BSC & BIT Numbering Systems. ITU Lecture 3b

Transcription

1 BSC & BIT Numbering Systems ITU Lecture 3b

2 Introduction We use a number to present a quantity (value) of any thing that can be quantified. Quantities are measured, monitored, recorded, or manipulated arithmetically. Two ways of representing a numerical value of quantities are: Analog and Digital Analog Representation: Analog Representation: One quantity is represented by another which is direct proportional to the first. Example: Automobile speedometer, which is reflection of the needle is proportional to the speed of the auto. Example: Room thermostat, in which the bending of the bimetallic strip is proportional to the room temperature. As the temperature of the room changes

3 Thermostat Speedometer

4 Note: Analogy values gradually over a continuous range of values Digital Representation: Quantities are represented in digits. Example digital clock, which provides the time of a day in the form of decimal digits which represent hours, minutes and seconds. Though the day change continuously, the digital clock does change continuously, instead it changes in steps of one per second (discrete steps)

5 Numbering Systems Computers use four numbering systems: Decimal, Binary, Octal and Hexadecimal. Each has advantages for different levels of digital processing. A number system defines how a number can be represented using distinct symbols. All number systems are positional, meaning that position of a symbol in relation to other symbols determine its value. Within a number, each symbol is called a digit (Decimal digit, binary digit, Octal digit, Hexadecimal digit).

6 Within a number, digits are arranged in a order of ascending values, moving from lowest value on the right to the Highest in the left. The left most digit is referred as Most Significant Digit (MSD) and the right most as the Least Significant Digit (LSD) A same quantity or value can be represented in different systems. For example, the two numbers (2A) 16 and (52) 8 both refer to the same quantity, (42) 10.

7 Digital Number Systems Many number systems are in use in Digital Technology. Common number system are Decimal, Binary, Octal and Hexadecimal Decimal system is the tool that we use every day quantifiable transactions.

8 The Decimal Number System (Base 10) The word decimal is derived from the Latin root deci (ten). Base b = 10. Ten symbols: S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} The symbols in this system are often referred to as decimal digits or just digits. Decimal number has evolved naturally as a result of a fact that a human being has 10 fingers. The word digit is the Latin word for Finger

9 In decimal system, each weight equal to 10 raised to the power of its position. The weight of the first position in decimal number system is 100, which is equal to 1. Each digit corresponds to a power of 10 based on its position in the number The powers of 10 increment from 0, 1, 2, etc. as you move right to left Example Weight & Value 5,489 = 5 * * * * 10 0

10 The Decimal Number System (Base 10) Integer values

11 The Decimal number system (Base 10) Real Values

12 The Binary Number System (Base 2) The word binary is derived from the Latin root bi (double). Base b = 2. Two symbols: S = {0, 1} The symbols in this system are often referred to as binary digits or just bits.

13 Weight and Value The binary system is also a weighted system. Each digit has a weight based on its position in the number. Weight in the binary system is 2 raised to the power represented by the position. The value of a specific digit is equal to its face value times the weight of its position.

14 The Binary Numbering System (Base 2) Integer Value

15 The Binary Number System (Base 2) Real Values

16 The Octal Number System (Base 8) The word Octal is derived from the Latin root Octo (eight). Base b = 8. Ten symbols: S = {0, 1, 2, 3, 4, 5, 6, 7} Example of Octal number

17 Weight and Value The Octal System is the weighted system. Each digit has a weight based on its position in the number. Weight in Octal is eight raised to the power presented by the position. The value represented by each weight is given in decimal terms. The value of a specific digit is equal to its face value times the weight of its position.

18 The Octal Number System (Base 8)

19 The Hexadecimal Number System (Base 16) The word hexadecimal is derived from the Greek root hex (six) and Latin root decem (ten). Base b = 16. Ten symbols: S = {0, 1,, 8, 9, A, B, C, D, E, F} The symbols in this system are often referred to as hexadecimal digits. Example of hexadecimal number 7DF59A

20 Weight and Value The Hexadecimal System is the weighted system. Each digit has a weight based on its position in the number. Weight in Octal is sixteen raised to the power presented by the position. The value represented by each weight is given in decimal terms. The value of a specific digit is equal to its face value times the weight of its position.

21 The Hexadecimal Number System (Base 16) Integer Values

22 Summary System Base Symbol Examples Decimal 10 {0,1,2,3,4,5,6,7,8,9} Binary 2 {0,1} Octal 8 {0,1,2,3,4,5,6,7} Hexadecimal 16 {0,1,2,3,4,5,6,7,8,9, A, B,C,D, E, F} A2C.A1

23 Summary of the Four Positional Number Systems

24 Significant Digit Binary: Most significant digit -MSD Least significant digit -LSD Decimal: Most significant digit -MSD Least significant digit-lsd

25 Positional Value System(Weighting Value) In a positional number system, the position a symbol occupies in the number determines the value it represents. In this system, a number represented as: As the value of : Note: Any number is the sum of the products of each digit value times its positional value

26 Conversion between number systems 1. Binary/Hex/Octal Decimal. 2. Decimal Binary/Hex/Octal. 3. Binary Hex/Octal

27 Conversion between number systems Binary/Hex/Octal Decimal.

28 Decimal Counting In counting decimal numbers, we start with 0 in the unit position and take each symbol (digit) in progression until we reach 9. In decimal counting, the unit position (LSD) changes upwards with each step in the count. The tens position changes 10 steps in the count. The hundreds position changes 100 steps in the count and so on. Decimal Places (Digits): With 2 decimal places (two character per number) we can count through 10 2 = 100 different numbers (i.e. 0 99). With 3 places we can count through 10 3 = 1000 (0-999) General : with N place you can count 10 N and the largest number will be 10 N -1

29 Binary Counting Binary counting is based on the number of bits. This can be demonstrated with 3 bit binary number To be explained in class Binary Places (Digits) To be explained in class

30 Octal Counting The largest octal digit is 7. We count from 0 to 7. Once it reaches 7, it recycles to 0 on the next count and cause the next high digit to be incremented. For example: 65,66, 67, 70, , 276, 277, 300 Octal Places (Digits) With N octal digits, we can count from 0 up to 8 N 1, for total of 8 N different counts. Example: With 3 octal digits we can count from to Which is a total of 8 3 = 512 different octal numbers.

31 Counting in Hexadecimal Counting in hexadecimal is such that each digit can be incremented by one (from 0 to F). Once a digit position reaches the value of F, it resets to zero and the next digit position is incremented. Example 38, 39, 3A, 3B, 3C, 3D, 3E, 3F, 40, 41, 42 6F8, 6F9, 6FA, 6FB, 6FC, 6FD, 6FE, 6FF, 700 Hexadecimal Places (Digits) Any Hexadecimal number with N digits, we can count from 0 to 16 N 1, for total of 16 N different values Example. Hexadecimal number with 3 digits, we can count from 000 to FFF, which 4096 = 16 3 to total values

32 Conversion between number systems (110.11) 2 = (6.75) X 10

33 Example Octal Decimal X X 10

34 Example Hexadecimal Decimal B9CF 16 X 10 FE64.3F 16 X 10

35 General approach for converting from Decimal Other number systems (Binary, Octal & Hexadecimal)

36 Decimal Binary X X 2

37 Decimal Octal X X 8

38 Decimal Hexadecimal X X 16

39 Binary Octal X 8

40 Binary Hexadecimal X 16