Download PowerPnt

Agenda

Data Representation


Numbering Systems


Purpose
Binary, Octal, Hexadecimal (Hex)
Numbering System Conversions


Binary to Octal, Octal to Binary
Binary to Hex, Hex to Binary
Data Representation
Manipulating binary data on the computer:
C Programming: Convert lowercase to uppercase.
Network communication.
Unix:
Set access permissions for directories, files
Webpage Programming:
Set the colour of
webpage background, text and links
Before you can program at the “machine level” you need
to understand how computers store data such as numbers
and characters.
Binary Code


Each digit in the binary code (i.e “0” or “1”) is
called a bit. A certain number of bits are used
to represent a character or a digit (part of a
large number) called a byte.
How many of these binary digits (bits)
should be put together to form a code (byte)
to represent all the characters / numbers on a
computer system?
Binary Code

A mathematical formula is used to determine the total number of
character / number possibilities for binary code:
2^n
The “2” represents the binary code (i.e. “0” and
“1” in the code). The “n” represents the total
number of digits in the binary code.
Example: 2^8 = 256
Therefore, an eight-digit binary code would allow
a total of 256 characters, or positive integers from
0 to 255, or integers from –128 to 127.
Remember that zero takes up one of those
numbers as well!

Earlier computers used binary codes that consisted of either 7 or 8 bits.
Newer computers use binary codes consisting of multiples of 8 (e.g. 8,
16, 32, 64 bit computers).
Numbering Systems




Humans are most comfortable with the decimal
numbering system (numbers with digits 0 - 9).
Early in computer development, humans were forced
to communicate with (and program) computers in
“binary” which was time-consuming.
To compromise, humans used other numbering
systems (such as octal & hexadecimal) to provide a
condensed (“short-hand”) method when working with
computers.
Binary, octal and hexadecimal numbers are
sometimes still used in programming commands as
well as Unix commands.
Numbering Systems

Decimal (Base 10)


Binary (Base 2)
numbering system for computers
Octal(Base 8)/ Hexadecimal(Base16)
 Short-hand (short-cut) method of
representing binary numbers


numbering system for humans
Decimal Numbers (Base 10)
Look at number 3,572 as sums of powers of 10:
+
+
+
=
3x10^3
5x10^2
7x10^1
2x10^0
=
=
=
=
3x1000
5x100
7x10
2x1
= 3000
= 500
= 70
=2
3,572
Binary Numbering System

You can use the same method to
convert a binary number to a decimal
number:


