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Understanding Floating Point Data Types and ASCII Codes in Computer Systems

This chapter from COMP 2610 by Dr. James Money dives into the fundamentals of floating point data types, particularly the IEEE standard for representing floating point numbers using 32 bits. It covers the structure including the sign bit, exponent, and fraction bits. The section also explains ASCII codes, the eight-bit standard for character representation, and how hexadecimal notation relates to binary, dispelling common confusions regarding numeral bases. This comprehensive overview is essential for understanding data types and their applications in computing.

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Understanding Floating Point Data Types and ASCII Codes in Computer Systems

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  1. 1 Bits, Data types, and Operations: Chapter 2 COMP 2610 Dr. James Money COMP 2610

  2. Floating Point Data Type • Most ISAs have a data type called float, which is 32 bits arranged as follows • 1 bit for sign • 8 bits for range of exponent • 23 bits for the precision or fraction • This is called the IEEE Standard for Floating Point Arithmetic

  3. Floating Point Data Type

  4. Floating Point Data Type • Recall the bits in the fractional part are of the form 1.b-1b-2…b-j • The corresponding decimal number is 1x20 + b-1x2-1+b-2x2-2+… + b-j 2-j

  5. Floating Point Data Type • If the exponent=00000000, then we can represent tiny numbers. • In this case, we assume the leading digit is zero and not 1 and exp=-126. That is, it is of the form -1s 0.fraction x 2-126

  6. Floating Point Data Type • Consider the floating point value 0 00000000 0000100000000000000000 • This is + 2-5 x 2-126 = 2-131

  7. Floating Point Data Type • Interpret the floating point values • 0 11111110 1111111111111111111111 • 1 00001101 0111000000000000000000 • 1 00000000 0000000000001000000000

  8. ASCII Codes • Another standard of representation is one for transferring character codes • This is an eight bit code referred to ASCII • ASCII stands for American Standard Code for Information Exchange • It simplifies the interface between I/O devices among companies

  9. ASCII Codes • Each key on a keyboard is identified by a unique ASCII code • The digit 3 is (00110011)2 = (41)10, digit 3 is (00110010)2 = (40)10 • The letter ‘e’ is (01100101)2 and carriage return is (00001101)2

  10. ASCII Codes • The list of codes are at the back of the book for all 256 ASCII codes • Some codes are associate with multiple keys, such as ‘e’ and ‘E’

  11. Hexadecimal Notation • One form that is common for reading values on the computer is called hexadecimal notation • Hexadecimal notation is the base 16 representation of the number

  12. Hexadecimal Notation • We use 0-9 for the same numbers in hexadecimal • What about 10-15? • We use the letters A-F

  13. Hexadecimal Notation • A – 10 • B – 11 • C – 12 • D – 13 • E – 14 • F - 15

  14. Hexadecimal Notation • Consider the binary string 0011110101101110 • This can be broken into groups of 4 bits: 0011 1101 0110 1110 • Now, recall that the range of 4 bits is 0 -15, just like hexadecimal notation

  15. Hexadecimal Notation • Hence, the numbers represent • We typically prefix this by x or 0x to indicate hexadecimal form • So our binary number is 0x3D6E

  16. Hexadecimal Notation • What is the number 0x5A6C in binary form? • Thus, (5A6C)16 = (0101 1010 0110 1100)2

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