Understanding Computers: Today and Tomorrow, 13th Edition Numbering systems

Understanding Computers: Today and Tomorrow, 13th Edition


Numbering systems

Understanding Computers: Today and Tomorrow, 13th Edition2

Learning Objectives

Understand and identify a few of the coding systems .


Data and Program Representation

• Digital computers: Can only understand two states, off and on (0 and 1)

• Digital data representation: The process of representing data in digital form so it can be understood by a computer


Digital Data Representation

• Bit: The smallest unit of data that a binary computer can recognize (a single 1 or 0)

• Byte = 8 bits , Nibble =4 bits

• Byte terminology used to express the size of documents and other files, programs, etc.

• Prefixes are often used to express larger quantities of bytes: kilobyte (KB), megabyte (MB), gigabyte (GB), terabyte (TB), etc.


Quiz

I have a 2 Gb Flash memory. How many 512Kb files can I store on the Flash memory?

Answer

Convert 2Gb to Kb

2x1024=2048Mb

Convert from Mb to Kb

2048x1024= 2097152kb

Number of files= 2097152/ 512 = 4096 file

5

Note Mb is in common use for Megabit e.g. 200Megabit = 200Mb

MB is in common use for Megabyte , e.g. 200MegaByte=200MB


Numbering Systems

• A numbering system is a way of representing numbers.

• A number is represented as a string of digits, e.g., a number N with n digits represented by sequence dn-1, …, d1,d0


Numbering Systems

• Positional notation– A system of expressing numbers in which the

digits are arranged in succession, the position of each digit has a place value, and the number is equal to the sum of the products of each digit by its place value .

• Determination of a number require the following :– The digits or symbols;– The positional of the digit in the number;– The base of the number system.


Numbering Systems

• Base ( radix ) – The base of a number system is simply the number of

different digits, including zero that exist in the number system.

– A number system of radix r, typically has a set of r allowed digits belongs to {0,1,…, r-1}

• We write a number N in the number system of radix r by expressing it in the form (N)r or Nr – (12)10 1 and 2 are digits in base 10,

325 2 and 3 are digits in base 5 i.e., belongs to {0,1,..,4}


Numbering Systems

• a radix point is the symbol used in numerical representations to separate the integer part of a number (to the left of the radix point) from its fractional part (to the right of the radix point). – Ex. 13.62510,

• 13 is the integer to the left of the radix point, and

• 625 (i.e. 625/1000) is the fractional part to the right.


Understanding Decimal Numbers

• The numbering system we commonly use called the decimal numbering system because it uses 10 digits - symbols - (0,1,2,…,9) to represent all possible numbers.

• Why do we use 10 digits, anyway?

• Numbers greater than nine such as 7216 represented using combination of these 10 digits.


Understanding Decimal Numbers

• Decimal numbers are made of decimal digits: (0,1,2,3,4,5,6,7,8,9)

• But how many items does a decimal number represent?– 8653 = 8x103 + 6x102 + 5x101 + 3x100

• What about fractions?– 97654.35 = 9x104 + 7x103 + 6x102 + 5x101 + 4x100 +

3x10-1 + 5x10-2

– In formal notation -> (97654.35)10



Understanding Octal Numbers

• Octal numbers are made of octal digits: (0,1,2,3,4,5,6,7)• How many items does an octal number represent?

– (4536)8 = 4x83 + 5x82 + 3x81 + 6x80 = (1362)10

• What about fractions?– (465.27)8 = 4x82 + 6x81 + 5x80 + 2x8-1 + 7x8-2

• Octal numbers don’t use digits 8 or 9


Understanding Binary Numbers

• Binary numbers are made of binary digits (bits): – 0 and 1

• How many items does an binary number represent?– (1011)2 = 1x23 + 0x22 + 1x21 + 1x20 = (11)10

• What about fractions?– (110.10)2 = 1x22 + 1x21 + 0x20 + 1x2-1 + 0x2-2



Understanding Hexadecimal Numbers

• Hexadecimal numbers are made of 16 digits: – (0,1,2,3,4,5,6,7,8,9,A, B, C, D, E, F)

