1. Binary Numbers

2. Main Memory
Main memory holds information such as computer programs, numeric data, or documents created by a word processor.
Main memory is made up of capacitors.
If a capacitor is charged, then its state is said to be 1, or ON.
We could also say the bit is set.
If a capacitor does not have a charge, then its state is said to be 0, or OFF.
We could also say that the bit is reset or cleared.

3. Main Memory (con’t)
Memory is divided into cells, where each cell contains 8 bits (a 1 or a 0). Eight bits is called a byte.
Each of these cells is uniquely numbered.
The number associated with a cell is known as its address.
Main memory is volatile storage. That is, if power is lost, the information in main memory is lost.

4. Main Memory (con’t) Other computer components can
get the information held at a particular address in memory, known as a READ,
or store information at a particular address in memory, known as a WRITE.
Writing to a memory location alters its contents.
Reading from a memory location does not alter its contents.

5. Main Memory (con’t)
All addresses in memory can be accessed in the same amount of time.
We do not have to start at address 0 and read everything until we get to the address we really want (sequential access).
We can go directly to the address we want and access the data (direct or random access).
That is why we call main memory RAM (Random Access Memory).

6. Secondary Storage Media
Disks -- floppy, hard, removable (random access)
Tapes (sequential access)
CDs (random access)
DVDs (random access)
Secondary storage media store files that contain
computer programs
data
other types of information
This type of storage is called persistent (permanent) storage because it is non-volatile.

7. I/O (Input/Output) Devices
Information input and output is handled by I/O (input/output) devices.
More generally, these devices are known as peripheral devices.
Examples:
monitor
keyboard
mouse
disk drive (floppy, hard, removable)
CD or DVD drive
printer
scanner

8. Bits, Bytes, and Words A bit is a single binary digit (a 1 or 0).
A byte is 8 bits
A word is 32 bits or 4 bytes
Long word = 8 bytes = 64 bits
Quad word = 16 bytes = 128 bits
Programming languages use these standard number of bits when organizing data storage and access.

9. Number Systems
The on and off states of the capacitors in RAM can be thought of as the values 1 and 0, respectively.
Therefore, thinking about how information is stored in RAM requires knowledge of the binary (base 2) number system.
Let’s review the decimal (base 10) number system first.

10. The Decimal Number System
The decimal number system is a positional number system.
Example:
X 100 = 1
X 101 = 20
6 X 102 = 600
5 X 103 = 5000

11. The Decimal Number System (con’t)
The decimal number system is also known as base 10. The values of the positions are calculated by taking 10 to some power.
Why is the base 10 for decimal numbers?
Because we use 10 digits, the digits 0 through 9.

12. The Binary Number System
The binary number system is also known as base 2. The values of the positions are calculated by taking 2 to some power.
Why is the base 2 for binary numbers?
Because we use 2 digits, the digits 0 and 1.

13. The Binary Number System (con’t)
The binary number system is also a positional numbering system.
Instead of using ten digits, 0 - 9, the binary system uses only two digits, 0 and 1.
Example of a binary number and the values of the positions:

14. Converting from Binary to Decimal
X 20 = 1
X 21 = 0
1 X 22 = 4
20 = X 23 = 8
21 = X 24 = 0
22 = X 25 = 0
23 = X 26 = 64
24 =
25 = 32
26 = 64

15. Converting from Binary to Decimal (con’t)
Practice conversions:
Binary Decimal
11101
100111

16. Converting From Decimal to Binary (con’t)
Make a list of the binary place values up to the number being converted.
Perform successive divisions by 2, placing the remainder of 0 or 1 in each of the positions from right to left.
Continue until the quotient is zero.
Example: 4210
42/2 = R = 0
21/2 = R = 1
10/2 = R = 0
5/2 = 2 R = 1
2/2 = 1 R = 0
1/2 = 0 R = 1 =

17. Converting From Decimal to Binary (con’t)
Practice conversions:
Decimal Binary 59 82
175

18. Working with Large Numbers
= ?
Humans can’t work well with binary numbers; there are too many digits to deal with.
Memory addresses and other data can be quite large. Therefore, we sometimes use the hexadecimal number system.

19. The Hexadecimal Number System
The hexadecimal number system is also known as base 16. The values of the positions are calculated by taking 16 to some power.
Why is the base 16 for hexadecimal numbers ?
Because we use 16 symbols, the digits 0 and 1 and the letters A through F.

20. The Hexadecimal Number System (con’t)
Binary Decimal Hexadecimal Binary Decimal Hexadecimal A B C D E F

21. The Hexadecimal Number System (con’t)
Example of a hexadecimal number and the values of the positions:
3 C B

22. Example of Equivalent Numbers
Binary:
Decimal:
Hexadecimal: 50A716
Notice how the number of digits gets smaller as the base increases.

23. Goals of Today’s Lecture
Binary numbers
Why binary?
Converting base 10 to base 2
Octal and hexadecimal
Integers
Unsigned integers
Integer addition
Signed integers
C bit operators
And, or, not, and xor
Shift-left and shift-right
Function for counting the number of 1 bits
Function for XOR encryption of a message

24. 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
Sound
Can manipulate in many ways
Read and write
Logical operations
Arithmetic …

25. Base 10 and Base 2 Base 10 Each digit represents a power of 10
4173 = 4 x x x x 100
Base 2
Each bit represents a power of 2
10110 = 1 x x x x x 20 = 22
Divide repeatedly by 2 and keep remainders
12/2 = R = 0
6/2 = R = 0
3/2 = R = 1
1/2 = R = 1
Result =

