Download Document

Introduction to
Number
Representation
 Binary Numbers
 Sign/Magnitude
 2s Complement
F451 Year 10 Computing
Binary
Binary
All computer processing is
carried out digitally.
This means that the
processor handles
instructions as binary
codes – zeros and ones.
All data on a PC is
essentially 0’s and 1’s.
Converting binary into positive denary integers
 Whole positive denary (base ten) numbers are
converted into binary as follows:
 135 from denary into binary
128 + 4 + 2 + 1 = 135
MSB
LSB
128 64 32 16
1
0
0
0
8
4
2
1
0
1
1
1
The repeated division method
A method for converting denary to binary:
98 in denary into binary:
98 divide by 2 = 49 remainder 0
49 divide by 2 = 24 remainder 1
24 divide by 2 = 12 remainder 0
12 divide by 2 = 6 remainder 0
6 divide by 2 = 3 remainder 0
3 divide by 2 = 1 remainder 1
1 divide by 2 = 0 remainder 1
0 divide by 2 = 0 remainder 0
DIV
MOD
Read the binary code from the remainder from bottom to the top:
01100010 which equals 98
Binary Coded Decimal (BCD)
 BCD represents denary integers using blocks of four
binary digits.
 Each block of four is converted and the denary values
are then read off:
8
4
2
1
1
0
0
1
8
0
4
0
2
1
1
1
8
1
4
0
2
0
1
0
8+0+0+1
0+0+2+1
8+0+0+0
9
3
8
 Therefore 1001 0011 1000 in BCD = 938 in denary.
Uses of BCD
 BCD enables fast
conversions from
denary to binary for
applications such as
pocket calculators.
 Each digit on a
calculator corresponds
directly to a four-bit
block in BCD.
Storing Negative Integers
1 method is Sign/Magnitude
75
-75
MSB
128
+/- 64 32 16
0
1
1
0
0
8
4
2
1
1
0
1
1
1 is a Negative, 0 is a Positive
Sign/Magnitude
 This method has some limitations
 2 types of data in the same value (MSB is a sign)
 Makes calculations difficult by losing 1 bit
127 maximum number
+/- 64 32 16
0
Sign
1
0
0
8
4
2
1
1
0
1
1
Value or Magnitude
Storing Negative Integers
Another method is 2s Complement
-75
128 64 32 16
-128
1
0
1
1
8
4
2
1
0
1
0
1
-128+32+16+4+1=-75
2s Complement Conversion
-117
Stage 1 : work out 117 in binary
128
64
32
16
8
4
2
1
0
1
1
1
0
1
0
1
Stage 2 : Reverse the 0’s and 1’s
-128 64
1
0
32
0
16
0
Stage 3 : Plus 1
8
1
4
0
2
1
1
10
Representing characters
There are three main coding systems
that provide conversions of keyboard
characters into binary:
–EBCDIC
–ASCII
–UNICODE
EBCDIC
EBCDIC stands for Extended Binary
Coded Decimal Interchange Code.
It is an extension of BCD which includes
non-numeric characters, including all
the keyboard characters and special
characters.
It is commonly used to encode data
onto magnetic tape.
ASCII
 ASCII stands for the American Standard
Code for Information Interchange.
 It has been adopted as the industrystandard way of representing keyboard
characters as binary codes.
 Every keyboard character is given a
corresponding binary code.
 ASCII uses an 8-bit code to provide 256
characters.
UNICODE
 UNICODE is the new standard to emerge that
is replacing ASCII.
 It has been adopted by many of the big
businesses in the computing industry.
 It is designed to cover more of the characters
that are found in languages across the world.
 It has become important due to the increased
use of the Internet, as more data is being
passed around globally.