• How many items does an hex number represent?– (3A9F)16 = 3x163 + 10x162 + 9x161 + 15x160 = 1499910

• What about fractions?– (2D3.5)16 = 2x162 + 13x161 + 3x160 + 5x16-1 = 723.312510

• Note that each hexadecimal digit can be represented with four bits.– (1110) 2 = (E)16

• Groups of four bits are called a nibble.– (1110) 2


Representing Numerical Data

• Least Significant Digit (LSD) is the right most digit has the lowest value

• Most Significant Digit (MSD) is the left most digit has the highest value

• The most significant symbol can not be zeroExample:

Show how the value of the number (9375)10 is estimated

0 1 2 3 position

100 101 102 103 Weight

5 7 3 9 Digit

9375 5x1 7x10 3x100 9x1000 Value

5 is the least significant digit (LSD)9 is the most significant digit (MSD)



Commonly used numbering systemsA. Decimal number system

base =10, Ten allowed digits{0,1,…,9}

B. Binary number system

base= 2, Two allowed digits {0,1}

A. A Binary digit is referred to as Bit

B. Most significant Bit (MSB) is the leftmost bit has the highest value

C. Least significant Bit (LSB) is the rightmost bit has the lowest value

C. Octal number system

base =8, Eight allowed digits {0,1,2,…,7}

D. Hexadecimal number system

base = 16, Sixteen allowed digits {0,1,…,9, A,B,C,D,E,F}

Where, A= ten , B= eleven, C=Twelve, D= thirteen, E=fourteen, F= fifteen

Q. Why is the digit following 9 assigned the character A not 10 ?


Check point

Q. What is the largest value can be represented in three integral digits?-------------------------------------------------------------

For the decimal system, it is (- - - )10

For the octal system, it is (- - - )8

For the hex system, it is (- - - )16

For the binary system, it is (- - - )2

-------------------------------------------------------------Hint (The largest value results when all 3 positions are filled with the largest

digit in the number system)

Answer • For the decimal system, it is ( 999 )10

• For the octal system, it is ( 777 )8

• For the hex system, it is ( fff )16

• For the binary system, it is ( 111 )2


Conversion Between Number Bases

Hexadecimal

(base16)

Decimal(base 10)

Octal(base 8)

Binary(base 2)

° Learn to convert between bases.



Converting between numbering systems

1. Converting from decimal to any base

2. Converting from any base to decimal

3. Converting from any base to any base

4. A special conversion case number bases that are related (Shortcut method to/from binary )


Convert an Integer from Decimal to Another Base

1. Divide decimal number by the base (e.g. 2)

2. The remainder is the lowest-order digit

3. Repeat first two steps until no divisor remains.

For each digit position:

Example for (13)10:

IntegerQuotient

13/2 = 6 + ½ a0 = 1 6/2 = 3 + 0 a1 = 0 3/2 = 1 + ½ a2 = 1 1/2 = 0 + ½ a3 = 1

Remainder Coefficient

Answer (13)10 = (a3 a2 a1 a0)2 = (1101)2

LSB

MSB


Repeated Division-by-r

To convert a whole number to the new base r, use successive division by r until the quotient is 0. The remainders form the answer, with the first remainder as LSB or LSD and the last as MSB or MSD.

Example: (43)10 = (?)2 2 432 21 rem 1 LSB2 10 rem 12 5 rem 02 2 rem 12 1 rem 0

0 rem 1 MSB

(43)10 = (101011)2


Convert an Fraction from Decimal to Another Base

For each digit position:

1. Multiply decimal number by the base (e.g. 2)

2. The integer is the highest-order digit

3. Repeat first two steps until fraction becomes zero.

Example for (0.625)10:

Integer

0.625 x 2 = 1 + 0.25 a-1 = 10.250 x 2 = 0 + 0.50 a-2 = 00.500 x 2 = 1 + 0 a-3 = 1

Fraction Coefficient

Answer (0.625)10 = (0.a-1 a-2 a-3 )2 = (0.101)2

MSB

LSB


Repeated Multiplication-by-r

To convert decimal fractions to binary, repeated multiplication by r is used, until the fractional product is 0 (or until the desired number of digits of accuracy obtained). The carried digits, or carries, produce the answer, with the first carry as the MSB or MSD, and the last as the LSB or MSD.

