Post on 10-Jan-2017
Real Numbers & IntegersAll numbers belong to the set or group called real numbers.
Inside the set of real numbers is a set of all positive and negative whole numbers.
This set is called integers.
Real numbers
-4.6
2.46
13.7
Integers0
-7
5221
-31100.052
-52.4140.0014
Signed Bit RepresentationThe simplest way of representing a negative number in binary is to use the first bit of the number to represent whether the number is positive or negative:
011 = 3111 = -3
This is known as signed bit representation.
Signed Bit RepresentationThe simplest way of representing a negative number in binary is to use the first bit of the number to represent whether the number is positive or negative:
011 = 3111 = -3
This is known as signed bit representation.
The problem with signed bit representation is that there are 2 values for zero:
000 = 0100 = -0
Two’s Complement RepresentationA better way of representing negative numbers in binary is by using Two’s Complement.
Two’s Complement is designed so that:
Binary Decimal11111101 -311111110 -211111111 -100000000 000000001 100000010 200000011 3
1. the set of integers show symmetry about zero
Two’s Complement RepresentationTwo’s complement is designed so that:
00000010+ 1
00000011
2. adding 1 to any number produces the next number (ignoring carry bits)
Two’s Complement RepresentationTo find the Two’s Complement of a number (its opposite sign):
1. Change all the 1’s to 0 and 0’s to 1.
2. Add 1.
Two’s Complement RepresentationFor example, how would -5 be represented using Two’s Complement?
5 = 000000101
1. Change all the 1’s to 0 and 0’s to 1.
11111010
Two’s Complement RepresentationFor example, how would -5 be represented using Two’s Complement?
5 = 000000101
1. Change all the 1’s to 0 and 0’s to 1.
11111010
2. Add 1.
11111010 +111111011
So -5 as Two’s Complement = 11111011
Two’s Complement RepresentationExample 2 - find the Two’s Complement of 88
88 = 01011000
1. Change all the 1’s to 0 and 0’s to 1.
10100111
Two’s Complement RepresentationExample 2 - find the Two’s Complement of 88
88 = 01011000
1. Change all the 1’s to 0 and 0’s to 1.
10100111
2. Add 1.
10100111 +110101000
So -88 as Two’s Complement = 10101000
RangeThe number of integers which could be stored in one byte (8 bits) is
28 = 256
RangeThe number of integers which could be stored in one byte (8 bits) is
28 = 256 The range of integers which could be stored in one byte (8 bits) using Two’s Complement is
-128 to +127
Why does there seem to be one less positive number?
There are 255 numbers plus the value 0. So there are 256 numbers in all.
RangeWhat range of numbers could be stored in two bytes using twos complement?
216 = 65536
RangeWhat range of numbers could be stored in two bytes using twos complement?
216 = 65536The range of integers which could be stored in two bytes (16 bits) is
-32768 to +32767
RangeWhat range of numbers could be stored in two bytes using twos complement?
216 = 65536The range of integers which could be stored in two bytes (16 bits) is
-32768 to +32767
This method of representing large numbers is unsuitable because of the increased memory needed to store the large number of bits needed.
A solution to this is to use Floating Point Representation.
CreditsHigher Computing – Data Representation – Representation of Negative Numbers
Produced by P. Greene and adapted by R. G. Simpson for the City of Edinburgh Council 2004
Adapted by M. Cunningham 2010