2 data storage - ntnuweb.ntnu.edu.tw/~tcchiang/introcs/2_data storage (i).pdf · one’s complement...

Introduction to Computer Science, Fall, 2010

Data Storage

Instructor: Tsung-Che [email protected]

Department of Computer Science and Information EngineeringNational Taiwan Normal University


Outline

IntroductionStoring IntegersStoring RealsStoring TextsStoring AudiosStoring ImagesStoring VideosSummary


Introduction

Data typesData today comes in different forms including

numbers, text, audio, image and video(multimedia).

B. Forouzan and F. Mosharraf, Foundations of Computer Science, 2nd ed., 2008.


Introduction

Data inside the computerAll data types are

transformed into a uniform representation when theyare stored in a computer and

transformed back to their original form whenretrieved.

This universal representation is called a bitpattern.



Introduction

Data inside the computer



Storing Numbers

A number is changed to the binary systembefore being stored in the computermemory.

However, there are still two issues thatneed to be handled: how to store the sign of the number how to show the decimal point


Storing Integers

Integers are whole numbers (numbers without afractional part).

An integer can be thought of as a number in whichthe position of the decimal point is fixed.

For this reason, fixed-point representation is usedto store an integer. In this representation thedecimal point is assumed but not stored.



Storing Integers- Unsigned representation

An unsigned integer is an integer that can neverbe negative and can take only 0 or positive values.Its range is between 0 and positive infinity.

An input device stores an unsigned integer usingthe following steps: The integer is changed to binary. If the number of bits is less than n, 0s are added to the

left.

The largest number of the n-bit unsignedrepresentation is .2n –1



Example 3.1

Store 7 in an 8-bit memory location using unsignedrepresentation.

First change the integer to binary, (111)2.Add five 0s to make a total of eight bits, (00000111)2.Note that the subscript 2 is not stored in the computer.

Solution




Example 3.2

Store 258 in a 16-bit memory location.

Example 3.3

What is returned from an output device when it retrieves thebit string 00101011 stored in memory as an unsigned integer?

43.




C/C++ Program:#include <cstdio>#include <iostream>using namespace std;

int main(){

unsigned int num = 42;

printf("C: %u %o %x %X\n", num, num, num, num);

cout << "C++: " << showbase<< dec << num << ' '<< oct << num << ' '<< nouppercase << hex << num << ' '<< uppercase << hex << num << ' '<< noshowbase;

system("pause");return 0;

}

C: 42 52 2a 2AC++: 42 052 0x2a 0X2A 請按任意鍵繼續 . . .



OverflowWhat happens if we try to store an integer that

is larger than 24 −1 = 15 in a memory locationthat can only hold four bits.



void display(unsigned int num){

printf("C: %u %o %x %X\n", num, num, num, num);

cout << "C++: " << showbase<< dec << num << ' ‘<< oct << num << ' '<< nouppercase << hex << num << ' '<< uppercase << hex << num << ' '<< noshowbase << endl;

}

int main(){

unsigned int num = std::numeric_limits<unsigned int>::max(),overflow = num+2;

display(num);display(overflow);


}


#include <cstdio>#include <iostream>#include <limits>using namespace std;

C: 4294967295 37777777777 ffffffff FFFFFFFFC++: 4294967295 037777777777 0xffffffff 0XFFFFFFFFC: 1 1 1 1C++: 1 01 0x1 0X1請按任意鍵繼續 . . .



Applications CountingAddressingStoring other data types (text, images, audio, and

video)


Storing Integers- Sign-and-magnitude representation

In this method, the available range for unsignedintegers (0 to 2n −1) is divided into two equal sub-ranges. The first half represents positive integers,the second half, negative integers.


Note. There are two zeros.



Example 3.4

Store +28 in an 8-bit memory location using sign-and-magnituderepresentation.

The integer is changed to 7-bit binary.The leftmost bit is set to 0. The 8-bit number is stored.

Solution




Example 3.5

Store -28 in an 8-bit memory location using sign-and-magnituderepresentation.

The integer is changed to 7-bit binary.The leftmost bit is set to 1. The 8-bit number is stored.


Solution



Example 3.6

Retrieve the integer that is stored as 01001101 insign-and-magnitude representation.

77.

