calculator and CAS support for various positional bases

calculator and CAS support for various positional bases

A typical basic

calculator

may do its

computations

in binary, but is usually only capable of displaying the results in decimal. Many (but not all)

scientific calculators

are capable of accepting input and showing output in binary, octal, decimal and

hexadecimal

, usually accessed with shifted keys (e.g., Shift-+ for binary, Shift-

-

for octal, Shift-

×

for decimal, Shift-

÷

is one possible layout, such as on the Sharp EL-305V), less commonly by a mode change. A value displayed in one may be taken to another, subject to the following pretty-much-standard caveats:

a) Only

integers

are allowed in binary, octal and hexadecimal. For example, taking

π

to binary will change it to just plain 11. The calculator will usually ignore the

fractional part

, simply discarding it, though a few might

round up

if the fractional part is more than 0.5 or some arbitrary limit (the

Windows Calculator

5.1 for Windows XP will change 3.9 to 3 in hexadecimal, but 3.999999999999999999999999999999 becomes 4 in hexadecimal).

b)

Scientific notation

is understandably unavailable in binary, but it’s not available in octal or hexadecimal either. Therefore, the number must be less than

b

k

-

1

-

1

, where

k

is the number of digits in the display (ignoring the two digits the

exponent

for scientic notation), usually 10 in most scientific calculators. This means the highest numbers that may be displayed in binary, octal and hexadecimal are usually 511, 134217727 and 68719476735, respectively. The Windows Calculator allows one to choose between byte, word, double word and quadruple word for display limit.

c)

Negative numbers

from decimal are converted to two’s complement. So, for example, taking

-

1

to hexadecimal with quadruple word size turns it to FFFFFFFFFFFFFFFF, which back to decimal is the

Mersenne number

18446744073709551615.

In the

Mac OS X Calculator

, binary, octal and hexadecimal have been removed from the Scientific mode and shifted to the new Programmer mode. The limitation to

positive integers

still applies.

Computer algebra systems

like Maple and

Mathematica

have more built-in

support

for various positional bases. For example, Mathematica can handle anything from binary to base 36, providing not only entry of operands and display of output in those bases, but also string manipulation (e.g., reinterpreting the vigesimal digits of a number backwards). Floating point values can be displayed in bases other than decimal (e.g.,

π

≈

3.243

⁢

F

⁢

6

⁢

A

⁢

888

16

). Negative numbers are displayed with

negative

signs, but two’s complement is also available. If one needs support for other bases, it can be programmed, but of course one must make a decision about symbols.

Title

calculator and CAS support for various positional bases

Canonical

name

CalculatorAndCASSupportForVariousPositionalBases

Date of creation

2013-03-22 16:54:57

Last modified on

2013-03-22 16:54:57

Owner

PrimeFan (13766)

Last modified by

PrimeFan (13766)

Numerical id

4

Author

PrimeFan (13766)

Entry type

Example

Classification

msc 11-01

Classification

msc 11A63

Related topic

Calculator

Related topic

AlgebraicComputation

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