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CSC 2400: Computer Systems

Bits, Bytes and Data Types

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Goals for this Week

• Binary number system • Why binary? • Converting between decimal and binary • … and octal and hexadecimal number systems

• Finite representations of binary integers • Unsigned and signed integers • Integer addition and subtraction

• Bitwise operators • AND, OR, NOT, and XOR • Shift-left and shift-right

Analog/Analogue Systems

• Analogue Systems • V(t) can have any value between its minimum and maximum value

V(t)

Digital Systems • Digital Systems

• V(t) takes a value selected from a set of values • Binary digital systems form the basis of almost all hardware

systems currently

1 0 1 0 1

V(t)

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Why Bits (Binary Digits)?

• Computers are built using digital circuits • Inputs and outputs can have only two values • True (high voltage) or false (low voltage) • Represented as 1 and 0

• Can represent many kinds of information • Boolean (true or false) • Numbers (23, 79, …) • Characters (‘a’, ‘z’, …) • Pixels, sounds • Internet addresses

• Can manipulate in many ways • Read and write • Logical operations • Arithmetic

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Coding

• A single binary input can have two values: 1 or 0

• More bits = more combinations

• How many values can you represent on 3 bits?

• What about n bits?

0 0 0 1 1 0 1 1

Memorize Powers of 2

Powers of 2 23 = 8 24 = 16 25 = 32 26 = 64 27 = 128 28 = 256 29 = 512

Powers of 10 210 = 1Kilo (1024) 103

= 1,000

220 = 1Mega (10242 = 1,048,576 ) 106 = 1,000,000 230 = 1Giga (10243 = 1,073,741,824) 109

= 1,000,000,000 240 = 1Tera (10244 = 1,099,511,627,776) 1012

= 1,000,000,000,000

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Base 10 and Base 2 • Decimal (base 10)

• Each digit represents a power of 10 • 417310 = 4 x 103 + 1 x 102 + 7 x 101 + 3 x 100

• Binary (base 2) • Each bit represents a power of 2 • 10102 = 1 x 23 + 0 x 22 + 1 x 21 + 0 x 20 = 1010

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Binary to Decimal

• Sum up the place values of all 1-bits:

256 128 64 32 16 8 4 2 1

1 0 0 0 0 0 1 1

1 0 1 0 0 1 1 0 0

100000112 = _________________________________10 ? 1010011002 = _________________________________10 ?

Place Value

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Decimal to Binary

• Use the Placement Method

1024 512 256 128 64 32 16 8 4 2 1

Powers of 2

15510 = _________________________________ 2 ?

• 128 goes into 155 once leaving 27 to be placed

1 ? ? ? ? ? ? ?

• 64 and 32 are too big (make them 0) • 16 goes in once leaving 11

1 0 0 1 ? ? ? ?

• and so on

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You Try It …

Working w/ Bits is Tedious for People

• Express large binary numbers in hex using ¼ fewer digits:

• This is the hexadecimal system. Memorize the tables above. 12

Hex Binary Decimal

0 0000 0

1 0001 1

2 0010 2

3 0011 3

4 0100 4

5 0101 5

6 0110 6

7 0111 7

Hex Binary Decimal

8 1000 8

9 1001 9

A 1010 10

B 1011 11

C 1100 12

D 1101 13

E 1110 14

F 1111 15

1. Based on the lookup tables on the previous page, convert the hex value 0x7A8BF7D6 into its binary equivalent:

2. Based on the lookup tables on the previous page, convert the binary 10 0110 1110 1001 0100 1100 0101 11112 to hex:

Exercises

0111 1010 1000 1011 1111 0111 1101 0110

0010 0110 1110 1001 0100 1100 0101 1111

6 E 9 4 B 5 F 2

7 A 8 B F 7 D 6

Why hexadecimal? • Widely used in assembly language programming

• Used in network programming and debugging

• Used in graphic design programs – e.g., for the red, green and blue components of a color:

FF0000 represents red, for example.

How many bits are used to represent each color?

How many different colors can be represented?