26. Writing Bits is Tedious for People
Octal (base 8)
Digits 0, 1, …, 7
In C: 00, 01, …, 07
Hexadecimal (base 16)
Digits 0, 1, …, 9, A, B, C, D, E, F
In C: 0x0, 0x1, …, 0xf
0000 = 0
1000 = 8
0001 = 1
1001 = 9
0010 = 2
1010 = A
0011 = 3
1011 = B
0100 = 4
1100 = C
0101 = 5
1101 = D
0110 = 6
1110 = E
0111 = 7
1111 = F
Thus the 16-bit binary number
converted to hex is
B2A9

27. Representing Colors: RGB
Three primary colors
Red
Green
Blue
Strength
8-bit number for each color (e.g., two hex digits)
So, 24 bits to specify a color
In HTML, on the course Web page
Red: <font color="#FF0000"><i>Symbol Table Assignment Due</i>
Blue: <font color="#0000FF"><i>Fall Recess</i></font>
Same thing in digital cameras
Each pixel is a mixture of red, green, and blue

28. Storing Integers on the Computer
Fixed number of bits in memory
Short: usually 16 bits
Int: 16 or 32 bits
Long: 32 bits
Unsigned integer
No sign bit
Always positive or 0
All arithmetic is modulo 2n
Example of unsigned int
 1
 15
 16
 33
 255

29. Adding Two Integers: Base 10
From right to left, we add each pair of digits
We write the sum, and add the carry to the next column
0 1 1
Sum
Carry
1 9 8
Sum
Carry 4 6 1 2 1 1 1 1

30. Binary Sums and Carries
a b Sum a b Carry
XOR
AND 69
103
172

31. Fractional Numbers
Examples: = 4 x x x x x 10-2
= 1 x x x x x x 2-2
= / ¼
= =
Conversion from binary number system to decimal system
Examples: = 1 x x x x x 2-2
= / ¼ =
Examples:
½ ¼ 1/8 x

32. Fractional numbers 4 2 1 Examples: 7.7510 = (?)2
Conversion of the integer part: same as before – repeated division by 2
7 / 2 = 3 (Q), 1 (R)  3 / 2 = 1 (Q), 1 (R)  1 / 2 = 0 (Q), 1 (R) = 1112
Conversion of the fractional part: perform a repeated multiplication by 2 and extract the integer part of the result
0.75 x 2 =1.50  extract 1
0.5 x 2 =  extract = 0.112
 stop
 Combine the results from integer and fractional part, =
How about choose some of
Examples: try 5.625
write in the same order 4 2 1
1/2
1/4
1/8
=0.5
=0.25
=0.125

33. Fractional Numbers (cont.)
Exercise 2: Convert (0.6)10 to its binary form
Solution:
Exercise 1: Convert (0.625)10 to its binary form
Solution: x 2 = 1.25  extract 1
0.25 x 2 = 0.5  extract 0
0.5 x 2 = 1.0  extract 1
0.0  stop
 (0.625)10 = (0.101)2
0.6 x 2 = 1.2  extract 1
0.2 x 2 = 0.4  extract 0
0.4 x 2 = 0.8  extract 0
0.8 x 2 = 1.6  extract 1
0.6 x 2 = 
 (0.6)10 = ( …)2

34. Fractional Numbers (cont.)
Exercise 3: Convert (0.8125)10 to its binary form
Solution: x 2 =  extract 1
0.625 x 2 = 1.25  extract 1
0.25 x 2 = 0.5  extract 0
0.5 x 2 = 1.0  extract 1
0.0  stop
 (0.8125)10 = (0.1101)2

35. Fractional Numbers (cont.)
Errors
One source of error in the computations is due to back and forth conversions between decimal and binary formats
Example: (0.6)10 + (0.6)10 = 1.210
Since (0.6)10 = ( …)2
Lets assume a 8-bit representation: (0.6)10 = ( )2 , therefore 
Lets reconvert to decimal system:
( )b= 1 x x x x x x x x x 2-8
= 1 + 1/8 + 1/16 + 1/128 =
 Error = 1.2 – =

36. One’s and Two’s Complement
One’s complement: flip every bit
E.g., b (i.e., 69 in base 10)
One’s complement is
That’s simply
Subtracting from is easy (no carry needed!)
Two’s complement
Add 1 to the one’s complement
E.g., (255 – 69) + 1  b
one’s complement

37. Putting it All Together
Computing “a – b” for unsigned integers
Same as “a – b”
Same as “a + (255 – b) + 1”
Same as “a + onecomplement(b) + 1”
Same as “a + twocomplement(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 base 10

38. Signed Integers Sign-magnitude representation
Use one bit to store the sign
Zero for positive number
One for negative number
Examples
E.g.,  44
E.g.,  -44
Hard to do arithmetic this way, so it is rarely used
Complement representation
One’s complement
Flip every bit
E.g.,  -44
Two’s complement
Flip every bit, then add 1
E.g.,  -44

39. Overflow: Running Out of Room
Adding two large integers together
Sum might be too large to store in the number of bits allowed
What happens?

40. Bitwise Operators: AND and OR
Bitwise AND (&)
Mod on the cheap!
E.g., h = 53 & 15;
Bitwise OR (|)
& 1 | 1 1 53 1 15 1 5

41. 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
For unsigned integer, fill in blanks with 0
What about signed integers? Varies across machines…
Can vary from one machine to another! 1 53
53<<2 53 1 1 1
53>>2 1 1 1

42. XOR Encryption
Program to encrypt text with a key
Input: original text in stdin
Output: encrypted text in stdout
Use the same program to decrypt text with a key
Input: encrypted text in stdin
Output: original text in stdout
Basic idea
Start with a key, some 8-bit number (e.g., )
Do an operation that can be inverted
E.g., XOR each character with the 8-bit number ^ ^

43. Conclusions 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, and XOR
Shift left and shift right
Useful for efficient and concise code, though sometimes cryptic