1. Number Systems & Binary Arithmetic
Slides adapted from:
Dr. Wood, Univ. of Maryland
Prof. Orr, Indiana Univ.

2. Encoding Information
We use encoding schemes to represent and store information. • Roman Numerals: I, IV, XL, MMIV • Acronyms: UMW, CPSC-125, SSGN-726 • Postal codes: 22401, M5W 1E6 Encoding schemes are only useful if the stored information can be retrieved.

3. Introduction to Numbering Systems
We are all familiar with the decimal number system (Base 10). Some other number systems that we will work with are:
Binary  Base 2
Octal  Base 8
Hexadecimal  Base 16

4. Characteristics of Numbering Systems
The digits are consecutive.
The number of digits is equal to the size of the base.
Zero is always the first digit.
The base number is never a digit.
When 1 is added to the largest digit, a sum of zero and a carry of one results.
Numeric values determined by the number have implicit positional values of the digits.

5. Decimal Notation
base 10 or radix uses 10 symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 • Position represents powers of 10 • or 5473 (5 * 103) + (4 * 102) + (7 * 101) + (3 * 100) • or (7 * 101) + (3 * 100) + (1 * 10-1) + (9 * 10-2)

6. Why we use a Decimal Number System

7. Binary Notation
Binary Notation • base uses only 2 symbols 0, 1 • Position represents powers of 2 • (1 * 24) + (1 * 23) + (0 * 22) + (1 * 21) + (0 * 20) • (1 * 21) + (0 * 20) + (1 * 2-1) + (1 * 2-2)

8. Binary Number System Also called the “Base 2 system”
The binary number system is used to model the series of electrical signals computers use to represent information
0 represents the no voltage or an off state
1 represents the presence of voltage or an
on state

9. Why do we need the Binary System?
“On” 1 True
“Off” 0 False

10. Octal Notation
base uses 8 symbols 0, 1, 2, 3, 4, 5, 6, 7 • Position represents power of 8 • (1 * 83) + (5 * 82) + (2 * 81) + (3 * 80) • (5 * 81) + (6 * 80) + (7 * 8-1) + (2 * 8-2)

11. Hexadecimal Notation
base 16 or ‘hex’ ... uses 16 symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F • Position represents powers of 16 • B65F16 or 0xB65F (11 * 163) + (6 * 162) + (5 * 161) + (15 * 160) • F3.A916 or 0xF3.A9 (15 * 161) + (3 * 160) + (10 * 16-1) + (9 * 16-2)

12. Why do we need Hex and Octal ?
• Humans are accustomed to decimal
• Computers use binary
• So why ever use octal or hexadecimal?
Binary numbers can be long with a lot of digits
Longer number make for more confusion
Copying longer numbers allows greater chance for error
Octal represents binary in 1/3 the number of digits
Hexadecimal represents binary in 1/4 the number of digits

13. Conversions Binary , Octal , Hex to Decimal
Decimal to Binary , Octal , Hex
Binary to Octal , Octal to Binary
Binary to Hex , Hex to Binary

14. Base B to Decimal (Base 10)
The easiest method for converting a number in base B to its decimal equivalent is to use the Multiplication Algorithm
Multiply the binary digits by increasing powers of B, starting from the right
Then, to find the decimal number equivalent, sum those products

15. Binary to Decimal Example
Convert ( )2 to its decimal equivalent:
Binary
Positional Values x x x x x x x x 27 26 25 24 23 22 21 20
Products
17310

16. Octal to Decimal Example
Convert 6538 to its decimal equivalent:
Octal Digits x x x
Positional Values
Products
42710

17. Hex to Decimal Example
Convert 3B4F16 to its decimal equivalent:
Hex Digits
3 B F x x x x
Positional Values
Products
15,18310

18. Decimal (Base 10) to Base B
The easiest way to convert a decimal number to its base B equivalent is to use the Division Algorithm
This method repeatedly divides a decimal number by B and records the quotient and remainder 
The remainder digits form the Base B equivalent in least significant to most significant digit sequence