Example: (0.3125)10 = (.0101)2

Carry

0.31252=0.625 0 MSB

0.6252=1.25 1

0.252=0.50 0

0.52=1.00 1 LSB


Note

For a number that has both integral and fractional parts,

• conversion is done separately for both parts, and

• then the result is put together with a system point in between both parts.


2-Any base r-to-Decimal Conversion

• Base-r to decimal: multiply digits with their corresponding weights.

• Examples (1101.101)2 = 123 + 122 + 120 + 12-1 + 12-3 =

8 + 4 + 1 + 0.5 + 0.125

= (13.625)10

(572.6)8 = 582 + 781 + 280 + 68-1

= 320 + 56 + 2 + 0.75 = (378.75)10

Convert 3b216 to decimal.

N = 3*162 + 11*161 + 2*160 = 94610

Convert 3708 to decimal.

N = 3*82 + 7*81 + 0*80 = 192 + 56 + 0 = 24810


3-Conversion between Bases

In general, conversion between bases can be done via decimal:

First convert given number to decimal then convert decimal number to the new base.

Base-2 Base-2

Base-3 Base-3

Base-4 Decimal Base-4

… ….

Base-R Base-R


Converting Between Base 16 and Base 2

° Conversion is easy!

Determine 4-bit value for each hex digit

° Note that there are 24 = 16 different values of four bits

° Easier to read and write in hexadecimal.

° Representations are equivalent!

3A9F16 = 0011 1010 1001 11112

3 A 9 F


Converting Between Base 16 and Base 8

1. Convert from Base 16 to Base 2

2. Regroup bits into groups of three starting from right

3. Ignore leading zeros

4. Each group of three bits forms an octal digit.

352378 = 011 101 010 011 1112

5 2 3 73

3A9F16 = 0011 1010 1001 11112

3 A 9 F


A special conversion case number bases that are related (Shortcut method to/from binary )

• Grouping of several digits in the smaller number base corresponds exactly to single digit in the larger number base

• Examples Binary Octal: Partition in groups of 3

(10 111 011 001 . 101 110)2 = (2731.56)8

Octal Binary: reverse

(2731.56)8 = (10 111 011 001 . 101 110)2

Binary Hexadecimal: Partition in groups of 4

(101 1101 1001 . 1011 1000)2 = (5D9.B8)16

Hexadecimal Binary: reverse

(5D9.B8)16 = (101 1101 1001 . 1011 1000)2


Arithmetic operation on numbering systems

• Addition• Subtraction• Multiplication• division


Decimal Addition Example

3 7 5 8

+ 4 6 5 7

1) Add 8 + 7 = 15Write down 5, carry 1

1

8

1 1

4 1 5 4) Add 3 + 4 + 1 = 8Write down 8

3) Add 7 + 6 + 1 = 14Write down 4, carry 1

2) Add 5 + 5 + 1 = 11Write down 1, carry 1

Add 3758 to 4657:

33


Decimal Addition Explanation

1 1 1

3 7 5 8

+ 4 6 5 7

8 4 1 5

What just happened?

1 1 1 (carry)

3 7 5 8

+ 4 6 5 7 8 14 11 15 (sum)

- 10 10 10 (subtract the base)

8 4 1 5

So when the sum of a column is equal to or greater than the base, we subtract the base from the sum, record the difference, and carry one to the next column to the left.

