2. Computer_Organization_unit_ 1_win.ppt
- 1. Computer Organization Representation of Information
- 2. Number Systems • The organization of any computer depends considerably on how it represents numbers, characters, and control information. • The most basic unit of information in a digital computer is called a bit, which is a contraction of binary digit (0 / 1). – a bit is nothing more than a state of "on" or "off" (or "high" and "low") within a computer circuit. • A group of 8 bits as the basic unit of addressable computer storage - byte. • Computer words consist of two or more adjacent bytes that are sometimes addressed and almost always are manipulated collectively. The word size represents the data size that is handled most efficiently by a particular architecture. Words can be 16 bits, 32 bits, 64 bits, or any other size that makes sense within the context of a computer's organization (including sizes that are not multiples of eight). • Eight-bit bytes can be divided into two 4-bit halves called nibbles (or nybbles). Because each bit of a byte has a value within a positional numbering system, the nibble containing the least-valued binary digit is called the low-order nibble, and the other half the high-order nibble.
- 3. Number Systems • A number system of base, or radix is a system that uses distinct symbols for r digits. • To determine the quantity that number represents, it is necessary to multiply each digit by an integer power of r and then form sum of all the weighted digits. • This is often referred to as a weighted numbering system because each position is weighted by a power of the radix. • To distinguish among numbers in different radices, we use the radix as a subscript, such as in 3310 to represent the decimal number 33. • Example : Three numbers represented as powers of a radix. 243.5110 = 2 x 102 + 4 x 101 + 3 x 100 + 5 x 10-1 + 1 x 10-2 2123 = 2 x 32 + 1 x 31 + 2 x 30 = 2310 101102 = 1 x 24 + 0 x 23 + 1 x 22 + 1 x 21 + 0 x 20 = 2210 • The two most important radices in computer science are binary (base 2), and hexadecimal (base 16). Another radix of interest is octal (base 8). – The binary system uses only the digits 0 and 1; – the octal system, 0 through 7. – The hexadecimal system allows the digits 0 through 9 with A, B, C, D, E, and F being used to represent the numbers 10 through 15.
- 4. Binary Number System Binary number system has only two digits 0, 1 Also called base-2 system Counting in binary system 0, 1, 10, 11, 100, 101, 110, 111, 1000,…. Binary number in expanded notation (1011)2 = 1*2^3 + 0*2^2 + 1*2^1 + 1*2^0 = 1*8 + 0*4 + 1*2 + 1*1 = (11)10 Hexadecimal Hexadecimal number system has 16 digits • 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F • Also called base-16 system Counting in Hexadecimal 0,1,…,F,10,11,…,1F,20,…FF,100,… Hexadecimal number in expanded notation (FF)16 = 15*16^1 + 15*16^0 = (255)10
- 5. Why Binary? • Early computer design was decimal – Mark I and ENIAC • John von Neumann proposed binary data processing (1945) – Simplified computer design – Used for both instructions and data • Natural relationship between on/off switches and calculation using Boolean logic On Off True False Yes No 1 0
- 6. Why Binary? •A computer is a Binary machine •It knows only ones and zeroes •Easy to implement in electronic circuits •Reliable •Cheap Bit and Byte BIT = Binary digIT, “0” or “1” State of on or off ( high or low) of a computer circuit Kilo 1K = 2^10 = 1024 ≈ 10^3 Mega 1M = 2^20 = 1,048,576 ≈ 10^6 Giga 1G = 2^30 = 1,073,741,824 ≈ 10^9 Byte is the basic unit of addressable memory 1 Byte = 8 Bits The right-most bit is called the LSB - Least Significant Bit The Left-most bit is called the MSB - Most Significant Bit
