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Mathematical table

 
Mathematical Table

 
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Mathematical table
 
 
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Before calculator
Calculator

A calculator is a device for performing mathematical calculations, distinguished from a computer by having a limited problem solving ability and an interface optimized for interactive calculation rather than programming....
s were cheap and plentiful, people would use mathematical tables —lists of numbers showing the results of calculation with varying arguments— to simplify and drastically speed up computation
Computation

Computation is a general term for any type of information processing. This includes phenomena ranging from human thinking to calculations with a more narrow meaning....
. Tables of logarithms and trigonometric functions were common in math and science textbooks.

A simple but common example is the multiplication table
Multiplication table

In mathematics, a multiplication table is a mathematical table used to define a multiplication binary operation for an algebraic system.The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the sun, as it lays the foundation for arithmetic operations with our base-ten numbers....
, which most people know from their early mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 classes:



To find the result of 7×8, one looks in the left column to seven, then across the "seven-line" to eight.






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Before calculator
Calculator

A calculator is a device for performing mathematical calculations, distinguished from a computer by having a limited problem solving ability and an interface optimized for interactive calculation rather than programming....
s were cheap and plentiful, people would use mathematical tables —lists of numbers showing the results of calculation with varying arguments— to simplify and drastically speed up computation
Computation

Computation is a general term for any type of information processing. This includes phenomena ranging from human thinking to calculations with a more narrow meaning....
. Tables of logarithms and trigonometric functions were common in math and science textbooks.

A simple but common example is the multiplication table
Multiplication table

In mathematics, a multiplication table is a mathematical table used to define a multiplication binary operation for an algebraic system.The decimal multiplication table was traditionally taught as an essential part of elementary arithmetic around the sun, as it lays the foundation for arithmetic operations with our base-ten numbers....
, which most people know from their early mathematics
Mathematics

Mathematics is the study of quantity, structure, space, change, and related topics of pattern and form. Mathematicians seek out patterns whether found in numbers, space, natural science, computers, imaginary abstractions, or elsewhere....
 classes:

× 1 2 3 4 5 6 7 8 9 10 11 12
1 1 2 3 4 5 6 7 8 9 10 11 12
2 2 4 6 8 10 12 14 16 18 20 22 24
3 3 6 9 12 15 18 21 24 27 30 33 36
4 4 8 12 16 20 24 28 32 36 40 44 48
5 5 10 15 20 25 30 35 40 45 50 55 60
6 6 12 18 24 30 36 42 48 54 60 66 72
7 7 14 21 28 35 42 49 56 63 70 77 84
8 8 16 24 32 40 48 56 64 72 80 88 96
9 9 18 27 36 45 54 63 72 81 90 99 108
10 10 20 30 40 50 60 70 80 90 100 110 120
11 11 22 33 44 55 66 77 88 99 110 121 132
12 12 24 36 48 60 72 84 96 108 120 132 144


To find the result of 7×8, one looks in the left column to seven, then across the "seven-line" to eight. The easily found answer is 56. To find 9×3, one would swap the factors and find the equal product 3×9 (27) by the same technique.

History and use

Abramowitz&stegun
Tables of trigonometric function
Trigonometric function

In mathematics, the trigonometric functions are function s of an angle. They are important in the trigonometry of Triangle and modeling Periodic function, among many other applications....
s were first known to be made by Hipparchus
Hipparchus

Hipparchus, the common Latinization of the Greek Hipparkhos, can mean:* Hipparchus, the ancient Greek astronomer** Hipparchic cycle, an astronomical cycle he created...
. Tables of common logarithm
Common logarithm

The common logarithm is the logarithm with base 10. It is also known as the decadic logarithm, named after its base. It is indicated by log10, or sometimes Log with a capital L ....
s and antilogarithms were used to do rapid multiplications, divisions, and exponentiations, including the extraction of nth roots. Tables of special functions are still used. For example, the use of tables of values of the cumulative distribution function
Cumulative distribution function

In probability theory and statistics, the cumulative distribution function or just distribution function, completely describes the probability distribution of a real-valued random variable X....
 of the normal distribution
Normal distribution

The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields....
 remains commonplace today, especially in schools.

