Bài giảng Kiến trúc máy tính - Chương 3, Phần 1: Data representation

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COMPUTER ARCHITECTURE

CSE Fall 2014

Faculty of Computer Science and Engineering

Department of Computer Engineering

BK

TP.HCM

Vo Tan Phuong

http://www.cse.hcmut.edu.vn/~vtphuong

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Chapter 3

Data Representation

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Presentation Outline

• Positional Number Systems

• Binary and Hexadecimal Numbers

• Base Conversions

• Binary and Hexadecimal Addition

• Binary and Hexadecimal subtraction

• Carry and Overflow

• Character Storage

• Floating Point Number

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Positional Number Systems

Different Representations of Natural Numbers

XXVII Roman numerals (not positional)

27 Radix-10 or decimal number (positional)

11011₂Radix-2 or binary number (also positional)

Fixed-radix positional representation with k digits

Number N in radix r = (d_k–1d_k–2. . . d₁d₀)_r

Value = d_k–1×r ^k–1+ d_k–2×r ^k–2+ … + d₁×r + d₀

Examples: (11011)₂= 1×2⁴+ 1×2³+ 0×2²+ 1×2 + 1 = 27

(2103)₄= 2×4³+ 1×4²+ 0×4 + 3 = 147

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Binary Numbers

• Each binary digit (called bit) is either 1 or 0

• Bits have no inherent meaning, can represent

– Unsigned and signed integers

– Characters

Most

Significant Bit

Least

Significant Bit

– Floating-point numbers

– Images, sound, etc.

7

6

5

4

3

2

1

0

1 0 0 1 1 1 0 1

• Bit Numbering

2⁷2⁶2⁵2⁴2³2²2¹2⁰

– Least significant bit (LSB) is rightmost (bit 0)

– Most significant bit (MSB) is leftmost (bit 7 in an 8-bit number)

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Converting Binary to Decimal

• Each bit represents a power of 2

• Every binary number is a sum of powers of 2

• Decimal Value = (d_n-1 2^n-1) + ... + (d₁ 2¹) + (d₀ 2⁰)

• Binary (10011101)₂= 2⁷+ 2⁴+ 2³+ 2²+ 1 = 157

7

6

5

4

3

2

1

0

1 0 0 1 1 1 0 1

2⁷2⁶2⁵2⁴2³2²2¹2⁰

Some common

powers of 2

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Convert Unsigned Decimal to Binary

• Repeatedly divide the decimal integer by 2

• Each remainder is a binary digit in the translated value

least significant bit

37 = (100101)₂

most significant bit

stop when quotient is zero

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Hexadecimal Integers

• 16 Hexadecimal Digits: 0 – 9, A – F

• More convenient to use than binary numbers

Binary, Decimal, and Hexadecimal Equivalents

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Converting Binary to Hexadecimal

 Each hexadecimal digit corresponds to 4 binary bits

 Example:

Convert the 32-bit binary number to hexadecimal

1110 1011 0001 0110 1010 0111 1001 0100

 Solution:

E

B

1

6

A

7

9

4

1110 1011 0001 0110 1010 0111 1001 0100

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Converting Hexadecimal to Decimal

• Multiply each digit by its corresponding power of 16

Value = (d_n-1 16^n-1) + (d_n-2 16^n-2) + ... + (d₁ 16) + d₀

• Examples:

(1234)₁₆= (1  16³) + (2  16²) + (3  16) + 4 =

Decimal Value 4660

(3BA4)₁₆= (3  16³) + (11  16²) + (10  16) + 4 =

Decimal Value 15268

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Converting Decimal to Hexadecimal

 Repeatedly divide the decimal integer by 16

 Each remainder is a hex digit in the translated value

least significant digit

most significant digit

stop when

quotient is zero

Decimal 422 = 1A6 hexadecimal

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Integer Storage Sizes

8

Byte

Half Word

Word

Storage Sizes

16

32

Double Word

64

Storage Type

Unsigned Range

0 to 255

Powers of 2

Byte

0 to (2⁸– 1)

0 to (2¹⁶– 1)

0 to (2³²– 1)

0 to (2⁶⁴– 1)

Half Word

Word

0 to 65,535

0 to 4,294,967,295

Double Word

0 to 18,446,744,073,709,551,615

What is the largest 20-bit unsigned integer?