- 7. Positional Notation: Binary Place 27 26 25 24 23 22 21 20 Value 128 64 32 16 8 4 2 1 Evaluate 1 x 128 1 x 64 0 x 32 1 x16 0 x 8 1 x 4 1 x 2 0 x 1 Sum for Base 10 128 64 0 16 0 4 2 0 1101 01102 = 21410
- 8. Positional Notation: Octal 6248 = 40410 Place 82 81 80 Value 64 8 1 Evaluate 6 x 64 2 x 8 4 x 1 Sum for Base 10 384 16 4 64’s place 8’s place 1’s place
- 9. Positional Notation: Hexadecimal 6,70416 = 26,37210 Place 163 162 161 160 Value 4,096 256 16 1 Evaluate 6 x 4,096 7 x 256 0 x 16 4 x 1 Sum for Base 10 24,576 1,792 0 4 4,096’s place 256’s place 1’s place 16’s place
- 10. Conversion Among Bases • The possibilities: Hexadecimal Decimal Octal Binary
- 11. Conversions • Binary to Decimal – The binary number system uses the radix or base 2. The two digit symbol used are 0 and 1 – As only two digits are used, the weights are power of 2. – 2^0 = 1 2^1 = 2 2^2=4 2^3=8 2^4=16 so on – The decimal equivalent of binary number equals the sum of all digits multiplied by their weights. • Eg (11001)2= (25)10 1*2^4 + 1*2^3+0*2^2+0*2^1+1*2^0 =16+8+0+0+1 = 25
- 12. Conversions • Decimal to Binary (base or radix is 10) – We begin with the base conversion of unsigned numbers. Conversion of signed numbers (numbers that can be positive or negative) is more complex, and it is important that you first understand the basic technique for conversion before continuing with signed numbers. – Conversion between base systems can be done by using either repeated subtraction or a division-remainder method. The subtraction method is cumbersome. – The division-remainder method is faster and easier than the repeated subtraction method. It employs the idea that successive divisions by the base. The remainders that we get when we sequentially divide by the base end up being the digits of the result, which are read from bottom to top.
- 13. Decimal to Binary (321)10 = ?2 quotient remainder 321 / 2 = 160 1 160 / 2 = 80 0 80 / 2 = 40 0 40 / 2 = 20 0 20 / 2 = 10 0 10 / 2 = 5 0 5 / 2 = 2 1 2 / 2 = 1 0 1 / 2 = 0 1 Reading the remainders from bottom to top, we have (321)10 = (101000001)2
- 14. • Converting Fractions – Fractions in any base system can be approximated in any other base system using negative powers of a radix. Radix points separate the integer part of a number from its fractional part. In the decimal system, the radix point is called a decimal point. Binary fractions have a binary point. – Fractions that contain repeating strings of digits to the right of the radix point in one base may not necessarily have a repeating sequence of digits in another base. – Because the remainder method works with positive powers of the radix for conversion of integers, it stands to reason that we would use multiplication to convert fractions. – Convert 0.3437510 to binary with 4 bits to the right of the binary point. – Reading from top to bottom, 0.3437510 = 0.01012 to four binary places.
- 15. Conversions Between Number Systems Decimal to Base r • Same as Decimal to Binary • Divide the number by r • Record the quotient and remainder • Divide the new quotient by r again • ….. • Repeat until the newest quotient is 0 • Read the remainder from bottom to top
- 16. Octal to Decimal Hexadecimal Decimal Octal Binary
- 17. Octal to Decimal • Technique – Multiply each bit by 8n , where n is the “weight” of the bit – The weight is the position of the bit, starting from 0 on the right – Add the results
- 18. Example 7248 => 4 x 80 = 4 2 x 81 = 16 7 x 82 = 448 __________ 46810
- 19. Octal to Binary Hexadecimal Decimal Octal Binary
- 20. Octal to Binary • Technique – Convert each octal digit to a 3-bit equivalent binary representation
- 21. Example 7058 = ?2 7 0 5 111 000 101 7058 = 1110001012
- 22. Decimal to Octal Hexadecimal Decimal Octal Binary
- 23. Decimal to Octal • Technique – Divide by 8 – Keep track of the remainder