Multiply digits with appropriate power of 2
(i.e. 2^0, 2^1, 2^2, 2^3, etc…)
Add results together to determine decimal
number.
Binary Numbers (Base 2)
Convert binary number 10100100 to a decimal number:
+
+
+
+
+
+
+
1x2^7
0x2^6
1x2^5
0x2^4
0x2^3
1x2^2
0x2^1
0x2^0
=
=
=
=
=
=
=
=
1x128
0x64
1x32
0x16
0x8
1x4
0x2
0x1
= 128
=0
= 32
=0
=0
=4
=0
=0
164 (Decimal number)
Therefore the binary code 10100100 represents the decimal
number 164.
Octal Numbers (Base 8)
Convert the octal number 264 to a decimal number:
+
+
=
2x8^2 = 2x64 = 128
6x8^1 = 6x8 = 48
4x8^0 = 4x1 = 4
180 (Decimal number)
Therefore the octal number 264 represents the decimal
number 180.
Hexadecimal Numbers (Base 16)
Convert the Hexadecimal number A1C to a decimal
number:
Note - In Hex:
+
+
=
A = 10
B = 11
C = 12
D = 13
E = 14
F = 15
Ax16^2 = 10x16^2 = 10x256 = 2560
1x16^1 = 1x16^1 = 1x16 = 16
Cx16^0 = 12x16^0 = 12x1 = 12
2588 (Decimal number)
Therefore the hexadecimal number A1C represents the decimal
number 2588.
Numbering System Shortcuts
There is a relation between binary digits and other
number systems like octal and hexadecimal (hex):
1 Octal digit = 3 binary digits
1 Hex digit = 4 binary digits
Some Unix operations require working at the binary
level. Since working at the binary level is tedious,
certain Unix commands allow for the use of octal
numbers to efficiently represent binary code.
Binary to Octal Conversion
Convert the binary number 111110000 to an octal
number:
=
x
i.e.
=
1 1 1
1 1 0
0 0 0
2^2 2^1 2^0
(4) (2) (1)
1x4+ 1x2+ 1x1
2^2 2^1 2^0
(4) (2)
(1)
1X4+ 1x2+ 0x1
2^2 2^1 2^0
(4)
(2) (1)
0X4+ 0x2+ 0x1
7
6
0
Remember:
1 octal number is
equal to 3 binary
numbers. Group
binary numbers into
groups of 3’s starting
from the right and
multiply each digit by
the appropriate power
of 2, then add.
Therefore, the binary number 111110000 represents 760 as an
octal number. This code can be used to represent directory and
file permissions (you will learn how to set permissions next
week).
Octal to Binary Conversion
Similar to previous calculation, but in reverse:
Convert octal number 760 to binary.
7
=
6
0
(4)(2)(1)
(4)(2)(1)
(4)(2)(1)
1 1 1
1 1 0
0 0 0
111110000
“Spread-out” octal
number to make room
for binary number
result.
Determine digits
(0’s or 1’s) that are
required when
multiplied by
appropriate power of 2
to add up to octal digit.
Binary to Hex Conversion
=
Convert the binary number 111110000 to a hexadecimal
number:
Note:
0 0 0 1
1 1 1 1
0 0 0 0
1 hexadecimal
(8) (4) (2) (1)
1
1
(8) (4) (2) (1)
15
F
(8) (4) (2) (1)
0
0
number is equal to 4
binary numbers.
Group binary numbers
into groups of 4’s
starting from the
right. Add leading
zeros if last group of
digits is less than 4.
To obtain result,
multiply digit by
appropriate power of
2, then add.
Therefore, the binary number 111110000 represents 1F0 as a
hexadecimal number.
Hex to Binary Conversion
Similar to previous calculation, but in reverse:
Convert hexadecimal number 1F0 to binary.
1
1
(8)(4)(2)(1)
0 0 0 1
=
F
15
0
0
(8)(4)(2)(1) (8)(4)(2)(1)
1 1 1 1
0 0 0 0
000111110000 = 111110000
“Spread-out” hex
number to make room
for binary number
result.
Determine digits
(0’s or 1’s) that are
required when
multiplied by
appropriate power of 2
to add up to
hexadecimal digit.
Hexadecimal Numbers
Example

When creating a webpage, you can set the background colour
by using a formatting tag with a hexadecimal number.
Example:
<body bgcolor=“#66ff99” text=“#66cccc” link=“#0000ee”>

Each 2 digit pairs indicate a colour intensity.
Example of bgcolor attribute:
66 – Red colour setting
ff - Blue colour setting
99 - Green colour setting
The combination of red, blue and green
colour settings will create any colour
desired. Check the “webpage colour
selector” link contained in UNX122 course
notes!
Representing Signed
Numbers in Binary
An interesting technique is used to represent signed (positive
and negative) numbers:



The first bit is used to represent a negative number.
The remaining bits are positive.
The negative (or positive number if the first bit is 0) is determined by adding sums of
the power of 2.
Eg. 11100101 = -(1x2^7)+1x2^6+1x2^5+0x2^4+0X2^3+1x2^2+0x2^1+1x2^0
= -27 (if recognized as a signed integer)
Eg. 01100101 = -(0x2^7)+1x2^6+1x2^5+0x2^4+0X2^3+1x2^2+0x2^1+1x2^0
= +101 (if recognized as a signed integer)
The computer uses special mathematical tricks such as
“bit-shifting” and “two’s complement” to perform common
mathematical operations other than simply add numbers.