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subtraction
summary
This subchapter looks at subtraction.
free computer programming text book projecttable of contents

music 
OSdata.com 
This subchapter looks at subtraction.
free computer programming text book projecttable of contents

This subchapter is a stub section. It will be filled in with instructional material later. For now it serves the purpose of a place holder for the order of instruction.
Professors are invited to give feedback on both the proposed contents and the propsed order of this text book. Send commentary to Milo, PO Box 1361, Tustin, California, 92781, USA.
This subchapter looks at subtraction.
Here is a little bit of college level work on subtraction.
A system of numbers os considered to be closed under an operation if it reproduces itself.
S is a set of numbers. S is closed with respect to subtraction if for any two numbers a and b in the set S, the difference a  b is also a number in the set S. A set closed with respect to subtraction is called a modul.
The set of all integers {0, ±1, ±2, ±3, …} is a modul. The set of natural numbers {1, 2, 3, …} is not a modul.
The set of all even integers {0, ±2, ±4, ±6, …} is a modul.
The set of all rational numbers is a modul.
The set of all real numbers is a modul.
The set of all complex numbers is a modul.
The set of all purely imagiunary numbers ib is a modul.
A theorem regarding moduls:
(a) A modul always contains the number zero (0).
(b) If a modul contains the number a then it also contains the number a.
(c) A modul is always closed with respect to addition.
Another theorem regarding moduls:
Any modul M containing only integers includes all multiples of the greatest common divisor of the numbers in M.
For most processors, integer arithmetic is faster than floating point arithmetic. This can be reversed in special cases such digital signal processors.
On many processors, floating point arithmetic is in an optional unit or optional coprocessor rather than being included on the main processor. This allows the manufacturer to charge less for the business machines that don’t need floating point arithmetic.
The basic four integer arithmetic operations are addition, subtraction, multiplication, and division. Arithmetic operations can be signed or unsigned (unsigned is useful for effective address computations). Some older processors don’t include hardware multiplication and division. Some processors don’t include actual multiplication or division hardware, instead looking up the answer in a massive table of results embedded in the processor.
A specialized, but common, form of subtraction is an decrement instruction, which subtracts one from the contents of a register or memory location. For address computations, “decrement” may mean the subtraction of a constant other than one. Some processors have “short” or “quick” subtraction instructions that extend decrement to include a small range of values.
See also Integer Arithmetic Instructions in Assembly Language and Floating Point Arithmetic Instructions in Assembly Language
Coding example: I am making heavily documented and explained open source code for a method to play music for free — almost any song, no subscription fees, no download costs, no advertisements, all completely legal. This is done by building a frontend to YouTube (which checks the copyright permissions for you).
View music player in action: www.musicinpublic.com/.
Create your own copy from the original source code/ (presented for learning programming).
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Created: October 31, 2010
Last Updated: November 24, 2010
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