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Binary Addition Rules

Rules:– 0 + 0 = 0– 0 + 1 = 1– 1 + 0 = 1 (just like in decimal)

1 + 1 = 210

= 102 = 0 with 1 to carry

1 + 1 + 1 = 310

= 112 = 1 with 1 to carry

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Binary Addition Example 1

1 1 0 1 1 1 + 0 1 1 1 0 0

1

1

111

0 1 0 0 11

Example 1: Add binary 110111 to 11100

Col 1) Add 1 + 0 = 1 Write 1

Col 2) Add 1 + 0 = 1 Write 1

Col 3) Add 1 + 1 = 2 (10 in binary) Write 0, carry 1

Col 4) Add 1+ 0 + 1 = 2 Write 0, carry 1

Col 6) Add 1 + 1 + 0 = 2 Write 0, carry 1

Col 5) Add 1 + 1 + 1 = 3 (11 in binary) Write 1, carry 1

Col 7) Bring down the carried 1 Write 1

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Binary Addition Explanation

1 1 0 1 1 1 + 0 1 1 1 0 0 - .

1

1

111

0 1 0 0 11

In the first two columns, there were no carries.

In column 3, we add 1 + 1 = 2 Since 2 is equal to the base, subtract

the base from the sum and carry 1.

In column 4, we also subtract the base from the sum and carry 1.



In column 7, we just bring down the carried 1

22 2 23

222

What is actually happened when we carried in binary?

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Binary Addition Verification

Verification

1101112 5510

+0111002 + 2810

8310

64 32 16 8 4 2 1

1 0 1 0 0 1 1

= 64 + 16 + 2 +1

= 8310

1 1 0 1 1 1 + 0 1 1 1 0 0 1 0 1 0 0 11

You can always check your answer by converting the figures to decimal, doing the addition, and comparing the answers.

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Binary Addition Example 2

Verification

1110102 5810

+0011112 + 1510

7310

64 32 16 8 4 2 1

1 0 0 1 0 0 1

= 64 + 8 +1

= 7310

1 1 1 0 1 0 + 0 0 1 1 1 1

1

1

11

0 0 1 0 10

Example 2:Add 1111 to 111010.

11

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Decimal Subtraction Example (Borrow method)

8 0 2 5- 4 6 5 7

Subtract 4657 from 8025:

7 9 11

11

8633

1) Try to subtract 5 – 7 can’t.Must borrow 10 from next column.

4) Subtract 7 – 4 = 3

3) Subtract 9 – 6 = 3

2) Try to subtract 1 – 5 can’t. Must borrow 10 from next column.

But next column is 0, so must go to column after next to borrow.

Add the borrowed 10 to the original 0. Now you can borrow 10 from this column.

Add the borrowed 10 to the original 5.Then subtract 15 – 7 = 8.

Add the borrowed 10 to the original 1..Then subract 11 – 5 = 6

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Decimal Subtraction Explanation

So when you cannot subtract, you borrow from the column to the left. The amount borrowed is 1 base unit, which in decimal is 10. The 10 is added to the original column value, so you will be able to

subtract.

8633

8 0 2 5- 4 6 5 7

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Binary Subtraction Example 1

1 1 0 0 1 1- 1 1 1 0 0

Example 1: Subtractbinary 11100 from 110011

20 0 2

12

1101

Col 1) Subtract 1 – 0 = 1

Col 5) Try to subtract 0 – 1 can’t. Must borrow from next column.


Col 3) Try to subtract 0 – 1 can’t. Must borrow 2 from next column.

But next column is 0, so must go to column after next to borrow.

Add the borrowed 2 to the 0 on the right. Now you can borrow from this column

(leaving 1 remaining).


Add the borrowed 2 to the original 0.Then subtract 2 – 1 = 1

1

Add the borrowed 2 to the remaining 0.Then subtract 2 – 1 = 1

Col 6) Remaining leading 0 can be ignored.

42


Binary Subtraction Verification

Verification

1100112 5110

- 111002 - 2810

2310

64 32 16 8 4 2 1

1 0 1 1 1

= 16 + 4 + 2 + 1

= 2310

1 1 0 0 1 1- 1 1 1 0 0

20 0 2

12

1101 1

Subtract binary 11100 from 110011:

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• Note

X = (1151)10 , y = (1102)10 which one is greater than the other ?

