introduction to number representation f451 year 10 computing binary numbers binary numbers...
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Introduction to Introduction to Number Number
RepresentationRepresentation
Introduction to Introduction to Number Number
RepresentationRepresentation
F451 Year 10 ComputingF451 Year 10 Computing
Binary NumbersBinary Numbers Sign/MagnitudeSign/Magnitude 2s Complement2s Complement
Binary NumbersBinary Numbers Sign/MagnitudeSign/Magnitude 2s Complement2s Complement
BinaryBinaryBinaryBinary
BinaryBinaryBinaryBinary
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 integersConverting binary into positive denary integersConverting binary into positive denary integersConverting 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 = 128 + 4 + 2 + 1 = 135135
128128 6464 3232 1616 88 44 22 11
MSBMSBMSBMSB LSBLSBLSBLSB
11 00 00 00 00 11 11 11
The repeated division methodThe repeated division methodThe repeated division methodThe repeated division methodA method for converting denary to binary:98 in denary into binary:
98 divide by 2 = 49 remainder 049 divide by 2 = 24 remainder 124 divide by 2 = 12 remainder 012 divide by 2 = 6 remainder 06 divide by 2 = 3 remainder 03 divide by 2 = 1 remainder 11 divide by 2 = 0 remainder 10 divide by 2 = 0 remainder 0
Read the binary code from the remainder from bottom to the top:
01100010 which equals 98
DIVDIV
MODMOD
DIVDIV
MODMOD
Binary Coded Decimal (BCD)Binary Coded Decimal (BCD)Binary Coded Decimal (BCD)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:
Therefore 1001 0011 10001001 0011 1000 in BCD = 938938 in denary.
88 44 22 11
11 00 00 11
8 + 0 + 0 + 18 + 0 + 0 + 1
99
88 44 22 11
00 00 11 11
0 + 0 + 2 + 10 + 0 + 2 + 1
33
88 44 22 11
11 00 00 00
8 + 0 + 0 + 08 + 0 + 0 + 0
88
Uses of BCDUses of BCDUses of BCDUses 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 IntegersStoring Negative IntegersStoring Negative IntegersStoring Negative Integers
1 method is Sign/Magnitude
6464 3232 1616 88 44 22 11128128+/-+/-
MSBMSBMSBMSB
00 11 00 00 11 00 11 11
75-75
11
1 is a Negative, 0 is a Positive1 is a Negative, 0 is a Positive1 is a Negative, 0 is a Positive1 is a Negative, 0 is a Positive
Sign/MagnitudeSign/MagnitudeSign/MagnitudeSign/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
6464 3232 1616 88 44 22 11+/-+/-
00 11 00 00 11 00 11 11
127 maximum number127 maximum number127 maximum number127 maximum number
SignSignSignSignValue or MagnitudeValue or MagnitudeValue or MagnitudeValue or Magnitude
Storing Negative IntegersStoring Negative IntegersStoring Negative IntegersStoring Negative Integers
Another method is 2s Complement
6464 3232 1616 88 44 22 11128128-128-128
11 00 11 11 00 11 00 11
-75
-128+32+16+4+1=-75
2s Complement Conversion2s Complement Conversion2s Complement Conversion2s Complement Conversion -117-117Stage 1 : work out 117 in binaryStage 1 : work out 117 in binary
-128-128 6464 3232 1616 88 44 22 11
11 00 00 00 11 00 11
Stage 2 : Reverse the 0’s and 1’sStage 2 : Reverse the 0’s and 1’s
128128 6464 3232 1616 88 44 22 11
00 11 11 11 00 11 00 11
Stage 3 : Plus 1Stage 3 : Plus 1
1100
Representing charactersRepresenting charactersRepresenting charactersRepresenting characters
There are three main coding systems that provide conversions of keyboard characters into binary:
–EBCDIC
–ASCII
–UNICODE
EBCDICEBCDICEBCDICEBCDIC
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.
ASCIIASCIIASCIIASCII
ASCII stands for the American Standard Code for Information Interchange.
It has been adopted as the industry-standard 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.
UNICODEUNICODEUNICODEUNICODE
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.