Answer: 2²⁰– 1 = 1,048,575

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Binary Addition

• Start with the least significant bit (rightmost bit)

• Add each pair of bits

• Include the carry in the addition, if present

carry

1

0 0 1 1 0 1 1 0

0 0 0 1 1 1 0 1

(54)

(29)

+

0 1 0 1 0 0 1 1

(83)

bit position: 7

6

5

4

3

2

1

0

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Hexadecimal Addition

• Start with the least significant hexadecimal digits

• Let Sum = summation of two hex digits

• If Sum is greater than or equal to 16

– Sum = Sum – 16 and Carry = 1

• Example:

carry:

1 1

1

1 C 3 7 2 8 6 A

9 3 9 5 E 8 4 B

A + B = 10 + 11 = 21

Since 21 ≥ 16

Sum = 21 – 16 = 5

Carry = 1

+

A F C D 1 0 B 5

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Signed Integers

• Several ways to represent a signed number

– Sign-Magnitude

– Biased

– 1's complement

– 2's complement

• Divide the range of values into 2 equal parts

– First part corresponds to the positive numbers (≥ 0)

– Second part correspond to the negative numbers (< 0)

• Focus will be on the 2's complement representation

– Has many advantages over other representations

– Used widely in processors to represent signed integers

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^dceTwo's Complement Representation

2015

 Positive numbers

8-bit Binary Unsigned Signed

value

 Signed value = Unsigned value

00000000

00000001

00000010

0

1

2

0

+1

 Negative numbers

+2

 Signed value = Unsigned value – 2ⁿ

. . .

126

127

128

129

. . .

+126

+127

-128

-127

. . .

 n = number of bits

01111110

01111111

10000000

10000001

. . .

 Negative weight for MSB

 Another way to obtain the signed

value is to assign a negative weight

to most-significant bit

1 0 1 1 0 1 0 0

11111110

11111111

254

255

-2

-128 64 32 16

8

4

2

1

-1

= -128 + 32 + 16 + 4 = -76

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Forming the Two's Complement

00100100 = +36

11011011

starting value

step1: reverse the bits (1's complement)

step 2: add 1 to the value from step 1

sum = 2's complement representation

+

1

11011100 = -36

Sum of an integer and its 2's complement must be zero:

00100100 + 11011100 = 00000000 (8-bit sum)  Ignore Carry

Binary Value

Another way to obtain the 2's complement:

least

significant 1

= 00100100

2's Complement

= 11011100

Start at the least significant 1

Leave all the 0s to its right unchanged

Complement all the bits to its left

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Sign Bit

• Highest bit indicates the sign

• 1 = negative

Sign bit

• 0 = positive

1

0

1

0

1

0

1

0

1

0

1

0

Negative

Positive

For Hexadecimal Numbers, check most significant digit

If highest digit is > 7, then value is negative

Examples: 8A and C5 are negative bytes

B1C42A00 is a negative word (32-bit signed integer)

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Sign Extension

Step 1: Move the number into the lower-significant bits

Step 2: Fill all the remaining higher bits with the sign bit

• This will ensure that both magnitude and sign are correct

• Examples

– Sign-Extend 10110011 to 16 bits

10110011 = -77

11111111 10110011 = -77

– Sign-Extend 01100010 to 16 bits

00000000 01100010 = +98

01100010 = +98

• Infinite 0s can be added to the left of a positive number

• Infinite 1s can be added to the left of a negative number

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Two's Complement of a Hexadecimal

• To form the two's complement of a hexadecimal

– Subtract each hexadecimal digit from 15

– Add 1

• Examples:

2's complement of 6A3D = 95C2 + 1 = 95C3

2's complement of 92F15AC0 = 6D0EA53F + 1 = 6D0EA540

2's complement of FFFFFFFF = 00000000 + 1 = 00000001

• No need to convert hexadecimal to binary

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