication table uses non-standard pronunciations for numbers in some specific instances (such as the replacement of san roku with saburoku).

The Japanese multiplication table
×1 ichi2 ni3 san4 shi5 go6 roku7 shichi8 ha9 ku
1 inin'ichi ga ichiinni ga niinsan ga saninshi ga shiingo ga goinroku ga rokuinshichi ga shichiinhachi ga hachiinku ga ku
2 nini ichi ga nini nin ga shini san ga rokuni shi ga hachini go jūni roku jūnini shichi jūshini hachi jūrokuni ku jūhachi
3 sansan ichi ga sansan ni ga rokusazan ga kusan shi jūnisan go jūgosaburoku jūhachisan shichi nijūichisanpa nijūshisan ku nijūshichi
4 shishi ichi ga shishi ni ga hachishi san jūnishi shi jūrokushi go nijūshi roku nijūshishi shichi nijūhachishi ha sanjūnishi ku sanjūroku
5 gogo ichi ga gogo ni jūgo san jūgogo shi nijūgo go nijūgogo roku sanjūgo shichi sanjūgogo ha shijūgokku shijūgo
6 rokuroku ichi ga rokuroku ni nijūroku san jūhachiroku shi nijūshiroku go sanjūroku roku sanjūrokuroku shichi shijūniroku ha shijūhachirokku gojūshi
7 shichishichi ichi ga shichishichi ni jūshishichi san nijūichishichi shi nijūhachishichi go sanjūgoshichi roku shijūnishichi shichi shijūkushichi ha gojūrokushichi ku rokujūsan
8 hachihachi ichi ga hachihachi ni jūrokuhachi san nijūshihachi shi sanjūnihachi go shijūhachi roku shijūhachihachi shichi gojūrokuhappa rokujūshihakku shichijūni
9 kuku ichi ga kuku ni jūhachiku san nijūshichiku shi sanjūrokuku go shijūgoku roku gojūshiku shichi rokujūsanku ha shichijūniku ku hachijūichi

Warring States decimal multiplication bamboo slips

A bundle of 21 bamboo slips dated 305 BC in the Warring States period in the Tsinghua Bamboo Slips (清華簡) collection is the world's earliest known example of a decimal multiplication table. [9]

Qinghuajian Suan Biao example.svg
A modern representation of the Warring States decimal multiplication table used to calculate 12 × 34.5

Standards-based mathematics reform in the US

In 1989, the National Council of Teachers of Mathematics (NCTM) developed new standards which were based on the belief that all students should learn higher-order thinking skills, which recommended reduced emphasis on the teaching of traditional methods that relied on rote memorization, such as multiplication tables. Widely adopted texts such as Investigations in Numbers, Data, and Space (widely known as TERC after its producer, Technical Education Research Centers) omitted aids such as multiplication tables in early editions. NCTM made it clear in their 2006 Focal Points that basic mathematics facts must be learned, though there is no consensus on whether rote memorization is the best method. In recent years, a number of nontraditional methods have been devised to help children learn multiplication facts, including video-game style apps and books that aim to teach times tables through character-based stories.

See also

Related Research Articles

<span class="mw-page-title-main">Arithmetic</span> Elementary branch of mathematics

Arithmetic is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th century, Italian mathematician Giuseppe Peano formalized arithmetic with his Peano axioms, which are highly important to the field of mathematical logic today.

<span class="mw-page-title-main">Decimal</span> Number in base-10 numeral system

The decimal numeral system is the standard system for denoting integer and non-integer numbers. It is the extension to non-integer numbers of the Hindu–Arabic numeral system. The way of denoting numbers in the decimal system is often referred to as decimal notation.

<span class="mw-page-title-main">Multiplication</span> Arithmetical operation

Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a product.

A multiplication algorithm is an algorithm to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient than others. Efficient multiplication algorithms have existed since the advent of the decimal system.

