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NUMBER SYSTEM
Introduction to Numbering Systems
• We are all familiar with the decimal number
system (Base 10)
...
The number of digits is equal to the size of the base
...
The base number is never a digit
...
Numeric values determined by the have implicit positional
values of the digits
...
Pressure as the
height of column of mercury or as voltage produced by
a pressure transducer
These are ANALOG measurements
5
Digital Representation
Convert ANALOG to DIGITAL measurement by using a
scale of units
DIGITAL measurements
– In units – a set of symbolic values - digits
– Values larger than any symbol in the set use sequence
of digits – Units, Tens, Hundreds…
– Measured in discrete or whole units
– Difficult to measure something that is not a multiple of
units in size
...
Digital representation
7
Data Representation
Computers use digital representation
– Based on a binary system
(uses on/off states to represent 2 digits)
...
– Examples?
– ALL data (no matter how complex)
must be represented in memory as binary digits (bits)
...
Computer programs and data are often represented (outside
the computer) using octal and hexadecimal number systems
because they are a short hand way of representing binary
numbers
...
There are 10 distinct digits (0 to 9) to represent any
quantity
...
The weights are based on powers of 10
...
Decimal to Binary Conversion
• The easiest way to convert a decimal number to its binary
equivalent is to use the Division Algorithm
• This method repeatedly divides a decimal number by 2 and
records the quotient and remainder
– The remainder digits (a sequence of zeros and ones) form
the binary equivalent in least significant to most
significant digit sequence
Division Algorithm
Convert 67 to its binary equivalent:
6710 = x2
Step 1: 67 / 2 = 33 R 1
Step 2: 33 / 2 = 16 R 1
Step 3: 16 / 2 = 8 R 0
Step 4: 8 / 2 = 4 R 0
Step 5: 4 / 2 = 2 R 0
Step 6: 2 / 2 = 1 R 0
Step 7: 1 / 2 = 0 R 1
Divide 67 by 2
...
Two basic types - ones and twos complement, of which
2’s complement is the most widely used
...
Negative numbers begin with 1, positive with 0
...
e
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This is called overflow
...
However, calculations slower and more
difficult
...
So, how can
we encode numbers, images, sound, text ??
We need standard encoding systems for each type of data
...
Other standards are created by official industry bodies
where none previously existed
...
As each character is entered from a keyboard (or other
input device) it is converted into a binary code
...
Characters used as control codes