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Decimal Addition

• From right to left, we add each pair of digits

• We write the sum, and add the carry to the next column

1 9 8

+ 2 6 4

Sum

Carry

2

1

6

1

4

0

0

8 5 + E D

• Hex addition is similar to decimal addition except that each hex digit has a range of ‘0’ to ‘F’ instead of ‘0’ to ‘9’, and a “carry out” occurs when the sum of hex digits in a particular column exceeds ‘F’ (15 decimal).

• Examples:

Hex Addition

A

C 3 + 2 7

C 7 + 2 A

E 1

1

F 2

1

1 7

3 B 9 A + D 2 E 6

1

8

1

E 1 0

D 3 5 F - 4 3 B 6 /

• Hex subtraction is similar to decimal subtraction except that if the subtrahend is greater than the minuend, we must “borrow” 16 (10 hex) from the previous digit.

• Examples:

Hex Subtraction

D

B 3 - 3 6

C A - 2 5

7 5 A

/

9 A

1 A

/ 1 1

2 C

F 8

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Recall: Decimal Addition

• From right to left, we add each pair of digits

• We write the sum, and add the carry to the next column

1 9 8

+ 2 6 4

Sum

Carry

0 1 1

+ 0 0 1

Sum

Carry

2

1

6

1

4

0

Base 10 Base 2

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Finite Representation of Integers

• Fixed number of bits in memory • Usually 8, 16, or 32 bits • (1, 2, or 4 bytes)

• Unsigned integer • No sign bit • Always 0 or a positive number • All arithmetic is modulo 2n

• Examples of unsigned integers • 00000001 1 • 00001111 ____ • 00010000 ____ • 00100001 ____ • 11111111 ____

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Modulo Arithmetic

• Consider only numbers in a range • E.g., five-digit car odometer: 0, 1, …, 99999 • E.g., eight-bit numbers 0, 1, …, 255

• Roll-over when you run out of space • E.g., car odometer goes from 99999 to 0, 1, … • E.g., eight-bit number goes from 255 to 0, 1, …

• Adding 2n doesn’t change the answer • For eight-bit number, n=8 and 2n=256 • E.g., (37 + 256) mod 256 is ________

• This can help us do subtraction… • Suppose you want to compute a – b • Note that this equals a + (256 -1 - b) + 1

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One’s and Two’s Complement

• One’s complement: flip every bit • E.g., b is 01000101 (i.e., 69 in decimal) • One’s complement is ______________ • That’s simply 255-69

• Subtracting from 11111111 is easy (no carry needed!)

• Two’s complement • Add 1 to the one’s complement • E.g., (255 – 69) + 1 ____________

- 0100 0101 1111 1111

b

one’s complement

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Putting it All Together

• Computing “a – b” • Same as “a + 256 – b” • Same as “a + (255 – b) + 1” • Same as “a + onesComplement(b) + 1” • Same as “a + twosComplement(b)”

• Example: 172 – 69 • The original number 69: _________ • One’s complement of 69: _________ • Two’s complement of 69: _________ • Add to the number 172: _________ • The sum comes to: _________ • Equals: 103 in decimal

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Reading Two’s Complement Patterns

• When a two’s complement number has a highest bit 1, it indicates that the number is negative.

• To find the value, perform the same steps:

• Unknown value: 1011 1011 • One’s complement: _________ • Add 1 (two’s complement): _________ We get a value of 69, so the original pattern must have been -69.

Practice: Ch. 2, Exercise 2

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Signed Integers

• Sign-magnitude representation • Use one bit to store the sign

• Zero for positive number • One for negative number

• Examples • E.g., 0010 1100 44 • 1010 1100 -44

• Hard to do arithmetic this way, so it is rarely used

• Complement representation • One’s complement

• Flip every bit • E.g., 1101 0011 -44

• Two’s complement • Flip every bit, then add 1 • E.g., 1101 0100 -44

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Fill in the Table

Bit Pattern

Value (Sign Magnitude)

Value (One’s Complement)

Value (Two’s Complement)

000

001

010

011

100

101

110

111

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Question

• What value does 10011001 represent? [Ex. 6, page 69]

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Overflow: Running Out of Room

• Adding two large integers together • Sum might be too large to store in the number of bits available • What happens?

• Unsigned integers • All arithmetic is “modulo” arithmetic • Sum would just wrap around

• Signed integers • Can get nonsense values • Example with 16-bit integers

• Sum: 10000+20000+30000 • Result: -5536

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Exercise

• Assume only four bits are available for representing integers, and signed integers are represented in 2’s complement.

