1. Chapter 2 Bits, Data Types, and Operations

2. 2-2 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 BinaryHexDecimal 000000 000111 001022 001133 010044 010155 011066 011177 BinaryHexDecimal 100088 100199 1010A10 1011B11 1100C12 1101D13 1110E14 1111F15

3. 2-3 Converting from Binary to Hexadecimal Every four bits is a hex digit. start grouping from right-hand side 011101010001111010011010111 7D4F8A3 This is not a new machine representation, just a convenient way to write the number.

4. Converting from Hexadecimal to Binary Hexadecimal to binary conversion: Remember that hex is a 4-bit representation. 2-4 FA91 hex or xFA91 F A 9 1 1111 1010 1001 0001 2DE hex or x2DE 2 D E 0010 1011 1100

5. Convert Hexadecimal to Decimal Hexadecimal to decimal is performed the same as binary to decimal, positional notation. Binary to decimal uses base 2 Decimal is base 10 Hexadecimal is base 16 2-5 3AF4 hex = 3x16 3 + Ax16 2 + Fx16 1 + 4x16 0 = 3x16 3 + 10x16 2 + 15x16 1 + 4x16 0 = 3x4096 + 10x256 + 15x16 + 4x1 = 12,288 + 2,560 + 240 + 4 = 19,092 ten

6. 2-6 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 00101000.101 (40.625) +11111110.110 (-1.25) 00100111.011 (39.375) 2 -1 = 0.5 2 -2 = 0.25 2 -3 = 0.125 No new operations -- same as integer arithmetic.

7. 2-7 Fractions: Fixed-Point How is -6 5/8 represented in the floating point data type? Break problem into two parts  Whole: 6 = 1x2 2 + 1x2 1 + 0x2 0 => 110  Fraction: 5/8 = ½ (4/8) + 1/8 => 1x2 -1 + 0x2 -2 + 1x2 -3 =.101 -6 5/8 ten = - 110.101 two

8. 2-8 Very Large and Very Small: Floating-Point Large values: 6.023 x 10 23 -- requires 79 bits Small values: 6.626 x 10 -34 -- requires >110 bits Use equivalent of “scientific notation”: F x 2 E Need to represent F (fraction), E (exponent), and sign. IEEE 754 Floating-Point Standard (32-bits): SExponentFraction 1b8b23b

9. 2-9 Floating Point Example Single-precision IEEE floating point number: 10111111010000000000000000000000 Sign is 1 – number is negative. Exponent field is 01111110 = 126 (decimal). Fraction is 0.100000000000… = 0.5 (decimal). Value = -1.5 x 2 (126-127) = -1.5 x 2 -1 = -0.75. signexponentfraction

10. 2-10 Floating Point Example Single-precision IEEE floating point number: 00111111110010000000000000000000 Sign is 0 – number is positive. Exponent field is 01111111 = 127 (decimal). Fraction is 0.100100000000… = 0.5625 (decimal). Value = 1.5625 x 2 (127-127) = 1.5625 x 2 0 = 1.5625. signexponentfraction

11. 2-11 Floating Point Example Single-precision IEEE floating point number: 00000000011110000000000000000000 Sign is 0 – number is positive. Exponent field is 00000000 = 0 (decimal) special case. Fraction is 0.111100000000… = 0.9375 (decimal). Value = 0.9375 x 2 (-126) = = 0.9375 x 2 -126. signexponentfraction

12. 2-12 Text: ASCII Characters ASCII: Maps 128 characters to 7-bit code. both printable and non-printable (ESC, DEL, …) characters 00nul10dle20sp30040@50P60`70p 01soh11dc121!31141A51Q61a71q 02stx12dc222"32242B52R62b72r 03etx13dc323#33343C53S63c73s 04eot14dc424$34444D54T64d74t 05enq15nak25%35545E55U65e75u 06ack16syn26&36646F56V66f76v 07bel17etb27'37747G57W67g77w 08bs18can28(38848H58X68h78x 09ht19em29)39949I59Y69i79y 0anl1asub2a*3a:4aJ5aZ6aj7az 0bvt1besc2b+3b;4bK5b[6bk7b{ 0cnp1cfs2c,3c<4cL5c\6cl7c| 0dcr1dgs2d-3d=4dM5d]6dm7d} 0eso1ers2e.3e>4eN5e^6en7e~ 0fsi1fus2f/3f?4fO5f_6fo7fdel

13. 2-13 Interesting Properties of ASCII Code What is relationship between a decimal digit ('0', '1', …) and its ASCII code? x30 -> ‘0’, x31 -> ’1’, … x39 -> ’9’ What is the difference between an upper-case letter ('A', 'B', …) and its lower-case equivalent ('a', 'b', …)? Difference of x20 Given two ASCII characters, how do we tell which comes first in alphabetical order? Compare ASCII values, the lowest value is the first in alphabetical order Are 128 characters enough? (http://www.unicode.org/)

14. 2-14 Other Data Types Text strings sequence of characters, terminated with NULL (0) typically, no hardware support Image array of pixels  monochrome: one bit (1/0 = black/white)  color: red, green, blue (RGB) components (e.g., 8 bits each)  other properties: transparency hardware support:  typically none, in general-purpose processors  MMX -- multiple 8-bit operations on 32-bit word Sound sequence of fixed-point numbers

15. 2 - 15 Another use for bits: Logic Beyond numbers logical variables can be true or false, on or off, etc., and so are readily represented by the binary system. A logical variable A can take the values false = 0 or true = 1 only. The manipulation of logical variables is known as Boolean Algebra, and has its own set of operations - which are not to be confused with the arithmetical operations of the previous section. Some basic operations: NOT, AND, OR, XOR

16. 2-16 LC-3 Data Types Some data types are supported directly by the instruction set architecture. For LC-3, there is only one hardware-supported data type: 16-bit 2’s complement signed integer Operations: ADD, AND, NOT Other data types are supported by interpreting 16-bit values as logical, text, fixed-point, etc., in the software that we write.