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

music 
OSdata.com 
This subchapter looks at multiplication.
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 multiplication.
Here is a little bit of college level work on multiplication.
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 under multiplication if for any two numbers a and b in the set S, the product a · b is also a number in the set S. It follows that any product of two, three, four, or any finite number of elements in S also belong to S. A set closed under multiplication is called a ray.
The set of all natural numbers {1, 2, 3, …} is closed under multiplication.
The set of all integers {0, ±1, ±2, ±3, …} is closed under multiplication.
The set of all even integers {0, ±2, ±4, ±6, …} is closed under multiplication.
The set of all odd integers {±1, ±3, ±5, ±7, …} is closed under multiplication.
The set of all rational numbers is closed under multiplication.
The set of all real numbers is closed under multiplication.
The set of all complex numbers is closed under multiplication.
The set of all real numbers between zero (0) and one (1) is closed under multiplication.
A system of numbers that is closed under addition, subtraction, and multiplication is called a ring. Because all systems closed under subtraction are also closed under addition, a ring can be defined as a modul that is closed under multiplication.
Examples of rings are the set of all integers, the set of all even numbers, the set of all rational numbers, the set of all real numbers, the set of all complex numbers, the set of all complex numbers a + ib where a and b are integral, the set of all numbers of the form a + b√2 where a and b are integers, and the set of all numbers of the form a + b√D where D, a, and b are integers
PL/I PROD(x) — Pl/I builtin function that returns the total product of all of the values of the elements in an arithmetic array x. The PROD function is always carried out in floating point arithmetic. This may create small rounding errors for commercial programmers needing dollar and cents accuracy.
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.
See also Integer Arithmetic Instructions in Assembly Language, Advanced Math 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).
return to table of contents
free downloadable college text book
Because I no longer have the computer and software to make PDFs, the book is available as an HTML file, which you can convert into a PDF.
previous page  next page 
Tweets by @osdata 
free computer programming text book projectBuilding a free downloadable text book on computer programming for university, college, community college, and high school classes in computer programming. If you like the idea of this project, Supporting the entire project: If you have a business or organization that can support the entire cost of this project, please contact Pr Ntr Kmt (my church) free downloadable college text book on computer programming. 
This web site handcrafted on Macintosh computers using Tom Bender’s TexEdit Plus and served using FreeBSD .
†UNIX used as a generic term unless specifically used as a trademark (such as in the phrase “UNIX certified”). UNIX is a registered trademark in the United States and other countries, licensed exclusively through X/Open Company Ltd.
Names and logos of various OSs are trademarks of their respective owners.
Copyright © 2010 Milo
Created: October 31, 2010
Last Updated: December 12, 2010
return to table of contents
free downloadable college text book
previous page  next page 