index of tables

index of tables

Since early in the history of mathematics, tables have served to aid in various computational tasks. Even today, with the ready availability of powerful

calculators

, tables still serve an important pedagogical purpose. Tables are found in the main text of mathematical papers and books, to illustrate the

concepts

in question, and as the back matter of books.

1

Elementary tables

The most

elementary

tables in mathematics are those for the four basic

operations

:

addition

,

subtraction

,

multiplication

and

division

. In the old days, schoolchildren had to learn these up to 12. Nowadays teachers stop at 10.

•

Table of addition up to 12

•

Table of subtraction up to 12

•

Table of multiplication up to 12

•

Table of division up to 12

Before the age of

computers

, tables of

logarithms

were of paramount importance to most scientists, as well as tables of

square roots

.

2

Logical tables

Truth tables

can be made for some fairly complicated

expressions

, but the ones of most use to computer programming students are those for the basic expressions, a truth table for

logical AND

, a truth table for

logical OR

, a truth table for

logical XOR

, etc. These can be expressed as binary operations by simply replacing TRUEs with 1s and FALSEs with 0s.

3

Combinatorics tables

Some tables have garnered

interest

beyond just looking up values to become classics. This is perhaps truest of Pascal’s

triangle

, which has many interesting

properties

besides being a convenient way to look up

binomial coefficients

without having to compute large

factorials

.

4

Number theory tables

Books about

prime numbers

can be reliably counted upon to list the first

thousand

positive

prime numbers, or perhaps the first ten thousand, but certainly the first hundred. A book about prime numbers, or a book about

number theory

in general, might have as an appendix a table of

integer factorizations

for

0

<

n

<

1001

. A

table of Mersenne primes

generally gives the

exponent

rather than writing out the prime in base 10 since these numbers get very large very quickly. A book specifically on

Mersenne primes

might also have a

table of factors of small Mersenne numbers

. The

Fermat numbers

grow large even more quickly, and thus a table of factors of Fermat numbers is even less likely to write out Fermat numbers.

Similarly to a book about prime numbers, a book on the

Fibonacci numbers

might have a

list of Fibonacci numbers

(the appendices of Koshy’s book on the subject are actually tables of the factorizations of the Fibonacci and

Lucas numbers

).

For

functions

with a very small range of possible output values, such as the Möbius function

-

2

<

μ

⁢

(

n

)

<

2

or the

Liouville function

|

λ

⁢

(

n

)

|

=

1

, it makes sense for a table to pair them up with their

matching

summatory functions

. Thus we have a table of values of the Möbius function and the

Mertens function

and a table of values of the Liouville function and its summatory function.

There are also tables using

primitive roots

and index to solve

congruences

in

relation

to specific roots and indices.  A pair of tables concern

solutions

of

Diophantine equations

: table of

integer contraharmonic means

and table of

integer

harmonic means

.

In the study of

global fields

, a

table of some fundamental units

and a table of

class numbers of imaginary quadratic fields

is a useful aid to the student.

Moving on to the

complex plane

, just a few

complex multiplication

tables would be

sufficient

, as it would be easier for the student to learn the

formula

for complex multiplication rather than to try to memorize some multidimensional table which would be too large even if limited to a small range.

5

Geometry

To illustrate the

Pythagorean theorem

, older books might have an appendix of the first

primitive Pythagorean triplets

. Tables of values of the sine, cosine and

tangent functions

would not be out of place even in a basic

geometry

book.

6

Reference tables

For reference purposes, there are tables where one can look up the specific value of a

real number

or a

sequence

of numbers. For example, an

index of important irrational constants

such as Borwein’s dictionary of real numbers. For sequences of integers, there are the books by Neil Sloane, the

Handbook of

Integer Sequences

and the

Encyclopedia of Integer Sequences

, both

predecessors

to the

Online Encyclopedia of Integer Sequences

.

Title

index of tables

Canonical

name

IndexOfTables

Date of creation

2013-03-22 18:07:31

Last modified on

2013-03-22 18:07:31

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

8

Author

PrimeFan (13766)

Entry type

Topic

Classification

msc 00A20

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