Computer Systems and Elements of Programming

1
Computer Systems and Elements of Programming

• Lecturer Steve Maybank
• School of Computer Science and Information
  Systems
• sjmaybank_at_dcs.bbk.ac.uk
• http//www.dcs.bbk.ac.uk/sjmaybank
• Autumn 2009
• Week 5a Number Representations

2
Algorithm for Computing the GCD

• Define Floorxlargest integer n such that nltx,
  where x is a real number.
• Input integers m, n such that mgtngt0 and mgt0
• While n?0
• qFloorm/n
• rm-qn
• mn
• nr
• EndWhile
• Printm

3
Integers

• The integers are the whole numbers ,
  -2,-1,0,1,2,
• Natural numbers 0,1,2,3
• Strictly positive integers 1,2,3,
• Strictly negative integers -1,-2,-3
• If a, b are integers then exactly one of the
  following statements is true
• altb, a b, a gt b

4
Fractions

• Fractions are ratios of integers, e.g. 1/2,
  10/36,-4/5, 0/1,
• In the ratio a/b, a is the numerator and b is the
  denominator.
• The denominator cannot be zero.
• If rltgt0, then the number (r a)/(r b) is the same
  as the number a/b.
• The fractions are called rational numbers.

5
Arithmetic with Fractions

• Addition
• Multiplication

6
Examples

• 7/121/3
• 3/72/11
• 4/9-1/7
• Find the larger of the following two fractions
  2/7, 3/8.
• Find a fraction of the form a/7 that is as close
  as possible to 1/3.

7
Decimal Fractions

• The fraction a/b is a decimal fraction if b is a
  power of 10, b10n where n is an integer.
• Examples 1/10, 16/100, 283/1000.
• Notation 0.1, 0.16, 0.283.
• The point in 0.16 is called the radix point.

8
Dyadic Numbers

• The fraction a/b is a dyadic number if b is a
  power of 2, b2n where n is an integer.
• Examples ½, 16/26 , 47/27.
• Binary notation 0.1, 0.01, 0.0101111.
• The point in 0.1 is called the radix point.

9
Binary Fractions and Decimal Fractions

• BinaryFraction101.01
• BinaryFraction10101/100
• DecimalFraction21/4
• DecimalFraction5 ¼
• BinaryFraction101.01
• DecimalFraction1x22 0x21 1x20 0x2-1 1x2-2
  
• DecimalFraction411/4
• DecimalFraction5 ¼

10
Example Addition of Binary Fractions
carry
carry
carry
carry
11
Example Subtraction of Binary Integers
Repay in 2nd row
Borrow in top row
12
Examples

• Add the following two binary fractions 101.1,
  1.11
• Subtract the binary integer 1011 from the binary
  integer 11001.

13
Representations of Negative Integers

• Put a minus sign in front of the representation
  for a positive integer.
• Excess notation.
• Twos Complement notation the most popular
  representation for negative integers in computers.

14
Excess Notation

• Problem represent a set of positive and negative
  integers using bit strings with a fixed length n.
• Represent 0 by 100 (n bits).
• Represent positive numbers by counting up from
  100 in standard binary notation.
• Represent negative integers by counting down from
  100 in standard binary notation.

15
Example of Excess Notation
111 3 110 2 101 1 100 0 011
-1 010 -2 001 -3 000 -4

• n3

16
Twos Complement Notation

• Form the bit string 100 with n1 bits.
• Represent 0 by the last n bits of 100.
• Represent positive integers by counting up from
  100 in standard binary notation and using the
  last n bits.
• Represent negative integers by counting down from
  100 in standard binary notation and using the
  last n bits.

17
Example of Twos Complement Notation
0111 7 0110 6 0101 5 0100 4 0011
3 0010 2 0001 1 0000 0
1111 -1 1110 -2 1101 -3 1100 -4 1011
-5 1010 -6 1001 -7 1000 -8
n4 The left most bit indicates the sign.
18
Addition and Subtraction

• In the twos complement system subtraction
  reduces to addition.
• E.g. to evaluate 6-5 in 4 bit twos complement
  notation, add the tc bit strings for 6 and 5,
  then take the four rightmost bits.

0110 6 1011 -5 10001
1
19
Explanation

• The bit string for TC6 equals the rightmost
  four bits of the bit string for Binary246,
  etc.
• The bit strings TC6, TC-5 are added as if
  they were binary numbers. The rightmost four bits
  of the result equal the rightmost four bits of
• Binary(246)(24-5) Binary2424 1.
• The right most four bits of Binary2424 1
  equal the bit string for TC1.

20
Twos Complement Notation for m and -m

• Suppose TCm s 1 t, where t is a string
  of zeros.
• Then TC-mComplements1t.
• Proof the rightmost n bits of TCmTC-m are
  all zero.
• Example n4,
• TC30011, TC-31101.

21
Example

• Find the 5 bit twos complement representations
  for 5 and -5.