Mechanical special-purpose computers known as difference engine
Difference engine

The Difference Engine was an automatic, mechanical calculator designed to tabulate polynomial. Both logarithmic and trigonometric functions can be Taylor series by polynomials, so a difference engine can compute many useful sets of numbers....
s were proposed in the 19th century to tabulate polynomial approximations of logarithmic functions – i.e. to compute large logarithmic tables. This was motivated mainly by errors in logarithmic tables made by the human 'computers' of the time. Early digital computers were developed during World War II
World War II

World War II, or the Second World War , was a global military conflict which involved a Participants in World War II, including all of the great powers, organised into two opposing military alliances: the Allies of World War II and the Axis powers....
 in part to produce specialized mathematical tables for aiming artillery
Artillery

Artillery is a military Combat Arms which employs any apparatus, machine, an assortment of tools or instruments, a system or systems used as weapons for the discharge of large projectiles in combat as a major contribution of fire power within the overall military capability of an armed force....
. From 1972 onwards, with the launch and growing use of scientific calculator
HP-35

The HP-35 was Hewlett-Packard's first pocket calculator and the world's first scientific pocket calculator . Like some of HP's desktop calculators it used reverse Polish notation....
s, most mathematical tables went out of use.

Creating tables is a common code optimization
Optimization (computer science)

In computing, optimization is the process of modifying a system to make some aspect of it work more efficiently or use fewer resources. For instance, a computer program may be optimized so that it executes more rapidly, or is capable of operating with less Computer data storage or other resources, or draw less power....
 technique, and works as well for computers as humans. In computers, use of such tables is done in order to speed up calculations in those cases where a table lookup
Lookup table

In computer science, a lookup table is a data structure, usually an array or associative array, often used to replace a runtime computation with a simpler array indexing operation....
 is faster than the corresponding calculations (particularly if the computer in question doesn't have a hardware implementation of the calculations). In essence, one trades computing speed for the computer memory space
Space-time tradeoff

In computer science, a space-time or time-memory tradeoff is a situation where the computer storage use can be reduced at the cost of slower program execution, or vice versa, the computation time can be reduced at the cost of increased memory use....
 required to store the tables.

See also

  • Table (information)
    Table (information)

    A table is both a mode of visual communication and a means of arranging data. The use of tables is pervasive throughout all communication, research and data analysis....
  • Abramowitz and Stegun
    Abramowitz and Stegun

    Abramowitz and Stegun is the informal name of a mathematics reference work edited by Milton Abramowitz and Irene Stegun of the United States National Bureau of Standards ....
     Handbook of Mathematical Functions
  • Jurij Vega
    Jurij Vega

    Baron Jurij Bartolomej Vega was a Slovenes mathematician, physicist and artillery Commissioned officer....
  • Matrix
    Matrix (mathematics)

    In mathematics, a matrix is a rectangular array of numbers, as shown at the right. In addition to a number of elementary, entrywise operations such as matrix addition a key notion is matrix multiplication....
  • Random number table
    Random number table

    Random number tables have been used in statistics for tasks such as selected randomness samples. This was much more effective than manually selecting the random samples ....
    A Million Random Digits with 100,000 Normal Deviates
    A Million Random Digits with 100,000 Normal Deviates

    A Million Random Digits with 100,000 Normal Deviates is a 1955 book by the RAND Corporation. The book of tables was an important 20th century work in the field of statistics and random numbers....
  • Truth table
    Truth table

    A truth table is a mathematical table used in logic?specifically in connection with Boolean algebra , boolean functions, and propositional calculus?to compute the functional values of logical expression s on each of their functional arguments, that is, on each combination of values taken by their logical variables....


External links

  • -- generates addition, subtraction, multiplication, division, exponentiation, and trigonometric tables using JavaScript.