complete number system

NUMBER SYSTEMS

Number Systems

• Binary– 0, 1

• Octal– 0, 1, 2, 3, 4, 5, 6, 7

• Decimal– 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

• Hexadecimal– 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,

,8 ,Ooops! There is no 8 in octal

10, 11, 12, 13, 14, 15A, B, C, D, E, F

Data Representation in Computer Systems

• Bit– Binary digit– Either ON or OFF (high or low; 0 or 1)

• Byte– 8 bits– IBM System/360 mainframe computer established the use of 8 bits

as the basic unit of addressable computer storage• Word

– With two or more adjacent bytes– 16 bits, 32 bits, 64 bits

• Nibbles (nybbles)– the 4-bit halves of an eight-bit byte– Low-order nibble; high-order nibble

Positional Numbering Systems

In the decimal numbering system, what does 243 mean?

Positional Numbering SystemsGeneral Idea:A numeric value is represented through increasing power of a radix (or base)Examples:1) 243.5110 = 2*102

= 200 + 40 + 30 + 0.5 + 0.01

= 243.51Note: We are not doing a conversion here!

+ 4*101 + 3*100 + 5*10-1 + 1*10-2

Boardwork 1• Evaluate the following decimal numbers:

1.) 4310 2.) 52710 3.) 905110 4.) 1540110

5.) 12813610

Conversion from any Base to Base 10 Using Positional Numbering Systems

Examples (From any base to base 10):

2) 2123 to base 10= 2*32 + 1*31 + 2*30 = 18 + 3 + 2 = 2310

3)101102 to base 10= 1*24 + 0*23 + 1*22 + 1*21 + 0*20 = 16 + 0 + 4 + 2 + 0= 2210

• In general, the relationship between a digit, its position, and the base of the number system is expressed by the following formula:

Number Systems

Positional Numbering Systems

Board Work 2

• Convert the following to Base 10 (Decimal)

2. 138

4. 21738

3. 1000111102

5. 67F16

1. 4B16

ANSWERS

Conversion Base 10 (dec) to Any base

• Division Remainder Method– The remainders end up being the digits of the

result and are read from bottom to top

From Base 10 to Base 2

Most significant bit12 )

( 022 )

4210

1010102Answer:

( 152 )

( 02 )

(Least significant bit422 ) Quotient

210

Remainder

2 )10

( 1

( 10When quotient becomes 0,

division stops. Read bottom-up

From Base 10 to Base 16

5,73510

166716Answer:

016 )( 1 Most significant bit116 )( 62216 )( 635816 )( 7 Least significant bit5,73516 )

Quotient

Remainder

Boardwork 3

• Convert the following decimal numbers using division method:1. 104 to Base 32. 147 to Base 23. 803 to Base 164. 539 to Base 16

5. 1246 to base 166. 48 to base 27. 537 to base 88. 1576 to base 16

ANSWERS

Converting Fractions

• Multiplication Method– Multiply fractional part by radix– Set apart integer part– Continue multiplying fraction part by radix until it

becomes 0– Read integer part top to bottom

• Example: Convert 0.430410 to base 5.4304 * 5 = 2.1520 2.1520 * 5 = 0.7600 0.7600 * 5 = 3.8000 3.8000 * 5 = 4.0000 4 fractional part 0.

Done multiplying.

Answer: 0.20345

Boardwork 4

At least 3 bits to the right• Convert 0.37510 to binary• Convert 0.82510 to binary

ANSWERS

Conversion using Nibble Method

• Convert 1100100111012 to Octal and Decimal= 110 = 6= 62358

= 1100 = C = C9D16

010 011 101 2 3 5

1001 1101 9 D

Boardwork 5Convert the given base to base 21. EF16

2. 678

3. 9216

4. 1378

5. 1B116

Answers to Boardwork 2

1. 7510

2. 1110

3. 28610

4. 114710

5. 166310

Answers to Boardwork 3

• 102123

• 100100112

• 32316

• 21B16

• 4DE16

• 1100002

• 10318

• 62816

Answers to Boardwork 4

• 0.0112

• 0.1102

Checkup Quiz. Complete the table below. Show your solution(s).

