MadSci Network: Computer Science

Re: How do you combine adding, subtracting, multiplying and division on a calcu

Date: Mon Nov 20 21:04:45 2000
Posted By: Joe McLeod, Staff, Advanced Services, Charter Communications
Area of science: Computer Science
ID: 974334499.Cs

To answer your question, we need to delve into the world of digital 
logic.  All computing devices use collections of logic gates to implement 
airthmetic operations.
For a good grounding in logic gates and their conceptual operation, please 
Summary of Logic Gates at Queen's University in Kingston, Ontario, 

as well as

Dr. Jeffrey J. Gosper's  
Simple Logic Gates page at Brunel University.  There is also a 
simulator here that will allow you to see logic gates, and some of the 
basic assemblages of logic gates, at work.

Physically, logic gates work by either allowing or inhibiting a voltage 
through the gate.  For an overview on how TTL gates operate physically, 
Transistors and Digital Logic Gates, a web lecture from the Southern 
Polytechnic State University's Computer Science Department.

As for "mixing" arithmetic operations, that can be construed on two 
levels:  The digital logic level; and the algebraic level.

For the digital logic level, most modern computing devices have a 
component referred to as an Arithmetic Logic Unit (ALU) which is a 
collection of the logic gates described on Dr. Gosper's site, and which 
serve just about any arithmetic function.  This is known as combinational 
logic design.  A very good resource for combinational logic can be found 
in Arithmetic Circuits.

This site also describes the numbering systems used in Binary Coded 
Decimal (BCD) notation.

Also, as you progress through the material at this site, you will see 
references to control lines and function inputs.  These two concepts are 
the key to understanding how a multi-function array of logic gates 
operate.  In the case of an ALU, the individual logic gates will have more 
than two inputs (sometimes many more).  Included in these inputs will be a 
control line (or lines) which turns the gate "on".  That control line 
receives its instructions (which gates to turn on for an arithmetic 
operation, and which to leave off) from an encoder.  The encoder takes 
your input (i.e. you entered 2 + 3 on a keypad) and formats it for 
processing by the ALU.  The encoder is where function inputs are formatted 
in order to tell the ALU which gates and values to use for the operation. 

At the algebraic level, computing devices observe the same order of 
precedence as a person would when performing calculations on paper; the 
primary difference is that a person may not always enforce the precedence 
correctly, whereas the computing device will.  That order is, from left to 
right in an equation, multiplication, division, addition, and 
subtraction.  With that in mind, it is also the person's responsibility to 
ensure that an equation entered in a computing device is written in the 
correct order.  

As for rounding, most computing devices do not begin to round until they 
exceed their units of precision.  Rounding, in the digital logic sense, 
refers more to the closest number in a range of very small or very large 
numbers.  To accomplish rounding, a Floating-Point Unit(FPU) is employed.  
The FPU may or may not be incorporated into an ALU.

There is a freeware book online at Rutgers University entitled PRINCIPLES OF 
COMPUTER ARCHITECTURE by Miles Murdocca which covers almost every 
aspect of your question.  I have only given the book a cursory inspection, 
but it seems to be quite complete and correct.  You will need the Adobe 
Acrobat Reader to view the book, and it is available via a link on that 

For information on Floating Point logic, see Appendix A in the above 
referenced work.

I hope this is sufficient to answer your question.  If not, please feel 
free to contact me directly, or post a follow-up question on MAD Scientist.


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