1. Introduction to Programming with Java, for Beginners
Want to look at how computers work, but first, we’ll look at the representation of data.
Bits, Data Types, and Operations
Based on slides © McGraw-Hill
Additional material © 2004/2005 Lewis/Martin

2. Computer Organization
Application Program
Algorithms
Language
Software
Hardware
Computer organized as levels of transformation. A problem stated in natural language such as English is actually solved by electrons moving around inside the electronics of the computer.
Instruction Set Architecture
(and I/O Interfaces)
Microarchitecture
Circuits
Devices

3. What does the Computer Understand?
At the lowest level, a computer has electronic “plumbing”
Operates by controlling the flow of electrons through very fast tiny electronic devices called transistors
The devices react to presence or absence of voltage
Could react actual voltages but designing electronics then becomes complex
Symbolically we represent
1. Presence of voltage as “1”
2. Absence of voltage as “0”
Harder to measure the voltage through wall outlet. Easy to detect no voltage or voltage…

4. What does the Computer process & store?
An electronic device can represent uniquely only one of two things
Each “0” and Each “1” is referred to as a Binary Digit or Bit
Fundament unit of information storage
To represent more things we need more bits
E.g. 2 bits can represent four unique things: 00, 01, 10, 11
k bits can distinguish 2k distinct items
Combination binary bits together can represent some information or data. E.g can be
1. Decimal value 41
2. Alphabet (or character) ‘A’ in ASCII notation
3. Command to be performed e.g. Performing Add operation

5. Data What kinds of data do we need to represent? Data type:
Numbers – signed, unsigned, integers, real, floating point, complex, rational, irrational, …
Text – characters, strings, …
Images – pixels, colors, shapes, …
Logical – true, false
......
Data type:
Representation and operations within the computer
We’ll start with numbers…

6. 101 329 Unsigned Integers 22 21 20 102 101 100 Non-positional notation
Could represent a number (“5”) with a string of ones (“11111”)
Problems?
Weighted positional notation
Like decimal numbers: “329”
“3” is worth 300, because of its position, while “9” is only worth 9
Non-positional: Try adding (or, more extremely, dividing) two numbers using this number representation system). Try adding (or, more extremely, dividing) two numbers using this number representation system).
Question: What is an integer? Whole number (positive or negative number with no fractional part).
Aside: ENIAC was both non-positional (for each decimal digit) and positional
101 22 21 20
1x4 + 0x2 + 1x1 = 5
most
significant
least
base-10
(decimal)
329
102
101
100
base-2
(binary)
3x x10 + 9x1 = 329

7. Unsigned Integers (cont.)
An n-bit unsigned integer represents 2n values
From 0 to 2n-1 7 1 6 5 4 3 2
val 20 21 22

8. Operation on Unsigned Data
Base-2 addition – just like base-10!
Add from right to left, propagating carry
carry
10111
+ 111
(18)
(18)
(15)
(9)
(11)
(1)
Do example: in class 1 1 1 1 1 1 1 1 1
(27)
(29)
(16)
(23)
(7) 1 1 1 1
(30)

9. Signed Integers Positive integers Negative integers
Just like unsigned with zero in most significant bit = 5
Negative integers
Sign-magnitude: set high-order bit to show negative, other bits are the same as unsigned = -5
One’s complement: flip every bit to represent negative = -5
In either case, MS bit indicates sign: 0=positive, 1=negative
Both sign-magnitude and 1’s complement have problem
Indus Valley Civilization (Pakistan) “discovered” negative number c 2600BC. (India “discovered” 0 about the same time.)

10. E.g. One’s Complement Problem
Addition is complex
add two sign-magnitude numbers?
e.g., try (13)
Need to add add back the carry into least significant bit (LSB)
Which will result in the correct result, = 1
10011 (-12)
(13)
(0)

11. Two’s Complement + 11011 (-5) + (-9) 00000 (0) 00000 (0) 1 1 1 1 Idea
Find representation to make arithmetic simple and consistent
Specifics
For each positive number (X), assign value to its negative (-X), such that X + (-X) = 0 with “normal” addition, ignoring carry out
00101 (5) (9)
(-5) + (-9)
00000 (0) (0) 1 1 1 1

12. Two’s Complement (cont.)
If number is positive or zero
Normal binary representation, zeroes in upper bit(s)
If number is negative
Start with positive number
Flip every bit (i.e., take the one’s complement)
Then add one
Why does this work? 1’s comp A + !A > gives you a bunch of ones; add one more to turn them all to zeros!
00101 (5) (9)
11010 (1’s comp) (1’s comp)
11011 (-5) 1 1 1 1
(-9)

13. Two’s Complement Signed Integers
Range of an n-bit number: -2n-1 through 2n-1 – 1
Note: most negative number (-2n-1) has no positive counterpart 23 7 1 6 5 4 3 2 20 21 22 1 23 -1 -2 -3 -4 -5 -6 -7 -8 20 21 22

14. Converting Binary (2’s C) to Decimal
If leading bit is one, take two’s complement to get a positive number
Add powers of 2 that have “1” in the corresponding bit positions
If original number was negative, add a minus sign
1024 10
512 9
256 8
128 7 64 6 32 5 16 4 3 2 1 2n n
X = two
= =
= 104ten
Assuming 8-bit 2’s complement numbers.

