Information Science 1 - Representa*on of Data in Memory- Week 03 College of Information Science and Engineering Ritsumeikan University
Topics covered l Basic terms and concepts of The Structure of a Computer l Positional numbering systems decimal binary octal hexadecimal l Conversion among different bases - to decimal and from decimal - other conversions l Test 2
Recall Week 02 l Digital system l Data, Binary l Memory, RAM l Bit, Byte, Computer Word, Address l CPU l CU, ALU, MAR, MDR, IR, GPR, PC, PSW l Machine Cycle, Fetch, Execute, Automatic Sequence Control l Input, Output l Computer Bus 3
Objectives of this class l To understand the fundamentals of numerical data representation and manipulation in computers l To master the skill of representing decimal numbers in the binary, octal, and hexadecimal systems l To be able to speak aloud a number in any of the four bases l To be able to convert from decimal to the above three numbering systems 4
Recall: Computer as a digital system l l A bit is the most basic (and, hence, the smallest) unit of information in a computer - It is a state of on or off in a digital circuit - Sometimes these states are also called as high or low (voltage) A byte is a group of eight bits - A byte is the smallest (in principle) addressable unit of computer storage - Addressable means that a particular byte can be retrieved according to its location 5
Computer words and nibbles l A word is a contiguous group of bytes l - Words can be any number of bytes or bits (word sizes of 32 or 64 bits are now most common) - In a word-addressable system, a word is the smallest addressable unit of memory storage A group of four bits is called a nibble (or nybble) - A byte, therefore, consists of two nibbles: a high-order nibble, and a low-order nibble 00111100010110001010011000110111101 6
Positional numbering system l A positional numbering system (or positional notation system) is a numeral system in which each digit is related to the next by a constant multiplier called the base or radix of that system The value of each digit position is, therefore, the value of its digit multiplied by a power of the base, where the power is determined by the digit's position counted from the separator (which is usually a dot. or comma, ) The value of a number is then calculated as the sum of the values of all positions 7
Decimal system l Decimal numbers have radix (base) = 10 (in Latin, decima means a tenth part ) l Symbols used: l Position weights: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 10 3 10 2 10 1 10 0. 10-1 10-2 10-3 1000 100 10 1 1/10 1/100 1/1000 For example, the decimal number 36.250: 36.250 = 3*10 + 6*1+ 2*1/10 + 5* 1/100 +0* 1/1000 8
Binary system l Binary numbers have radix = 2 (in Latin, bini means two together ) l Symbols used: 0 and 1 l The radix ( 10) is denoted by a subscript (2 l Position weights: 2 3 2 2 2 1 2 0. 2-1 2-2 2-3 8 4 2 1 1/2 1/4 1/8 For example, the binary equivalent of 36.250: 36.250 = 1*32 + 1*4 + 1/4 = 100100.01 (2 9
10 Octal system l Octal numbers have radix = 8 (in Latin, octo means eight) l Symbols used: 0, 1, 2, 3, 4, 5, 6, and 7 l Convenient when bits are grouped in triplets l Position weights: 8 3 8 2 8 1 8 0. 8-1 8-2 2-3 512 64 8 1 1/8 1/64 1/512 For example, the octal equivalent of 36.250: 36.250 = 4*8 + 4*1 + 2*1/8 = 44.2 (8
11 Hexadecimal system l Hexadecimal numbers have radix = 16 (in Greek, hexa means six) l Symbols used: 0,, 9, A, B, C, D, E, and F l Convenient to represent nibbles l Position weights: 16 3 16 2 16 1 16 0. 16-1 16-2 16-3 4096 256 16 1 1/16 1/256 1/4096 For example, the hexadecimal equivalent of 36.250: 36.250 = 2*16 + 4*1 + 4*1/16 = 24.4 (16
12 Radix r numbers l For any radix r number represented with n +m+1 digits as Number (r = a n... a 1 a 0. a -1... a -m its decimal equivalent is calculated as follows: Number (10 = a n r n +... + a 1 r 1 + a 0 + a -1 r -1 +... + a -m r m n = a r i i = m i
Example: ASCII 13
Example: RGB color codes 14
Radix conversion l We can already perform three conversions (from radix r to decimal): Decimal Octal Hexadecimal Binary l There are nine other possible conversions (from decimal, and between radix 2, 8, 16) 15
16 Converting from decimal to l The algorithm: another radix Left of the separator (the decimal point): repeatedly divide the integer part by the radix and write the remainders (R) from right to left Right of the separator: repeatedly multiply the fractional part by the radix and write the integer portion (I) of the result left to right
17 Example: decimal to binary 22.8125 =? (2 22.8125 = 10110 (2. 11 R 0 5 R 1 2 R 1 1 R 0 0 R 1 22.8125 =.1101 (2 1.625 I 1 1.25 I 1 0.5 I 0 1.0 I 1 0 I 0 22.8125 = 10110.1101 (2
18 More examples 1234 =? (8 1234 =? (16 1234 = 2322 (8 154 R 2 19 R 2 2 R 3 0 R 2 1234 = 4D2 (16 77 R 2 4 R 13 = D 0 R 4
19 The remaining conversions l To convert between octal and hexadecimal to/from binary, memorize and use the conversion table (next slide): Octal: use 3 bits for each digit Hexadecimal: use 4 bits for each digit l To convert between octal and hexadecimal, convert to and then from binary 0110100110.1 (2 =? (8 =? (16 000 110 100 110.100 0001 1010 0110.1000 6 4 6.4 1 A 6.8 0110100110.1 (2 = 646.4 (8 = 1A6.8 (16
Conversion table Decimal Base 10 Binary Base 2 Octal Base 8 Hexadecimal Base 16 0 (10 0000 (2 0 (8 0 (16 1 (10 0001 (2 1 (8 1 (16 2 (10 0010 (2 2 (8 2 (16 3 (10 0011 (2 3 (8 3 (16 4 (10 0100 (2 4 (8 4 (16 5 (10 0101 (2 5 (8 5 (16 6 (10 0110 (2 6 (8 6 (16 7 (10 0111 (2 7 (8 7 (16 8 (10 1000 (2 10 (8 8 (16 9 (10 1001 (2 11 (8 9 (16 10 (10 1010 (2 12 (8 A (16 11 (10 1011 (2 13 (8 B (16 12 (10 1100 (2 14 (8 C (16 13 (10 1101 (2 15 (8 D (16 14 (10 1110 (2 16 (8 E (16 15 (10 1111 (2 17 (8 F (16 20
21 Key points of this lecture l Because binary numbers are the basis for all data representation in digital systems, it is important that you become proficient with the binary system to understand the operation of all computer components as well as the design of computer architectures l The binary system is the most important positional numbering system for computers
22 Key points (continued) l It is, however, difficult to read long strings of bits, and even a modestly-sized decimal number becomes a very long binary number For example, 1359510 = 11010100011011 (2 For compactness and ease of reading, binary values are usually expressed using the octal or hexadecimal system l To convert among the different systems, use the conversion algorithms and, when appropriate, the conversion table
23 l Read these slides again Homework l Read slides for the next lecture and do the self-preparation assignments l Learn the vocabulary l Consult, when necessary, the textbook
24 Next class l Representation of data (2) - Basic digital arithmetic - Other representation systems
25 Test 01 l l l l l Put away all electronic devices. Make sure your name and student ID are written at the top of the page. Begin immediately when you receive the test paper. Test papers will be collected for grading. Do not talk until all papers have been collected at the end of the test.