Example 3.7

Retrieve the integer that is stored as 10100001 insign-and-magnitude representation.

-33.



Overflow




Applications It is not commonly used to store integers, but

it is used to store part of a real number incomputers. (discussed later)

It is also used when we quantize an analog signal,such as audio.


Storing Integers- Two’s complement representation

Almost all computers use two’s complementrepresentation to store a signed integer.

In this method, the available range for an unsignedinteger of (0 to 2n −1) is divided into two equal sub-ranges. The first sub-range is used to represent nonnegative

integers, the second half to represent negative integers.




One’s complementThis operation simply reverses (flips) each bit.

A 0-bit is changed to a 1-bit, a 1-bit is changedto a 0-bit.




One’s complementWe get the original integer if we apply the one’s

complement operations twice.




Two’s complement (Method 1) First, we copy bits from the right until a 1 is

copied; then, we flip the rest of the bits.




Two’s complement (Method 2) First, take the one’s complement, and thenThen, add 1 to the result.

0 0 1 1 0 1 0 01 1 0 0 1 0 1 1

11 1 0 0 1 1 0 0

+

Original number

One’s complement

Two’s complement



Two’s complementWe get the original integer if we apply the

two’s complement operations twice.




To store an integer, the integer is changedto an n-bit binary. If the integer is positive or zero, it is stored as

it is. If it is negative, the computer takes the two’s

complement of the integer and stores it.



Example 3.12

Store the integer 28 in an 8-bit memory location using two’scomplement representation.

Example 3.13

Store the integer -28 in an 8-bit memory location using two’scomplement representation.




Example 3.14

Retrieve the integer that is stored as 00001101 in memory intwo’s complement format.




Example 3.15

Retrieve the integer that is stored as 11100110 in memoryusing two’s complement format.




Exercise

Store the integer -45 in an 8-bit memory location using two’scomplement representation.

00101101 11010011(45) (2’s complement)

Exercise

Retrieve the integer that is stored as 10101011 in memoryusing two’s complement format.

10101011 01010101 -85


void display(int num){

printf("C: %d %X\n", num, num);

cout << "C++: " << showbase<< dec << num << ' ‘<< uppercase << hex << num << ' '<< noshowbase << endl;

}

int main(){

int num = 28;

display(num);display(-num);


}


C: 28 1CC++: 28 0X1CC: -28 FFFFFFE4C++: -28 0XFFFFFFE4請按任意鍵繼續 . . .

0000 0000 0000 0000 0000 0000 0001 1100

1111 1111 1111 1111 1111 1111 1110 0100



Overflow




void display(int num){

cout << "C++: " << showbase<< dec << num << ' ‘<< uppercase << hex << num << ' '<< noshowbase << endl;

}

int main(){

int max = std::numeric_limits<int>::max(), pOverflow = max+2,min = std::numeric_limits<int>::min(), nOverflow = min-2;

display(max);display(pOverflow);display(min);display(nOverflow);


}

C++: 2147483647 0X7FFFFFFFC++: -2147483647 0X80000001C++: -2147483648 0X80000000C++: 2147483646 0X7FFFFFFE請按任意鍵繼續 . . .



template <class T> void display(T num){

cout << "C++: " << showbase << dec << num << ' ‘<< uppercase << hex << num << ' ' << noshowbase << endl;

}

int main(){

int signed_int = -1;unsigned int unsigned_int = signed_int;display(unsigned_int);

// unsigned constant tailing by 'u'unsigned int unsigned_large = 3*1000*1000*1000u;int signed_small = unsigned_large;display(signed_small);


}

C++: 4294967295 0XFFFFFFFFC++: -1294967296 0XB2D05E00請按任意鍵繼續 . . .



Applications It is the standard representation for storing

integers in computers today.We will see why this is the case when we see

the simplicity of operations using two’scomplement.


Comparison the Three Systems



Storing Reals

Fixed-point representation?

Real numbers with very large integral parts orvery small fractional parts should not be storedin fixed-point representation.

.

5 digits 5 digits

What about 1000000000?What about 0.000000001?


Storing Reals

Floating-point representationWe can have different numbers of digits to the

left or right of the decimal point. The range can increase tremendously.

A number if made up of three sections: The fixed-point section has only one digit to the left

of the decimal point.