15. More Examples X = 00100111two = 25+22+21+20 = 32+4+2+1 = 39ten
= =
= 39ten
1024 10
512 9
256 8
128 7 64 6 32 5 16 4 3 2 1 2n n
X = two
-X =
= =
= 26ten
X = -26ten
Assuming 8-bit 2’s complement numbers.

16. Converting Decimal to Binary (2’s C)
First Method: Division
Change to positive decimal number
Divide by two – remainder is least significant bit
Keep dividing by two until answer is zero, recording remainders from right to left
Append a zero as the MS bit; if original number negative, take two’s complement
What if we don’t append a 0 at end?
X = 104ten 104/2 = 52 r0 bit 0
52/2 = 26 r0 bit 1
26/2 = 13 r0 bit 2
13/2 = 6 r1 bit 3
6/2 = 3 r0 bit 4
3/2 = 1 r1 bit 5
1/2 = 0 r1 bit 6
X = two

17. Converting Decimal to Binary (2’s C)
1024 10
512 9
256 8
128 7 64 6 32 5 16 4 3 2 1 2n n
Second Method: Subtract Powers of Two
Change to positive decimal number
Subtract largest power of two less than or equal to number
Put a one in the corresponding bit position
Keep subtracting until result is zero
Append a zero as MS bit; if original was negative, take two’s complement
What if we tried to go right to left?
X = 104ten = 40 bit 6
= 8 bit 5
8 - 8 = 0 bit 3
X = two

18. Sign Extension To get correct results
Must represent numbers with same number of bits
What if we just pad with zeroes on the left?
Now, let’s replicate the MSB (the sign bit)
4-bit 8-bit
0100 (4) (still 4)
1100 (-4) (12, not -4)
4-bit 8-bit
0100 (4) (still 4)
1100 (-4) (still -4)

19. Operations: Arithmetic and Logical
Recall
A data type includes representation and operations
Operations for signed integers
Addition
Subtraction
Sign Extension
Logical operations are also useful
AND OR
NOT
And. . .
Overflow conditions for addition

20. Addition + 11110000 (-16) + 11110111 (-9) 1 01011000 (88) (-19)
2’s comp. addition is just binary addition
Assume all integers have the same number of bits
Ignore carry out
For now, assume that sum fits in n-bit 2’s comp. representation
Assuming 8-bit 2’s complement numbers
Ignore carry. ??
(104) (-10)
(-16) (-9) (88) (-19)
carry(discard)

21. Subtraction - 00010000 (16) - (-9) + 11110000 (-16) + (9)
Negate 2nd operand and add
Assume all integers have the same number of bits
Ignore carry out
For now, assume that difference fits in n-bit 2’s comp. representation
(104) (-10)
(16) - (-9)
(-16) + (9)
(88) (-1) 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Assuming 8-bit 2’s complement numbers.

22. Logical Operations Operations on logical TRUE or FALSE
Two states: TRUE=1, FALSE=0
View n-bit number as a collection of n logical values
Operation applied to each bit independently 1 A
A AND B B 1 A
A OR B B 1 A
NOT A

23. Examples of Logical Operations
AND
AND
Useful for clearing bits
AND with zero = 0
AND with one = no change OR
Useful for setting bits
OR with zero = no change
OR with one = 1
NOT
Unary operation -- one argument
Flips every bit OR
NOT

24. Overflow + 01001 (9) + 10111 (-9) 10001 (-15) 01111 (+15)
Overflow is said to occur if result is too large to fit in the number of bits used in the representation.
Sum cannot be represented as n-bit 2’s comp number
We have overflow if
Signs of both numbers are the same, and Sign of sum is different
If Positive number is subtracted from a Negative number, result is positive and vice versa
01000 (8) (-8)
(9) (-9)
10001 (-15) (+15)
Question: Can we have overflow when adding numbers with opposite sign? No, sum will never be more positive nor more negative than both the operands.
Question: Why does carry into MSB have to equal carry out? Just a shortcut to test sign of result differs from signs of operands. If both 0, result only 1 if carry-in is 1, no carry out (overflow). If both 1, result 0 if carry-in is 0, so carry-out is 1 (overflow). If one 0 and other 1, carry-in always results in carry-out being 1.