<span class="mw-page-title-main">Addition</span> Arithmetic operation

Addition is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or sum of those values combined. The example in the adjacent image shows two columns of three apples and two apples each, totaling at five apples. This observation is equivalent to the mathematical expression "3 + 2 = 5".

<span class="mw-page-title-main">Subtraction</span> One of the four basic arithmetic operations

Subtraction is one of the four arithmetic operations along with addition, multiplication and division. Subtraction is an operation that represents removal of objects from a collection. For example, in the adjacent picture, there are 5 − 2 peaches—meaning 5 peaches with 2 taken away, resulting in a total of 3 peaches. Therefore, the difference of 5 and 2 is 3; that is, 5 − 2 = 3. While primarily associated with natural numbers in arithmetic, subtraction can also represent removing or decreasing physical and abstract quantities using different kinds of objects including negative numbers, fractions, irrational numbers, vectors, decimals, functions, and matrices.

<span class="mw-page-title-main">Rote learning</span> A memorization technique based on repetition

Rote learning is a memorization technique based on repetition. The method rests on the premise that the recall of repeated material becomes faster the more one repeats it. Some of the alternatives to rote learning include meaningful learning, associative learning, spaced repetition and active learning.

A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" (one).

<span class="mw-page-title-main">Napier's bones</span> 1617 device for calculating products and quotients

Napier's bones is a manually-operated calculating device created by John Napier of Merchiston, Scotland for the calculation of products and quotients of numbers. The method was based on lattice multiplication, and also called rabdology, a word invented by Napier. Napier published his version in 1617. It was printed in Edinburgh and dedicated to his patron Alexander Seton.

<span class="mw-page-title-main">Positional notation</span> Method for representing or encoding numbers

Positional notation usually denotes the extension to any base of the Hindu–Arabic numeral system. More generally, a positional system is a numeral system in which the contribution of a digit to the value of a number is the value of the digit multiplied by a factor determined by the position of the digit. In early numeral systems, such as Roman numerals, a digit has only one value: I means one, X means ten and C a hundred. In modern positional systems, such as the decimal system, the position of the digit means that its value must be multiplied by some value: in 555, the three identical symbols represent five hundreds, five tens, and five units, respectively, due to their different positions in the digit string.

<span class="mw-page-title-main">Mental calculation</span> Arithmetical calculations using only the human brain

Mental calculation consists of arithmetical calculations using only the human brain, with no help from any supplies or devices such as a calculator. People may use mental calculation when computing tools are not available, when it is faster than other means of calculation, or even in a competitive context. Mental calculation often involves the use of specific techniques devised for specific types of problems. People with unusually high ability to perform mental calculations are called mental calculators or lightning calculators.

In computer science, arbitrary-precision arithmetic, also called bignum arithmetic, multiple-precision arithmetic, or sometimes infinite-precision arithmetic, indicates that calculations are performed on numbers whose digits of precision are limited only by the available memory of the host system. This contrasts with the faster fixed-precision arithmetic found in most arithmetic logic unit (ALU) hardware, which typically offers between 8 and 64 bits of precision.

<span class="mw-page-title-main">Suanpan</span> Chinese abacus

The suanpan, also spelled suan pan or souanpan) is an abacus of Chinese origin first described in a 190 CE book of the Eastern Han Dynasty, namely Supplementary Notes on the Art of Figures written by Xu Yue. However, the exact design of this suanpan is not known. Usually, a suanpan is about 20 cm (8 in) tall and it comes in various widths depending on the application. It usually has more than seven rods. There are two beads on each rod in the upper deck and five beads on each rod in the bottom deck. The beads are usually rounded and made of a hardwood. The beads are counted by moving them up or down towards the beam. The suanpan can be reset to the starting position instantly by a quick jerk around the horizontal axis to spin all the beads away from the horizontal beam at the center.

