CSI 333 Lecture 1 Number Systems

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1 CSI 333 Lecture 1 Number Systems 1 1 / 23

2 Basics of Number Systems Ref: Appendix C of Deitel & Deitel. Weighted Positional Notation: 192 = General: Digit sequence : d n 1 d n 2... d 1 d 0 Value : Decimal system: n 1 ( di 10 i) i=0 Base (or Radix) : 10 Digits : 0, 1, 2,..., / 23

3 Number Systems (continued) Base r system: Digits : 0, 1,..., r 1 Digit sequence : s n 1 s n 2... s 1 s 0 Value : n 1 ( si r i) i=0 s Least significant digit s n Most significant digit 1 3 / 23

4 Number Systems (continued) Common Values for Radix r: r = 2 : Binary - - Digits: 0, 1. r = 8 : Octal - - Digits: 0, 1..., 7. r = 10 : Decimal - - Digits: 0, 1,..., 9. r = 16 : Hexadecimal (Hex) - - Digits: 0, 1,..., 9, A, B, C, D, E, F (where A = 10, B = 11,..., F = 15). Convention: Base written as subscript (e.g ). 1 4 / 23

5 Converting from any base to Decimal Example 1: Find the decimal value of = = = Example 2: Find the decimal value of 3F F 4 16 = = = / 23

6 Converting to Decimal (continued) Example 3: Find the decimal value of Convention in C: Decimal Octal 0x Hex = = = Machine hardware: Uses binary. 1 6 / 23

7 Example: = / 23 Octal and Hexadecimal Number Systems Note: Examples: Each octal digit bits Each hex digit bits 6 8 = C 16 = Converting from Binary to Octal: 1 Moving from right to left, form groups of three bits. (May need to add leading zeros.) 2 Write octal equivalent for each group.

8 Octal and Hexadecimal (continued) Converting from Binary to Hex: Similar to binary to octal conversion, except that we form groups of 4 bits. Example: = 56D 16 Converting Octal or Hex to Binary: Express each digit in binary. Examples: = = Note: Octal and hex are short forms for representing binary numbers. 1 8 / 23

9 Converting from Decimal to Other Bases Example 1: Convert to binary. So, = Division Quotient Remainder Remark: In the binary representation, the remainders are written in bottom-to-top order. 1 9 / 23

10 Converting from Decimal... (continued) Example 2: Convert to hex. So, = 34C 16. Division Quotient Remainder = C Remark: Again, the remainders are written in bottom-to-top order / 23

11 2 s Complement form for Negative Integers Sign Convention in Binary: Most significant bit = 0 for non-negative integers. Most significant bit = 1 for negative integers. An Algorithm for Finding 2 s Complement Form: 1 Compute the binary representation for the corresponding positive integer. 2 Complement each bit. 3 Add / 23

12 2 s Complement form... (continued) Recall: Binary addition table. Inputs Sum Carry / 23

13 2 s Complement form... (continued) Problem: Find the 2 s complement representation of 15 using 8 bits. Solution: +15 in 8-bit binary: Complement each bit: Add 1 : <-- Answer 1 13 / 23

14 2 s Complement form... (continued) Problem: Find the 2 s complement representation of 64 using 8 bits. Solution: +64 in 8-bit binary: Complement each bit: Add 1 : <-- Carries <-- Answer 1 14 / 23

15 2 s Complement form... (continued) Another method for 2 s complement: 1 Start with the binary representation of the positive value. 2 Copy bits from right to left, until the first 1 has been copied. 3 Complement every bit thereafter. Problem: Find the 2 s complement representation of 64 using 8 bits. Solution: +64 in 8-bit binary: s complement for -64: ^ (First 1 copied) 1 15 / 23

16 2 s Complement form... (continued) Problem: What is the 2 s complement of ? Solution: Given: Complement: Add 1 : <--- Carries <--- Answer (Value = 0) 1 --> Carry out of sign bit (should be ignored) 1 16 / 23

17 Representing Real Numbers Remark: Unlike integers, some real numbers cannot be represented exactly. Weighted Positional Notation (for Reals): Example: Integer part: Fractional part: = = / 23

18 Representing Reals (continued) General formula for base r: Sequence : s n 1 s n 2... s 1 s 0. s 1 s 2... s p Value : n 1 ( si r i) + i=0 p ( s i r i) i=1 Example 1: Convert to decimal = = = / 23

19 Representing Reals (continued) Example 2: Convert to decimal = = = Example 3: Convert to decimal = = (non-terminating) = / 23

20 Decimal to Binary Conversion for Real Numbers Real number: Integer part. Fractional part Method: Example 1: Convert each part separately. Convert to binary = (decimal to binary for integer part). To convert to binary: = 0.25 : d 1 = = 0.5 : d 2 = = 1.0 : d 3 = 1 So, = Therefore, = / 23

21 Decimal to Binary for Reals (continued) Example 2: Convert to binary = (decimal to binary for integer part). To convert to binary: = : d 1 = = 1.25 : d 2 = = 0.5 : d 3 = = 1.0 : d 4 = 1 So, = Therefore, = / 23

22 Decimal to Binary for Reals (continued) Example 3: Convert to binary = 1.4 : d 1 = = 0.8 : d 2 = = 1.6 : d 3 = = 1.2 : d 4 = = 0.4 : d 5 = = 0.8 : d 6 = 0. The part 0110 repeats indefinitely; so, = Later: IEEE Standard for representing real (or floating point) numbers / 23

23 Suggestions and Questions to Think About Try additional base conversion problems, including those involving real numbers. Try converting negative integers into 2 s complement form. Recall that converting a real number from base 3 to base 10 may lead to a non-terminating representation. Can this happen when we do the following? Convert a real number in base 5 to base 10. Convert a real number in base 10 to base / 23

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