Celestin Apprentice 4/Apprentice-Release4.iso/Utilities/Programming/Micro Fox Tools/Numbering Systems Folder/About Numbering Systems

Text File | 1995-12-03 | 5.6 KB | 110 lines | [TEXT/ttxt]

Numbering Systems   Version: 1.0    Nov 1995
Author:     Jesse Carneiro
            Micro Fox Software
Address:    P.O. Box 14932 / Columbus, Ohio 43214-0932 / U.S.A.
Internet:   [e-mail]
Web Page:   http://www.infinet.com/~microfox
Shareware:  Send a postcard of your town, if you use it.

  The application Numbering Systems serves as a little utility for
  converting Decimal into Hexadecimal and Binary systems and vice
  versa.

  It shows the relationship of numbers in different systems. If you
  enter a decimal number and press the Enter or Return key, it will
  show on the Hexadecimal and Binary fields the representation of the
  entered number in those systems. Now if you press the Up or Down
  arrow keys, you’ll add or subtract one to that number and see its
  progression on the other fields. Notice how they change as you keep
  adding. Observe specially when you change from decimal 255 to 256.

Representing Numbers
  We grew up quite familiar with the decimal numbering system. It
  seems so simple and quite adequate for what we need. However, when
  you start to learn programming, soon you’ll encounter other systems.
  At first these other numbering systems seem hard to understand (at
  least it seemed to me). But soon, you’ll understand the reason for
  their use.

  Computers, in general, use memory in which every byte (eight
  consecutive bits of zeros or ones) has a number associated with it.
  This number associated with every byte is said to be the address of
  that byte. A way to get to it. It starts at zero and move on up to
  however many megabytes of memory the user has installed on his
  machine. Pretty soon, using the decimal system, we would need many
  digits just to represent most of these bytes. If they are after the
  million byte mark, we will need six digits, etc., but using the
  hexadecimal numbering system, we could refer to these bytes with
  only a few digits, which is more convenient for programmers. On the
  other hand, sometimes a single byte (composed of 8 bits) could have
  some of its bits represent “flags," that is, the state of something
  (usually something that can be "ON" or "OFF" — often referred to as
  that bit being set [1] or not being set [0]).

  However, as the accompanying program, Numbering Systems, will show
  you, turning bits ON (changing them from zero to one) on a byte,
  will accommodate numbers only up to 255*. As such, the number 255
  represent all 8 bits being set to one. Add one more 1 to it and you
  will need another byte to represent it. Now we have a lot more bits
  to set. But soon you will run into a number limit when again, all of
  its bits are set to one. That number is 65535 in decimal, FFFF in
  hexadecimal or 1111 1111 1111 1111 in the binary system (Please
  Note: I am only referring to unsigned numbers, i.e., whole positive
  numbers). It represents the maximum number that can fit on two bytes.
  To represent larger numbers you need more bytes. It is obvious by
  now that the more bits (or bytes, if you group them) that you have
  to represent a number, the larger the number can be.

  Just like we group letters together to form words and then group
  words together to form sentences, and then group these to form
  paragraphs, etc., in computer programming we group 8 bits together
  and call it a Byte. We group two bytes and call it an Integer. We
  group four bytes and call it a Long Integer, group a mixture of
  these and you can call it a struct (in C) or a Record (in Pascal).
  In the programming languages Pascal, C, etc. that is the norm, but
  note that on some programming languages, an integer and a long
  integer may both have four bytes. They are used to accommodate
  numbers that can be fit within their limits.

Addresses
  On the Macintosh, as with other computers, by virtue of its hardware
  design, the address of the bytes in memory is given 32 bits (4 bytes)
  for the representation of this address number. As such, you can fit
  a number as larger as decimal 4,294,967,296 (over four billion),
  FFFFFFFF in hexadecimal or 32 ones in binary. That would be the
  maximum number of bytes that such computer could access. It is an
  amazingly high number. Can you imagine 4 gigabytes of RAM memory?

  Just remember that all of this is only different ways to represent
  numbers. Decimal numbers makes more sense to us. Hexadecimal is very
  handy when working with a programming language. And zeros and ones
  is only what the CPU of a computer understands anyway. But they all
  represent the same thing: a number.

  I hope this helps a bit (no pun intended).

    Jesse Carneiro
    Micro Fox Software
    P.O. Box 14932
    Columbus, OH 43214-0932
    U.S.A.
    Phone or Fax [phone]
    Internet: [e-mail]   (support, comments, etc.)
    Web Page: http://www.infinet.com/~microfox


  *PS - The program Numbering Systems only deals with positive numbers
        that can fit on an integer (two bytes). In reality, a byte can
        accommodate negative numbers also. The difference being that
        instead of the range being 0 to 255 as is on an unsigned integer,
        the range becomes –127 to +128 on a signed integer. The same
        treatment is given to signed long integer.


  ©1995 Jesse Carneiro
  The usual disclaimers apply: Jesse Carneiro and Micro Fox Software
  accept no responsibility for any incidental or consequential damage
  of any sort as a direct or indirect result of using this software.
  The software is provided “as-is,” and no warranties are made as to
  its fitness for a particular purpose.  I would be very interested to
  hear of any problems, though.