19. Decimal to Binary Example
Convert 67 to its binary equivalent:
6710 = x2
Step 1: 67 / 2 = 33 R Divide 67 by 2. Record quotient in next row
Step 2: 33 / 2 = 16 R Again divide by 2; record quotient in next row
Step 3: 16 / 2 = 8 R Repeat again
Step 4: 8 / 2 = 4 R Repeat again
Step 5: 4 / 2 = 2 R Repeat again
Step 6: 2 / 2 = 1 R Repeat again
Step 7: 1 / 2 = 0 R 1 STOP when quotient equals 0

20. Decimal to Octal Example
Convert to its octal equivalent:
427 / 8 = 53 R3 Divide by 8; R is LSD
53 / 8 = 6 R5 Divide Q by 8; R is next digit
6 / 8 = 0 R Repeat until Q = 0
6538

21. Decimal to Hex Example 33E16
Convert to its hexadecimal equivalent:
830 / 16 = 51 R14
51 / 16 = 3 R3
3 / 16 = 0 R3
= E in Hex
33E16

22. Binary to Octal Example
Binary to Octal – Group in 3’s • • (12) (1002) (1102) (1112) • (18) (48) (68) (78) • 14678

23. Octal to Binary Example
Octal to Binary – Write each octal digit as a 3 bit binary equivalent
• 14678
• (18) (48) (68) (78)
• (0012) (1002) (1102) (1112) •

24. Binary to Hex Example
Binary to Hex – Group in 4’s • • (10112) (01102) (01012) (11112) • (B16) (616) (516) (F16) • B65F16

25. Hex to Binary Example
Hex to Binary – write each hex digit as a 4 bit binary equivalent
• B65F16
• (B16) (616) (516) (F16)
• (10112) (01102) (01012) (11112) •

26. Computing Systems
• Computers were originally designed for the primary purpose of doing numerical calculations. Abacus, counting machines, calculators • We still do numerical operations, but not necessarily as a primary function. • Computers are now “information processors” and manipulate primarily nonnumeric data. Text, GUIs, sounds, information structures, pointers to data addresses, device drivers

27. Codes • A code is a scheme for representing information.
• Computers use codes to store types of
• Examples of codes:
Alphabet
DNA (biological coding scheme)
Musical Score
Mathematical Equations

28. Codes
• Two Elements (always) 1) A group of symbols 2) A set of rules for interpreting these symbols • Place-Value System (common) - Information varies based on location - Notes of a musical staff - Bits in a binary number

29. Computers and Codes
Computers are built from two-state electronics. • Each memory location (electrical, magnetic, optic) has two states (On or Off) • Computers must represent all information using only two symbols • On (1) • Off (0) • Computers rely on binary coding schemes

30. Computers and Binary
• Decimal 109 • Binary • Computer 16-bit word size • Computer 32-bit word size bit words are now common

31. zeroes are frequently dropped.
Binary Addition
4 Possible Binary Addition Combinations:
(1) (2) 0
(3) (4) 1
Note that leading
zeroes are frequently dropped.
Carry
Sum

32. Negative Numbers 4 ways to represent negative numbers
Sign Magnitude Format
One’s Complement Format
Two’s Complement Format
Excess – N Format
We study the first 3 here

33. Sign Magnitude format
Sign-Magnitude Format • Uses the highest order bit as a ‘sign’ bit. All other bits are used to store the absolute value (magnitude) • Negative numbers have the sign bit set • in 16-bit representation using a sign bit

34. Consequences of Sign Magnitude
Two ways to represent 0
+0 =
-0 =
Reduces the range of values that can be stored.

35. One’s Complement
One’s Complement Format • Exact opposite of the sequence for the positive value. Each bit is “complemented” or “flipped” • 109 in 16-bit representation • -109 in 16-bit One’s Complement

36. Consequences of One’s Complement
Two ways to represent 0
+0 =
-0 =
• This makes mathematical operations difficult

37. Two’s Complement
Two’s Complement Format • Add 1 to One’s Complement • 109 in 16-bit representation • -109 in 16-bit One’s Complement • -109 in 16-bit Two’s Complement

38. Consequences of Two’s Complement
In two's-complement, there is only one zero ( )
Mathematical operations are much simpler

39. Next Time : Programming Languages