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Problem Solving using computers

Data..Representation & storage

Representation of Numeric data

The Binary System

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What Types Of Numbers ?

Number A representation of the count of things

• Natural Numbers– The Number 0;– Any number obtained by repeatedly adding a count of 1 to the

previous, starting with 0;– Think of them as ‘counting up’ from the 0;

• Negative Numbers– A value less than 0;– Represents a ‘count down’ from the 0;

• IntegerA Natural Number, a negative of a natural number, or 0.

An integer number system is a system for ‘counting things’ in a simple systematic way.

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Problem statementDevelop a scheme for representation of numbers

using bits(lets focus on positive integer numbers).

• Can we represent numbers as we did for text data ? (i.e. represent each digit appearing in the number by its corresponding 8-bit ASCII code (Click here ).

• For example,The representation of 37 will be: 00110011 00110111 3 7 • Disadvantage? Non economical. if we are restricted to use bit

patterns of length 16 for each number , how many possible numbers can we represent ?

Shockingly … only the numbers from 00 to 99 , why?• Representing numbers by the ASCII codes of their digits is

grossly inefficient, any alternatives?

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Lets imitate what we know;The Decimal System

• How is a positive integer represented in the decimal system?

• For example why is 375 equal to three hundred and seventy five?

375 = 3 *100 + 7 * 10 + 5*1

= 3*102 + 7 * 101 + 5*100

• Another look at 375

33 77 55

101000101011101022Position weightsPosition weights

Number digitsdigits

5*105*100 0 = 5= 5

7*107*1011 = 70 = 70

3*103*1022 = 300 = 300

+

+

375375

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Lets imitate what we know;Decimal System principles

• A number is a sequence of digits . Any digit must be in the set {0,1,2,3,4,5,6,7,8,9} (that’s why its is called base 10 system).

• Each digit contributes to value the number represents.

• The value contributed by a digit equals the product of the digit times the weight of the position of the digit in the number.

• Position weights are powers of 10; The weight of the right most (least significant position) is 100

.

• The weight of any position 10x, where x is the number of positions to the right of this position.

10

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10

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10

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331

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3

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Lets imitate what we know;From Decimal to binary

• What if all we had was bits ? Meaning that the only possible digits we can write into a bit are {0,1}?

• Lets apply all the principles of base 10 system to construct a “base 2 system”

220022112222Position weightsPosition weights

digitsdigits

22332244

1101

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Converting a number in base 2 to its base 10 equivalent

Example

• The number 100101 represented in base 2 system

(written (100101)2 ) =

1*20+

0*21+

1*22+

0*23+

0*24+

1*25

(37)(37)1010

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Converting a number in base 10 to its base 2 equivalent

• (125)10 = (? )2 (i.e. the number 125 in base 10 is equal to what in base 2?)

• There is a standard algorithm we can apply to answer this question:– (1) Start with your number, call it n, in base 10;– (2) Divide n by 2 and record the remainder;– (3) If the quotient = 0 stop, else Assign the quotient value to n, and go to step 2;

• When you stop the remainders recorded will be the digits of the number in base 2 in the positions from rightmost to leftmost;

• Lets apply this algorithm to an example.

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Converting a number in base 10 to its base 2 equivalent

• (125)10 = (? )2

• Thus (125)10 = (1111101)2

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Base 2 System Versus

ASCII encoding of digits• Assume can only use 16 bits to represent each positive integer, how

many different numbers can we represent:– Using the ASCII encoding of digits? – Using base 2 system?

• Answer– Using ASCII encoding (only the number from 0 – 99 can be

represented; a total of 100 distinct numbers)

– Using Base 2 system • We have 216 distinct bit patterns; each can be used to

represent a distinct number; a total of 65536 distinct numbers

• The numbers will be in the range: zero (0000000000000000)2 to 65535 (1111111111111111)2

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Base 2 System Arithmetic Addition

• We represent Numbers because we need to do Arithmetic operations on them;

• Can we do arithmetic on numbers represented in base 2 ? Yes

• Consider Addition; When we first did it in elementary school we had to learn and memorize addition facts

• For the binary system the following are the addition facts

• 0 + 0 = 0

• 0 + 1 = 1

• 1 + 0 = 1

• 1 + 1 = 10 (i.e. 0 and carry 1)