Ans. Compare each digit from left to right

digit 1 in x = digit 1 in y goto the next digit

digit 2 in x = digit 2 in y goto the next digit

digit 3 in x > digit 3 in y stop x greater than y

Quiz: (110101)2 , (11001)2 which one is greater than the other ?

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Binary multiplication

• Binary multiplication uses the same technique as decimal multiplication.

• Example: multiplying 1102 by 102.

110x 10--------- 000+ 110 --------1100

6x 2------12

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Division

• Division rule– Set quotient to zero – Repeat while dividend is greater than or equal to

divisor • Subtract divisor from dividend • Add 1 to quotient

– End of repeat block – quotient is correct, dividend is remainder – STOP

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Binary division

• Division as repeated subtractionExample in decimal 84 ÷ 21 = ??

I subtracted 21 four times, so 84 ÷ 21 = 4

84− 21------ 63− 21------- 42− 21 ------ 21− 21------- 0

1

1

1

1

----4

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• Example: Find 1100 ÷ 0101 • Ans.

1100 - 0101 = 111

111-101= 010

Hence result is 2 and with remainder 2 ( (10)2 )

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quotient 0

1

1-----2


ONE’S COMPLEMENT AND TWO’S COMPLEMENT REPRESENTATIONS


One’s Complement Representation

• The one’s complement of a binary number involves inverting all bits.

• 1’s comp of 00110011 is 11001100

• 1’s comp of 10101010 is 01010101


Two’s Complement

• Algorithm 1 – Simply complement each bit and then add 1 to the result.– Finding the 2’s complement of (01100101)2 and of

its 2’s complement…

N = 01100101 [N] = 10011011

10011010 01100100

+ 1 + 1

--------------- ---------------

10011011 01100101


Two’s Complement Representation

Algorithm 2 – Starting with the least significant bit, copy all of the bits up to and including the first 1 bit and then complementing the remaining bits.

N = 0 1 1 0 0 1 0 1[N] = 1 0 0 1 1 0 1 1


LOGICAL OPERATION ON BINARY


Logical OR Rules

OR Operations

• OR Results in 1 if either or both of the operands are 1.

• OR Table

0 OR 0 = 0

0 OR 1 = 11 OR 0 = 11 OR 1 = 1

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Logical OR Operation

To perform the OR operation, take one column at a time and perform the OR operation using the OR table.

Ex 1: 1 0 0 1 0 0 1 1

OR 0 0 0 0 1 1 1 1

1 0 0 1 1 1 1 1

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Logical XOR Rules

XOR Operations • The exclusive OR. Similar to OR except that

it now gives 0 when both its operands are 1.

Rules.0 XOR 0 = 00 XOR 1 = 11 XOR 0 = 11 XOR 1 = 0

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Logical XOR Examples

Ex 1: 1 0 0 1 1 0 0 1

XOR 0 0 0 0 1 1 1 1

1 0 0 1 0 1 1 0

Ex 2: 0 1 1 1

XOR 0 0 1 0

0 1 0 1

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Logical AND Rules

AND Operations

• AND yields 1 only if both its operands are 1.

Rules.0 AND 0 = 00 AND 1 = 01 AND 0 = 01 AND 1 = 1

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Logical AND Examples

Ex 1: 1 1 0 1 0 0 1 1

AND 0 0 0 0 1 1 1 1 0 0 0 0 0 0 1 1

Ex 2: 0 1 1 1 AND 1 0 0 1 0 0 0 1

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Logical NOT

NOT Operations

• NOT is a separate operator for flipping the bits.

Rules.NOT 0 = 1NOT 1 = 0

Example. NOT 1 0 1 0 = 0 1 0 1

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THANK YOU

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Understanding Computers: Today and Tomorrow, 13th Edition Numbering systems