1 / 16

Chapter 2 Bits, Data Types, and Operations

Chapter 2 Bits, Data Types, and Operations. 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.

malia
Download Presentation

Chapter 2 Bits, Data Types, and Operations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 2Bits, Data Types,and Operations

  2. Hexadecimal Notation • It is often convenient to write binary (base-2) numbersas 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

  3. Converting from Binary to Hexadecimal • Every four bits is a hex digit. • start grouping from right-hand side 011101010001111010011010111 3 A 8 F 4 D 7 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. FA91hex or xFA91 FA91 1111101010010001 2DEhex or x2DE 2DE 001010111100

  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 3AF4hex = 3x163 + Ax162 + Fx161 + 4x160 = 3x163 + 10x162 + 15x161 + 4x160 = 3x4096 + 10x256 + 15x16 + 4x1 = 12,288 + 2,560 + 240 + 4 = 19,092ten

  6. 2-1 = 0.5 2-2 = 0.25 2-3 = 0.125 00101000.101 (40.625) + 11111110.110(-1.25) 00100111.011 (39.375) Fractions: Fixed-Point • How can we represent fractions? • Use a “binary point” to separate positivefrom negative powers of two -- just like “decimal point.” • 2’s comp addition and subtraction still work. • if binary points are aligned No new operations -- same as integer arithmetic.

  7. Fractions: Fixed-Point • How is -6 5/8 represented in the floating point data type? • Break problem into two parts • Whole: 6 = 1x22 + 1x21 + 0x20 => 110 • Fraction: 5/8 = ½ (4/8) + 1/8 => 1x2-1 + 0x2-2 + 1x2-3 = .101 -6 5/8 ten = - 110.101two

  8. Very Large and Very Small: Floating-Point • Large values: 6.023 x 1023 -- requires 79 bits • Small values: 6.626 x 10-34 -- 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): 1b 8b 23b S Exponent Fraction

  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. sign exponent fraction

  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 20 = 1.5625. sign exponent fraction

  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. sign exponent fraction

  12. Text: ASCII Characters • ASCII: Maps 128 characters to 7-bit code. • both printable and non-printable (ESC, DEL, …) characters

  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. 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. 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. LC-3 Data Types • Some data types are supported directly by theinstruction 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 interpreting16-bit values as logical, text, fixed-point, etc.,in the software that we write.

More Related