25. Number System People like to use decimal numbers
Computers use binary numbers
Java translates decimal numbers into binary
The computer does all its arithmetic in binary
Java translates binary results back into decimal
You occasionally have to use numbers in other number systems
In Java, you can write numbers as octal, decimal, or hexadecimal (but not binary)
Colors are usually specified in hexadecimal notation: #FF0000, #669966,

26. Hexadecimal Notation
It is often convenient to write binary (base-2) numbers as hexadecimal (base-16) numbers instead
Fewer digits: four bits per hex digit
Less error prone: easy to corrupt long string of 1’s and 0’s 7 6 5 4 3 2 1
Hex
0111
0110
0101
0100
0011
0010
0001
0000
Decimal
Binary F E D C B A 9 8
Hex 15
1111 14
1110 13
1101 12
1100 11
1011 10
1010
1001
1000
Decimal
Binary
Why is hexadecimal so convenient for representing binary numbers? 4 binary digits represent 16 values, the same as 1 hex digit; not true of, e.g., decimal.

27. Hexadecimal Conversions
Binary to Hex
Every group of four bits is a hex digit
Start grouping from right-hand side 7 D 4 F 8 A 3
What if we group from right to left?
Hex to Decimal
1AC16 = 1 x x x 160 = 42810
This is not a new machine representation, just a convenient way to write the number.

28. Octal Numbers 3 digits per every octal group Octal Decimal Binary 7 65 4 3 2 1
Octal
111
110
101
100
011
010
001
000
Decimal
Binary

29. Octal to Binary Conversion
One octal digit equals three binary digits
Octal to Decimal
1738 = 1 x x x 80
= 12310

30. Fractions: Fixed-Point
How can we represent fractions?
Use a “binary point” to separate positive from negative powers of two (just like “decimal point”)
2’s comp addition and subtraction still work
If binary points are aligned
(40.625)
(-1.25)
(39.375)
2-1 = 0.5
2-2 = 0.25
2-3 = 0.125
No new operations -- same as integer arithmetic

31. Very Large and Very Small: Floating-Point
Problem
Large values: x requires 79 bits
Small values: x requires >110 bits
Use equivalent of “scientific notation”: F x 2E
Need to represent F (fraction), E (exponent), and sign
IEEE 754 Floating-Point Standard (32-bits):
Note: Avagadro’s number and Planck’s constant
Special casing exponent = 0, allows for the representation of 0. 1b 8b
23b S
Exponent
Fraction

32. Floating Point Example
Single-precision IEEE floating point number
Sign is 1: number is negative
Exponent field is = 126 (decimal)
Fraction is … = 2-1 = 1/2 = 0.5 (decimal)
Value = -1.5 x 2( ) = -1.5 x 2-1 = -0.75
sign
exponent
fraction

33. Text: ASCII Characters
ASCII: Maps 128 characters to 7-bit code.
Both printable and non-printable (ESC, DEL, …) characters
del 7f o 6f _ 5f O 4f ? 3f / 2f us 1f si 0f ~ 7e n 6e ^ 5e N 4e
> 3e . 2e rs 1e so 0e } 7d m 6d ] 5d M 4d = 3d - 2d gs 1d cr 0d | 7c l 6c \ 5c L 4c
< 3c , 2c fs 1c np 0c { 7b k 6b [ 5b K 4b ; 3b + 2b
esc 1b vt 0b z 7a j 6a Z 5a J 4a : 3a * 2a
sub 1a nl 0a y 79 i 69 Y 59 I 49 9 39 ) 29 em 19 ht 09 x 78 h 68 X 58 H 48 8 38 ( 28
can 18 bs 08 w 77 g 67 W 57 G 47 7 37 ' 27
etb 17
bel 07 v 76 f 66 V 56 F 46 6 36
& 26
syn 16
ack 06 u 75 e 65 U 55 E 45 5 35 % 25
nak 15
enq 05 t 74 d 64 T 54 D 44 4 34 $ 24
dc4 14
eot 04 s 73 c 63 S 53 C 43 3 33 # 23
dc3 13
etx 03 r 72 b 62 R 52 B 42 2 32
" 22
dc2 12
stx 02 q 71 a 61 Q 51 A 41 1 31 ! 21
dc1 11
soh 01 p 70 ` 60 P 50 @ 40 30 sp 20
dle 10
nul 00
American Standard Code for information Interchange
Java uses Unicode Standard for characters (

34. Storage Space for Numerics
Numeric types in Java are characterized by their size: how much memory they occupy
1 byte = 8 bits
Integral types
Floating point types
size
range
byte
1 byte
-128: 127
short
2 bytes
-32768:32767
char
0:65535
int
4 bytes :
long
8 bytes …
smallest > 0
4.9E-324
1.4E-45
1.7E308
8 bytes
double
3.4E38
4 byte
float
largest
size

35. Other Data Types Text strings Image
Sequence of characters, terminated with NULL (0)
Image
Array of pixels
Monochrome: one bit (0/1 = black/white)
Color: red, green, blue (RGB) components (e.g., 8 bits each)