Data, Representation & Numeracy, Slides of Computer Science

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Lecture 1 – Part B

Data, Representation &

Numeracy

Data, Representation & Numeracy

• In order to understand how data is processed by computers it

is necessary to understand

• what is data
• how is data represented in computers
• how is data manipulated and processed

Size terminology

• The following prefix terms are used to denote timing interval

size (seconds):

• 10

   milli 

• 10

   micro 

• 10

   nano 

• 10

   pico 

• 10

   femto 

• The following prefix terms are used to denote storage device

size (bytes):

• 2 10 kilo (1024)
• 2 20 mega ( ~ 10 6 )
• 2 30 giga ( ~ 10 9 )
• 2 40 tera ( ~ 10 12 )
• 2 50 peta ( ~ 10 15 )

Binary Coding

• Binary, or base-2, coding is used for all data in the computer
• 2-state switches
  • only 0 and 1 are used as values
  • each corresponds to a specific electrical (magnetic/optical) signal value
• Characters
  • the ASCII system is used to represent printable characters and also some non-printable signals used in inter-computer communication across networks
• Numbers
  • unsigned integer
  • signed integer
  • real, or floating point, numbers
  • decimal (base-10) numbers are also used in computers

Positional Representation (2)

• In a system based on a different multiplier, or radix (we say, “base

-R”) we may represent a general, positive number N in the form:

N = d

L-

d

L-

… d

2

d

1

d

0

. f

1

f

2

… f

M

where each d or f represents a single digit, but now the concept

and representation of digit takes on a new meaning.

• This can be rewritten in the form:

L-1 M

N = Sum dk * R

k

+ Sum fk * R

-k

; 0 ≤ dk , fk < R

k=0 k=

• This representation still permits us to perform the basic arithmetic

operations as we did for decimal arithmetic!

Unsigned Binary

• Non-negative integers (greater or equal to zero)
  • Range from 0 (smallest) to 2 L -1 (largest) for an L-bit representation
• We can construct an unsigned binary sequence using the

algorithm

• Build up starting from 0.
• Add 1 to the previous number.
• Repeat previous step until a string of 1’s of length L is reached. Binary Dec. Binary Dec. 0111 7 1111 15 0110 6 1110 14 0101 5 1101 13 0100 4 1100 12 0011 3 1011 11 0010 2 1010 10 0001 1 1001 9 0000 0 1000 8

Signed Binary - 1’s complement

• In most cases we require both positive and negative integers
• One idea for “designing” negative numbers uses complementation
  • change 0’s to 1’s and 1’s to 0’s
• Values range from - L- +1 (most negative) to 2 L- -1 (largest, most positive) for an L-bit representation.
• Permits two different representations of zero! Binary Dec. Binary Dec. 0111 7 1000 - 0110 6 1001 - 0101 5 1010 - 0100 4 1011 - 0011 3 1100 - 0010 2 1101 - 0001 1 1110 - 0000 0 1111 -

Signed Binary - 2’s complement

• We need a more efficient representation of positive and

negative integers using an L-bit form.

• One straightforward approach is to consider the operation:

X

+ Y

0 What representation of Y gives a zero sum?

**00000011

• bbbbbbb 00000000 3 Y**

Signed Binary - 2’s complement

• We need a more efficient representation of positive and

negative integers using an L-bit form.

• The 2’s complement scheme is based on the algorithm:
  • 1. Find the 1’s complement
  • 2. Add 1 to the result to find the 2’s complement form. **3 00000011 ~3 11111100

• 1 -3 11111101 2 1**

(r-1)’s and r’s complements

• In an arbitrary radix of fixed (finite) representation length L,

there are two related complements that are important.

• (r-1)’s complement form.
  • Given the number N in radix-r form, the (r-1)’s complement is defined as r L^ – 1 – N
  • Consider r = 10, L = 4 and N = 2468. Clearly, r 4 = 10000, thus r 4 – 1 = 9999. Thus, **9999
• 2468 7531 Easy!**
  • Note that the sum of N and the (r-1)’s complement of N is always a string of (r- 1)’s (9 in this example).
• r’s complement form
  • The r’s complement is found from the (r-1)’s complement by just adding 1.
  • Note that this guarantees that the sum of N and its r’s complement is just zero (0) to within the size L of the representation!

Subtraction of unsigned Numbers

• Example: M < N, L=5 ( M = 13250, N = 72532)

Borrowing: 13250 10’s Compl.: 13250

- 72532 + 27467

Find 10’s compl., add minus

Final Answer: - 59282

Converting between arbitrary bases

• Consider the problem of converting the number 15310 into its equivalent number in base-.
• Although this may seem strange at first, it is solved by repeatedly dividing 153 (and successive quotients) by 13 and collecting remainders, hence:
• 13 into 153 11 10 (A) 0 11 (B) ANSWER: BA
• Problem: What is the radix of the integer numbers (coefficients) below if the solution to the quadratic equation shown is x=5? x 2 **- 12 x + 39 = 0 Substituting x = 5 and expanding the positional representations of each coefficient: 5 2
• ( 1R + 2 ) * 5 + ( 3R + 9 ) = 25 – 5R – 10 + 3R + 9 = 24 – 2R = 0 It follows from the last step that R = 12. WARNING: Converting to arbitary bases may be quite difficult. If terms up to R 4 arise (a quartic equation) it can be solved exactly, but not for higher powers. Such problems require brute force.*

Floating Point

• Use scientific notation to represent the
  • sign
  • fraction (mantissa)
  • exponent sign exponent fraction 0 + (2’s compl.) 1 - When the leftmost (ie. most significant) fraction bit is one (1) the fraction is called normalized; otherwise it is unnormalized. Note that the exponent may be positive or negative. We use the 2’s complement form. The exponent is then called the characteristic.

Binary Coded Decimal (BCD)

• Many computers contain logic to process decimal

representations directly.

• The standard representation is based on an ASCII-compatible

scheme for each separate decimal digit. This is called the

BCD scheme.

• Note that a minimum of 4 bits is required to represent all 10

decimal digits.

0 0000 5 0101 1 0001 6 0110 2 0010 7 0111 3 0011 8 1000 4 0100 9 1001