Storing Reals

Example 3.18 (Decimal)

The following shows the decimal number

7,452,000,000,000,000,000,000.00

in scientific notation (floating-point representation).

The three sections are the sign (+), the shifter (21) and thefixed-point part (7.425). Note that the shifter is theexponent.



Storing Reals

Example 3.19 (Decimal)

Show the number -0.0000000000000232in scientific notation (floating-point representation).

The three sections are the sign (-), the shifter (-14) and thefixed-point part (2.32). Note that the shifter is the exponent.



Storing Reals

Example 3.20 (Binary)

Example 3.21 (Binary)



Storing Reals

Normalization Leave only one non-zero digit to the left of the

decimal point. In the decimal system, the digit can be 1 to 9,

while in the binary system, it can only be 1.



Storing Reals

Sign, exponent, and mantissa

+1000111.0101 becomes


Note. The point and the bit 1 to the left of the decimal point are not stored.


Storing Reals

Sign It is stored by one bit.

Exponent It defines the shifting of the decimal point. Note that the power can be negative or positive. The

Excess representation is used to store it.

Mantissa It is the binary integer to the right of the decimal point. The mantissa and the sign is treated like an integer

stored in sign-and-magnitude representation. However, remember that the mantissa is not an integer.

It is a fractional part that is stored like an integer.


Storing Reals

The Excess SystemThe exponent is a signed number. In the Excess system, both positive and

negative integers are stored as unsignedintegers.

A positive integer (called a bias) is added toeach number to shift them uniformly to thenon-negative side.

The value of this bias is 2m−1 −1, where m is thesize of the memory location to store theexponent.


Storing Reals

Example 3.22

We can express sixteen integers in a number system with 4-bitallocation. The new system is referred to as Excess-7 (241 –1),or biased representation with biasing value of 7.



IEEE Standard

Single-precision & double-precision format



IEEE Standard

Example 3.23

Show the Excess_127 (single precision) representation of thedecimal number 6.75.

a. The sign is positive, so S = 0.b. Decimal to binary transformation: 6.75 = (110.11)2.c. Normalization: (110.11)2 = (1.1011)2 × 22.d. E = 2 + 127 = 129 = (10000001)2, M = 1011. We need to add

nineteen zeros at the right of M to make it 23 bits.e. The presentation is shown below:

Solution



IEEE Standard

Example 3.24

Show the Excess_127 (single precision) representation of thedecimal number –161.875.

a. The sign is negative, so S = 1.b. Decimal to binary transformation: 161.875= (10100001.111)2.c. Normalization: (10100001.111)2 = (1.0100001111)2 × 27.d. E = 7 + 127 = 134 = (10000110)2 and M = (0100001111)2.e. Representation:

Solution


Note. 這裡沒有什麼補數不補數了…


IEEE Standard

Example 3.25

Show the Excess_127 (single precision) representation of thedecimal number –0.0234375.

a. S = 1 (the number is negative).b. Decimal to binary transformation: 0.0234375 = (0.0000011)2.c. Normalization: (0.0000011)2 = (1.1)2 × 2−6.d. E = –6 + 127 = 121 = (01111001)2 and M = (1)2.e. Representation:

Solution



IEEE Standard

Example 3.26

The bit pattern (11001010000000000111000100001111)2 isstored in Excess_127 format. Show the value in decimal.

a. The sign is negative.b. The shifter = E −127 = 148 −127 = 21.c. This gives us (1.00000000111000100001111)2 × 221.d. The binary number is (1000000001110001000011.11)2.e. The absolute value is 2,104,378.75.f. The number is −2,104,378.75.

Solution


IEEE Standard

Storing zeroA real number with an integral part and the

fractional part set to zero, that is, 0.0, cannotbe stored using the steps discussed above.

To handle this special case, it is agreed that inthis case the sign, exponent and the mantissaare set to 0s.

Truncation errorsWhen a real number is stored using floating-

point representation, the value stored may notbe exactly as we expect it to be.


IEEE Standard

Online converter http://www.h-schmidt.net/FloatApplet/IEEE754.html http://babbage.cs.qc.edu/IEEE-754/

2 data storage - ntnuweb.ntnu.edu.tw/~tcchiang/introcs/2_data storage (i).pdf · one’s complement...

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