SIT153 Introduction to Computation

Transcription

1 SIT153 Introduction to Computation Lecture 1.4 Number Systems IV chapter 2 11 Feb 2016

2 Admin matters (1) for the next 2 weeks Study units: (1) Number systems (2) Data storage Week Date Day Activity Topic Txt. bk. chapter 2 8 Feb Mon 9 Feb Tues 10 Feb Wed Lectures: L Feb Thur Lectures: L1.4 Practical 1, Blue 3 Lab 12 Feb Fri Number systems ch Feb Mon 16 Feb Tues 17 Feb Wed Lectures: L Feb Thur Lectures: L2.2 Practical 2, Blue 3 Lab 19 Feb Fri Data storage ch 3 2

3 To be discussed in this lecture (1) Last lecture Conversion of fractions and reals between number systems This lecture Conversion between binary, octal & Hex number systems Alternative method of conversion dec->bin for small integer values Find the number of digits needed in each system to represent a particular value. 3

4 To be discussed in this lecture (2) Conversion of integers between number systems binary to octal, octal to binary binary to hexadecimal, hexadecimal to binary Conversion of real numbers between number systems binary to octal, octal to binary binary to hexadecimal, hexadecimal to binary 4

5 Binary to octal & octal to binary conversion (1) Binary to octal: Each group of 3 bits is translated into one octal digit. Octal to binary: Each octal digit is translated into a group of 3 binary digits. 2.5 Binary digits Octal digit

6 Binary to octal & octal to binary conversion (2) Example: Show the octal equivalent of the binary number ( ) 2. Solution Each group of three bits is translated into one octal digit The result is (562) 8. Class exercise: Show the octal equivalent of each of the binary numbers below: a. ( ) 2 this an 8-bit number 2.6 b. ( ) 2 this is a 16-bit number

7 Binary to octal & octal to binary conversion (2) Example: What is the binary equivalent of (24) 8? Solution Write each octal digit as its equivalent bit pattern to get and The result is (010100) 2. Class exercise: What is the binary equivalent: 2.7 a. (74) 8 as an 8-bit number b. (2437) 8 as a 16-bit number

8 Binary to hex & hex to binary conversion (1) Binary to hex: Each group of 4 bits is translated into one hex digit. Hex to binary: Each hex digit is translated into a group of 4 binary digits. 2.8 Binary digits Hex digit A B C D E F

9 Binary to hex & hex to binary conversion (2) Example: Show the hexadecimal equivalent of the binary number ( ) 2. Solution first arrange the binary number in 4-bit patterns (right-to-left): E The result is (4E2) 16. Class exercise: Show the hex equivalent of each of the binary numbers below: a. ( ) 2 this an 8-bit number 2.9 b. ( ) 2 this is a 16-bit number

10 Binary to hex & hex to binary conversion (2) Example: What is the binary equivalent of (24C) 16? Solution Each hexadecimal digit is converted to 4-bit patterns: C The result is ( ) 2. Class exercise: What is the binary equivalent: 2.10 a. (CD) 16 as an 8-bit number b. (7FFF) 16 as a 16-bit number

11 Binary to Octal & to hex conversion (fractions) Consider: Binary: ( ) 2 Group fraction digits from left to right & pad with zeros on the right if necessary. To Octal: ( ) 2 = (0.24) 8 Note: padding with zero to the right To Hex: ( ) 2 = (0.50) 16 Note: padding with zero to the right Checking the conversion: decimal Bin: Octal: Hex:

12 Octal to hex & hex to octal conversion: integers(1) order of conversion LEFT RIGHT NOTE: Grouping of binary digits is done from right to left. Class exercise: Convert the following octal numbers to hexadecimal: 2.12 a. (74) 8 b. (2437) 8

13 Octal to hex & hex to octal conversion: integer (2) order of conversion LEFT RIGHT AGAIN: Grouping of binary digits is done from right to left. Class exercise: Convert the following hex numbers to octal: 2.13 a. (CD) 16 b. (7FFF) 16

14 Conversion of real numbers Binary to Octal, Octal to binary Binary to Hex, Hex to binary Octal to hex, hex to octal Procedure: (1) Convert integer part (2) Convert fraction part (3) Combine digits 14

15 Alternative method of conversion An alternative method for converting a small decimal integer (usually less than 256) to binary is to break the number as the sum of numbers that are equivalent to the binary place values shown: Working: = 37 => digit for 2 7 is = 5 => digit for 2 5 is = 1 => digit for 2 2 is 1 & digit for 2 0 is Fractions: reading for the student

16 Number of digits required to store integers (1) Given: b is the base k is the number of digits and integer is: S k-1 S k-2...s 2 S 1 S 0 What is the largest integer that can be represented using k digits in base b? Answer:??? What is the decimal equivalent of this number? Answer:??? Base b number of digits k Largest integer in base b Decimal equivalent Note: b k = = = = FFFF =

17 Number of digits required to store integers (2) Given: b is the base k is the number of digits and integer is: S k-1 S k-2...s 2 S 1 S 0 What is the largest integer that can be represented using k digits in base b? Answer: number with k digits where each digit S i is (b-1) What is the decimal equivalent of this number? Answer: b k - 1 Base b number of digits k Largest integer in base b Decimal equivalent Note: b k = = = = FFFF =

18 Example 2.23 HOMEWORK: Find the minimum number of binary digits required to store decimal integers with a maximum of six digits. Solution k = 6, b 1 = 10, and b 2 = 2. Then x = k (logb 1 / logb 2 ) = 6 (1 / ) = 20. The largest 6-digit decimal number is 999,999 and the largest 20-bit binary number is 1,048,575. Note that the largest number that can be represented by a 19-bit number is , which is smaller than 999,999. We definitely need twenty bits. 2.18