- 24. Example 123410 = ?8 8 1234 154 2 8 19 2 8 2 3 8 0 2 123410 = 23228
- 25. Binary to Octal Hexadecimal Decimal Octal Binary
- 26. Binary to Octal • Technique – Group bits in threes, starting on right – Convert to octal digits
- 27. Example 10110101112 = ?8 1 011 010 111 1 3 2 7 10110101112 = 13278
- 28. Decimal to Hexadecimal Hexadecimal Decimal Octal Binary
- 29. Decimal to Hexadecimal • Technique – Divide by 16 – Keep track of the remainder
- 30. Example 123410 = ?16 123410 = 4D216 16 1234 77 2 16 4 13 = D 16 0 4
- 31. Hexadecimal to Decimal Hexadecimal Decimal Octal Binary
- 32. Hexadecimal to Decimal • Technique – Multiply each bit by 16n , where n is the “weight” of the bit – The weight is the position of the bit, starting from 0 on the right – Add the results
- 33. Example ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560 274810
- 34. Binary to Hexadecimal Hexadecimal Decimal Octal Binary
- 35. Binary to Hexadecimal • Technique – Group bits in fours, starting on right – Convert to hexadecimal digits
- 36. Example 10101110112 = ?16 10 1011 1011 2 B B 10101110112 = 2BB16
- 37. Hexadecimal to Binary Hexadecimal Decimal Octal Binary
- 38. Hexadecimal to Binary • Technique – Convert each hexadecimal digit to a 4-bit equivalent binary representation
- 39. Example 10AF16 = ?2 1 0 A F 0001 0000 1010 1111 10AF16 = 00010000101011112
- 40. Octal to Hexadecimal Hexadecimal Decimal Octal Binary
- 41. Octal to Hexadecimal • Technique – Use binary as an intermediary
- 42. Example 10768 = ?16 1 0 7 6 001 000 111 110 2 3 E 10768 = 23E16
- 43. Hexadecimal to Octal Hexadecimal Decimal Octal Binary
- 44. Hexadecimal to Octal • Technique – Use binary as an intermediary
- 45. Example 1F0C16 = ?8 1 F 0 C 0001 1111 0000 1100 1 7 4 1 4 1F0C16 = 174148
- 46. Quantities/Counting (1 of 3) Decimal Binary Octal Hexa- decimal 0 0 0 0 1 1 1 1 2 10 2 2 3 11 3 3 4 100 4 4 5 101 5 5 6 110 6 6 7 111 7 7 p. 33
- 47. Quantities/Counting (2 of 3) Decimal Binary Octal Hexa- decimal 8 1000 10 8 9 1001 11 9 10 1010 12 A 11 1011 13 B 12 1100 14 C 13 1101 15 D 14 1110 16 E 15 1111 17 F
- 48. Quantities/Counting (3 of 3) Decimal Binary Octal Hexa- decimal 16 10000 20 10 17 10001 21 11 18 10010 22 12 19 10011 23 13 20 10100 24 14 21 10101 25 15 22 10110 26 16 23 10111 27 17 Etc.
- 49. One Byte Equivalences • 00000001 1 1 1 • 00000010 2 2 2 • 00000100 4 4 4 • 00001000 10 8 8 • 00010000 20 16 10 • 00100000 40 32 20 • 01000000 100 64 40 • 10000000 200 128 80 Binary Octal Decimal Hex
- 50. One Byte Examples • Binary Decimal Hex • 00010001 17 11 • 00001010 10 0A • 10101010 170 AA • 11000000 192 C0 • 11000001 193 C1 • 11100000 224 E0 • 11110000 240 F0 • 11111111 255 FF
- 51. Questions Decimal to Binary • Convert 8 decimal to binary. • 00001000 • Convert 88 decimal to binary. • 01011000 • Convert 204 decimal to binary. • 11001100 • Convert 255 decimal to binary. • 11111111
- 52. Questions Binary to Decimal • Convert 00101100 binary to decimal. • 44 • Convert 10000000 binary to decimal. • 128 • Convert 00001000 binary to decimal. • 8 • Convert 01010101 binary to decimal. • 85
- 53. Questions Binary to Hex • Convert 00101100 binary to hex. • 2C • Convert 10000000 binary to hex. • 80 • Convert 00001000 binary to hex. • 8 • Convert 01010101 binary to hex. • 55
- 54. Questions Hex to Binary • Convert FF hex to binary. • 11111111 • Convert AC hex to binary. • 10101100 • Convert 22 hex to binary. • 00100010 • Convert 8 hex to binary. • 00001000
- 55. Questions Decimal to Hex • Convert 8 decimal to hex. • 8 • Convert 22 decimal to hex. • 16 • Convert 65 decimal to hex. • 41 • Convert 128 decimal to hex. • 80
- 56. Questions Hex to Decimal • Convert 22 hex to decimal. • 34 • Convert 34 hex to decimal. • 52 • Convert 6 hex to decimal. • 6 • Convert 82 hex to decimal. • 130