1. Computer Science 210 Computer Organization
Number Systems
Unsigned Integers

2. People Decimal Numbers (base 10) Sign-Magnitude (-324)
Decimal Fractions (23.27)
Letters for text

3. Computers Binary Numbers (base 2) Sign-magnitude
Binary fractions and floating point
ASCII codes for characters (A65)

4. Why binary? Information is stored in computer via voltage levels.
Using decimal would require 10 distinct and reliable levels for each digit.
This is not feasible with reasonable reliability and financial constraints.
Everything in computer is stored using bits: numbers, text, programs, pictures, sounds, videos, ...

5. Decimal: Non-negatives
Base 10
Uses decimal digits: 0, 1, . . ., 9
Positional System - position gives power
Example: = 3x x x x100
Positions: …543210

6. Binary: Non-negatives
Base 2
Uses binary digits (bits): 0,1
Positional system
Example: = 1x23 + 1x22 + 0x21 + 1x20

7. Conversions
External Internal (For people) (For computer) A
People want to see and enter numbers in decimal.
Computers must store and compute with bits.

8. Binary to Decimal Conversion
Algorithm:
Expand binary number using positional scheme.
Perform computation using decimal arithmetic.
Example:  1x24 + 1x23 + 0x22 + 0x21 + 1x = = = 2510

9. Decimal to Binary - Algorithm 1
Algorithm: While N  0 do Set N to N/2 (whole part) Record the remainder (1 or 0) Set A to remainders in reverse order

10. Decimal to binary - Example
Example: Convert to binary N Rem N Rem
32410 =

11. Decimal to Binary - Algorithm 2
Algorithm: Set A to 0 (all bits 0) While N  0 do Find largest P with 2P  N Set bit in position P of A to 1 Set N to N - 2P

12. Decimal to binary - Example
Example: Convert to binary N Power P A
32410 =

13. Octal Numbers Base 8 Digits 0,1,2,3,4,5,6,7
Not so many digits as binary
Easy to convert to and from binary
Often used by people who need to see the internal representation of data, programs, etc.

14. Octal Conversions Octal to Binary Binary to Octal
Simply convert each octal digit to a three bit binary number.
Example: =
Binary to Octal
Starting at right, group into 3 bit groups
Convert each group to an octal digit
Example = =

15. Hexadecimal Base 16 Digits 0,…,9,A,B,C,D,E,F Hexadecimal  Binary
Just like Octal, only use 4 bits per digit.
Example: 98C316 =
Example = = 34EB

16. Binary Addition One bit numbers: + 0 1 0 | 0 1 1 | 1 10
Example (53) (45) (98)

17. Overflow
In a given type of computer, the size of integers is a fixed number of bits.
16 or 32 bits are popular choices
It is possible that addition of two n bit numbers yields a result requiring n+1 bits.
Overflow is the term for an operation whose results exceeds the size allowed for a number.