A divisibility rule is a shorthand and useful way of determining whether a given integer is divisible by a fixed divisor without performing the division, usually by examining its digits. Although there are divisibility tests for numbers in any radix, or base, and they are all different, this article presents rules and examples only for decimal, or base 10, numbers. Martin Gardner explained and popularized these rules in his September 1962 "Mathematical Games" column in Scientific American.

Principles and Standards for School Mathematics (PSSM) are guidelines produced by the National Council of Teachers of Mathematics (NCTM) in 2000, setting forth recommendations for mathematics educators. They form a national vision for preschool through twelfth grade mathematics education in the US and Canada. It is the primary model for standards-based mathematics.

<span class="mw-page-title-main">Elementary arithmetic</span> Numbers and the basic operations on them

Elementary arithmetic is a branch of mathematics involving basic numerical operations, namely addition, subtraction, multiplication, and division. Due to its low level of abstraction, broad range of application, and position as the foundation of all mathematics, elementary arithmetic is generally the first critical branch of mathematics to be taught in schools.

<span class="mw-page-title-main">Fraction</span> Ratio of two numbers

A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction consists of an integer numerator, displayed above a line, and a non-zero integer denominator, displayed below that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction 3/4, the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates 3/4 of a cake.

Lattice multiplication, also known as the Italian method, Chinese method, Chinese lattice, gelosia multiplication, sieve multiplication, shabakh, diagonally or Venetian squares, is a method of multiplication that uses a lattice to multiply two multi-digit numbers. It is mathematically identical to the more commonly used long multiplication algorithm, but it breaks the process into smaller steps, which some practitioners find easier to use.

<span class="mw-page-title-main">Rod calculus</span>

Rod calculus or rod calculation was the mechanical method of algorithmic computation with counting rods in China from the Warring States to Ming dynasty before the counting rods were increasingly replaced by the more convenient and faster abacus. Rod calculus played a key role in the development of Chinese mathematics to its height in Song Dynasty and Yuan Dynasty, culminating in the invention of polynomial equations of up to four unknowns in the work of Zhu Shijie.

<i>Mirifici Logarithmorum Canonis Descriptio</i> First publication of complete tables of logarithms, 1614

Mirifici Logarithmorum Canonis Descriptio and Mirifici Logarithmorum Canonis Constructio are two books in Latin by John Napier expounding the method of logarithms. While others had approached the idea of logarithms, notably Jost Bürgi, it was Napier who first published the concept, along with easily used precomputed tables, in his Mirifici Logarithmorum Canonis Descriptio.

References

  1. Trivett, John (1980), "The Multiplication Table: To Be Memorized or Mastered!", For the Learning of Mathematics, 1 (1): 21–25, JSTOR   40247697 .
  2. 1 2 3 Qiu, Jane (January 7, 2014). "Ancient times table hidden in Chinese bamboo strips". Nature News. doi: 10.1038/nature.2014.14482 . S2CID   130132289.
  3. Wikisource:Page:Popular Science Monthly Volume 26.djvu/467
  4. for example in An Elementary Treatise on Arithmetic by John Farrar
  5. David E. Smith (1958), History of Mathematics, Volume I: General Survey of the History of Elementary Mathematics. New York: Dover Publications (a reprint of the 1951 publication), ISBN   0-486-20429-4, pp. 58, 129.
  6. David W. Maher and John F. Makowski. "Literary evidence for Roman arithmetic with fractions". Classical Philology, 96/4 (October 2001), p. 383.
  7. Leslie, John (1820). The Philosophy of Arithmetic; Exhibiting a Progressive View of the Theory and Practice of Calculation, with Tables for the Multiplication of Numbers as Far as One Thousand. Edinburgh: Abernethy & Walker.
  8. "「九九」は中国伝来…平城宮跡から木簡出土". Yomiuri Shimbun. December 4, 2010. Archived from the original on December 7, 2010.
  9. Nature article The 2,300-year-old matrix is the world's oldest decimal multiplication table

1 x 1 to 23 x 23