• Compute the value of the expression 7 + 7

Bitwise Operations

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Networking

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Encryption/ Decryption

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Compression

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Bitwise Operators: AND and OR

• Bitwise AND (&)

• Bitwise OR (|)

&

0

1

0 1 |

0

1

0 1

0 0 1 1 0 1 0 1

0 0 0 0 1 1 1 1

___

& 15

___

0 0 1 1 0 1 0 1

0 0 0 0 1 1 1 1

___

| 15

___

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Bitwise Operators: Not and XOR

• One’s complement (~) • Turns 0 to 1, and 1 to 0 • E.g., set last three bits to 0

• x = x & _____

• XOR (^) • 0 if both bits are the same • 1 if the two bits are different

^

0

1

0 1

Practice: Ch. 2, Exercise 9

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Bitwise Operators: Shift Left/Right

• Shift left (<<): Multiply by powers of 2 • Shift some # of bits to the left, filling the blanks with 0

• Shift right (>>): Divide by powers of 2 • Shift some # of bits to the right. Fill in blanks with sign bit.

0 0 1 1 0 1 0 1 53

53<<2

0 0 1 1 0 1 0 1 53

53>>2

1 0 1 1 0 1 0 1 -75

-75>>2

sign extension sign extension

Practice: Ch. 2, Exercises 10, 12, 13, 14

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• Used to change or query one or more bits in a variable.

• The bitmask indicates which bits are to be affected.

• Common operations:

Can extend to groups of bits

Bitmasks

Operation Expression C expression

Set Nth bit of x x = x OR 2N x = x |(1 << N);

Clear Nth bit of x x = x ________ x = _____________;

Read Nth bit of x ??? = x ______ ??? = x _________;

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Exercise

• Let x be an 8-bit integer.

• Set the 5th least significant bit of x:

• Clear the 5th least significant bit of x:

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Bitmask Example /* This program demonstrates setting a bit, clearing a bit, and ** reading a bit. (pg 64) */ #include <stdio.h> main() { char a; int i; a=17; a=a | (1 << 3); /* set 3rd bit */ printf("%d\n",a); a=a & (~(1<<4)); /* clear 4th bit */ printf("%d\n",a); for (i=7; i>=0; i--) printf("%d ",(a&(1<<i)) >> i); /* read i'th bit */ printf("\n"); }

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Exercise #1

• Write a small program that reads in an integer and prints out its binary representation.

Sample output: Please enter an integer: 1025 The binary representation of 1025 is 00000000 00000000 00000100 00000001

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Exercise #2

• Write a small program that reads in an integer prints out hexadecimal value of its bytes, separated by spaces, starting with the most significant byte.

Sample output: Please enter an integer: 1025 The four bytes (in hexadecimal) are: 00 00 04 01

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Example: Counting the 1’s

• How many 1 bits in a number? • E.g., how many 1 bits in the binary representation of 53?

• Four 1 bits

• How to count them?

0 0 1 1 0 1 0 1

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Counting the Number of ‘1’ Bits #include <stdio.h> #include <stdlib.h> int main(void) { unsigned n, count; printf(“Number: "); if (scanf("%u", &n) != 1) { fprintf(stderr, "Error: Expect number.\n"); exit(EXIT_FAILURE); } /* Enter code to count the 1 bits */ printf(“Number of 1 bits: %u\n”, count); return 0; }

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Practice

• Write a function getByte that extracts a byte from a word. /* * getByte(x, n) - Extract byte n from word x * Bytes numbered from 0 (LSB) to 3 (MSB) * Examples: getByte(0x12345678,1) = 0x56 * Legal operations: ~ & ^ | + << >> */ int getByte(int x, int n) { /* Add code here */ }

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Summary

• Computer represents everything in binary • Integers, floating-point numbers, characters, addresses, … • Pixels, sounds, colors, etc.

• Binary arithmetic through logic operations • Sum (XOR) and Carry (AND) • Two’s complement for subtraction

• Binary operations in C • AND, OR, NOT, XOR, shift left and shift right • Useful for efficient and concise code, though sometimes cryptic

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Required Reading

• Textbook, Chapter 2