Higher Computing Science Representing Negative Numbers

2. Real Numbers & Integers
All 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
Integers
0
-7
5
221
-31
100.052
-52.414
0.0014

3. Signed Bit Representation
The 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 = 3
111 = -3
This is known as signed bit representation.

4. Signed Bit Representation
The 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 = 3
111 = -3
This is known as signed bit representation.
The problem with signed bit representation is that there are 2
values for zero:
000 = 0
100 = -0

5. Two’s Complement Representation
A better way of representing negative numbers in binary is by using Two’s
Complement.
Two’s Complement is designed so that:
Binary Decimal
11111101 -3
11111110 -2
11111111 -1
00000000 0
00000001 1
00000010 2
00000011 3
1. the set of integers show symmetry about zero

6. Two’s Complement Representation
Two’s complement is designed so that:
00000010
+ 1
00000011
2. adding 1 to any number produces the next number (ignoring carry bits)

7. Two’s Complement Representation
To 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.

8. Two’s Complement Representation
For 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

9. Two’s Complement Representation
For 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
+1
11111011
So -5 as Two’s Complement = 11111011

10. Two’s Complement Representation
Example 2 - find the Two’s Complement of 88
88 = 01011000
1. Change all the 1’s to 0 and 0’s to 1.
10100111

11. Two’s Complement Representation
Example 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
+1
10101000
So -88 as Two’s Complement = 10101000

12. Range
The number of integers which could be stored in one byte (8 bits) is
28
= 256

13. Range
The 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.

14. Range
What range of numbers could be stored in two bytes using twos complement?
216
= 65536

15. Range
What range of numbers could be stored in two bytes using twos complement?
216
= 65536
The range of integers which could be stored in two bytes (16 bits) is
-32768 to +32767

16. Range
What range of numbers could be stored in two bytes using twos complement?
216
= 65536
The 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.

17. Credits
Higher 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