Decimal Binary Octal Hexadecimal

54.812510

1101111.0112

11.48

1FD.A216

NUMBER SYSTEMS (contd)

Binary Addition

KEEP IN MIND the following rules:

• 0 + 0 = 0• 0 + 1 = 1• 1 + 0 = 1• 1 + 1 = 10

ExamplesTake note where and when it is necessary to carry the 1.

Boardwork

• Do the following binary additions:

1 0 1 0 1 1 1 0 1 1

1 1 1 0 0 1 1 1 0 0 0 1

Binary Subtration

• Take note of the following terms:Minuend – Subtrahend = Difference

KEEP IN MIND the following rules:• 0 - 0 = 0• 1 - 1 = 0• 1 - 0 = 1• 0 - 1 = 1 (we use the borrow system)

Examples

• Subtract the binary numbers 101012 and 11102

1

11

1

0110

11

1 0 0 2

Final Answer: 1112

seatwork (1/2 crosswise)• Add the binary equivalent of the following

decimal numbers: 1) 21 and 1152) 48 and 73) 4401 and 264) 7, 6, 13, 5, and 14

• Obtain the Difference of the binary equivalent of the following decimal numbers: 5) 2 from 106) 11 from 237) 45 from 136

Show your solutions!!!

Negative Values in Binary Number System

• Signed Magnitude (S.M.)• One’s Complement (O.C.)• Two’s Complement (T.C.)– When working with the above, we restrict the

numbers in an n-bit system.

Signed Magnitude• Use the most significant bit as the sign bit, 0

for positive, 1 for negative.– Examples

001100 12

101100 -12

000101 5

100101 -5

Complement Systems

• Complements are used in digital computers for simplifying the subtraction operation (and for logical manipulations).

• Two types of complements:– (r-1)’s complement or 1’s complement– r’s complement or 2’s complement

Diminished Radix Complement (r-1)

• 1’S complement is obtained by subtracting each digit by from 1

• Or put simply, 1’s complement of a binary number is formed by changing 1’s to 0’s and vice versa

Diminished Radix Complement (r-1)

• Examples: (changing 1’s to 0s and vice versa)1) 1011000 =2) 0101101 =

Solution (long method, subtract from each digit from 1)1) 1111111 – 1011000 = 01001112) 1111111 – 0101101 = 1010010

01001111010010

Radix Complement (r)

• 2’s complement in binary is obtained by getting the 1’s complement of the number then add 1

• Or the 2’scomplement can be also be performed by leaving all least significant zeros and the first nonzero digit unchanged, and then replacing 1’sby 0’sand 0’sby 1’s in all other higher significant digits.

Radix Complement (r)

• Examples (first method):obtain 1’s thencomplement add 1

1) 101100 010011 + 1 = 0101002) 1101100 0010011 + 1 = 00101003) 0110111 1001000 + 1 = 1001001

Radix Complement (r)

• Example (2nd method):– The 2’scomplement of 101100 is 010100.– The 2’scomplement of 0.0110 is 0.1010.

Subtraction with Complements• Example: Subtract 1000011 from 1010100

using 2’s complement. (7-bit system)1010100

2’s complement + 01111011 0010001

Overflow bit in two’s complement are ignored.

10101002 8410

10000112 6710

8410 – 6710 = 17

Subtraction with Complements• Example: Subtract 1000011 from 1010100

using 1’s complement1010100

1’s complement + 011110010010000

Final answer is 00100012

+1End-around carry 0010001

Seatwork: Convert / Compute

Compute using the indicated number representations (signed number, 1’s complement, and 2’s complement), and assume 8-bit number.• 7 + (-5) signed magnitude• -13 + (10) two’s complement• -9 + (-1) one’s complement• 12 + (-12) two’s complement

Binary Multiplication

• Multiply 111 by 101111

X 10000

+ 111 1110

Remember:1. Copy the multiplicand when the

multiplier is 1. 2. Write a row of zeros when the

multiplier is 0.3. Shift your results one column to

the left as you move to a new multiplier digit.

4. Add the results together using binary addition to find the product.

Binary Division

Answers to seatwork 1

1) 10000100002

2) 1101112

3) 10001010010112

4) 1011012

5) 10002

6) 11002

7) 10110112