DOCSLIB.ORG

Bits and Bytes

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Bits and Bytes Bits and Bytes Computers are used to store and process data. Processed data is called information. You are used to see or hear information processed with the help of a computer: a paper you’ve just typed using Microsoft World, a digital photograph, a MP3 song you’ve downloaded on the Internet, etc. But how is the data stored in the computer? Whatever the medium in which it is stored, the most fundamental piece of data is always represented by a “two-value system”: low or high voltage, positive or negative polarity of a magnetic medium, absence or presence of “bumps” on a CD or DVD… These two values are logically represented by the digits 0 and 1. This is called a binary representation. Usually, the digit 0 is interpreted as the low value or absence of some kind of signal (such as no bump) and the digit 1 is interpreted as the high value of presence of some kind of signal. In other word, we encode the physical signal with the logical binary representation. This 0 or 1 digit is called a bit (short for binary digit). Obviously, just one bit cannot give much information since there are only two possible values. Hence, to store more useful data we must combine bits. However, even combining several (even many) bits will only give a finite numbers of possible objects. For example combining four bits gives the sequences 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111. These sixteen sequences can be selected to represent any sixteen objects of our choice but only sixteen objects. So if we want to represent integers, we can decide they represent the integers 0 to 15; if we represent letters they could represent the letters a, b, c, d, e, f, g, h, i, j, k, l, m, and n (or any other combination of sixteen letters of our choice); if we represent colors they could represent 16 predefined colors of our choice. We say that the data stored on a computer, and represented logically by combinations of bits, is discrete: it is finite and made of separable units. Analog versus digital Many different kinds of data can be represented in a computer: • numbers: whole numbers, positive and negative integers, real numbers; • text: made of letters, characters, digits; • audio: speech, music, sound tracks; • images and graphics; • video, etc. Some of these data types, such as text or integers on a bounded range, are discrete and it will be rather straightforward to convert these data to a binary representation that can be stored in the computer. But most data is analog, which means that it is continuous in nature. Let’s look at some examples. Pictures Continuous in space Infinite number of colors Sound Continuous in time Infinite number of frequencies 1 Real numbers Continuous even if bounded: always a number between two given numbers Because most data is analog, it must be broken in discrete pieces and those pieces must be represented by a binary representation to be stored in the computer. When this process is performed, we say that we digitize the data. Note that because digitizing data transforms continuous infinite data into finite discrete data, some data will be lost. For example we cannot really represent all the colors of the color spectrum, we cannot represent all the real numbers between 0 and 1. In this unit of the course, we will study how some types of data are represented and discuss some of the issues that arise from the conversion from analog to digital data. We will also use some tools to manipulate data. Binary Numbers Because the representation of more complex data such as sound or color is based on the representation of quantities (i.e. numbers), we must start our study with the representation of whole numbers. As seen above the sixteen first whole numbers (0 to 15) can be represented by all possible four-bit sequences. We will generalize this to longer bit sequences, but we will show how to derive a meaningful way to convert from a decimal number (the “regular” number) to a binary number and vice versa. The number system we use is the number system in base 10. Base 10 number system We use 10 digits, 0 to 9. In a number, the value of each digit depends of its position in the number. Each position corresponds to a value that is a power of 10. Table 1 shows the decomposition of the number 2324 in the number system of base 10 (also called decimal number system). 2 3 2 4 2⋅103 3⋅102 2⋅101 4⋅100 thousands hundreds tens units Table 1 This can also be written as . Base 2 number system The binary number system is the number system in base 2. We use 2 digits, 0 and 1, and each position in a number corresponds to a power of 2. Table 2 shows the decomposition of the binary number 1110 in the number system of base 2. 1 1 1 0 1⋅23 1⋅22 1⋅21 0⋅20 eights fours twos units Table 2 Therefore the value of the binary number is . When necessary, to avoid confusion we will append the subscript 2 to binary numbers. So the equality means that the number with binary representation 1110 is the same as the number with decimal representation 14 (we obviously cannot write 1110 = 14!). 2 We can check that the 16 sequences given in page 1 do match the decimal numbers 0 to 15 according to our definition. Binary Number Decomposition in sum of powers of 2’s Decimal Value 0000 0 0001 1 0010 2 0011 3 0100 4 0101 5 0110 6 0111 7 1000 8 1001 9 1010 10 1011 11 1100 12 1101 13 1110 14 1111 15 Table 3 We can notice the pattern: , , , . In general, , where n is the number of zero following the 1. This is, of course, consistent with the definition of the binary number: there are n + 1 digits; each digit in this binary number corresponds to a power of 2, from for the left-most digit to for the right-most digit; all powers of 2’s except the left most ( ) are multiplied by 0. Recreation: Magic cards1 A magician has four magic cards; each card holds eight numbers between 1 and 15. 1 3 5 2 3 6 4 5 6 8 9 10 7 9 11 7 10 11 7 12 13 11 12 13 13 15 14 15 14 15 14 15 Blue Green Red Yellow She instructs each member of the audience to choose a number between 1 and 15 and hands out the cards to one person at random. The selected person returns only the cards that contain the number she/he had chosen (and kept secret). The magician guesses the number chosen. Suppose for example the person hands out the blue, green and yellow cards. Then the magician says “Eleven”! And she is right! How does this work? The cards are based on the binary representation of the numbers 1 to 15. The blue card contains all the numbers that have a 1 as right-most bit (i.e., the bit that corresponds to the units). The green card 1 Adapted from Aha! by Martin Gardner (1) 3 contains all the numbers that have a 1 as second digit from the right (i.e., the bit that corresponds to 21). The red card contains all the numbers that have a 1 as the second digit from the left (i.e., the bit that corresponds to 22 = 4). And finally the yellow card contains all the numbers that have a 1 as left-most digit (i.e., the bit that corresponds to 23 = 8). So, in our example, when the magician is given the blue, green and yellow cards, he knows that the number is 1011 in binary and therefore 8 + 2 + 1 = 11 in decimal. Note that the magician does not have to know binary numbers; she just needs a clever aide who made the cards for her! She just has to add the number that is at the top left corner of the cards that the member of the audience gave her. Exercises: all about patterns… 1. Table 3 shows clearly that the larger the number the more bits are needed to express a number in its binary representation. For example, 6 can be expressed with only 3 bits: 1102 but 12 cannot since 12 is 11002. It is permissible to add or remove leading zeros (i.e., zeros with no 1’s to their left), but it is not permissible to remove any other digits from the binary representation without altering its value. This is similar as 01060 being the same as 1060 in the decimal number system, but certainly different from 0106 or 0160. With one bit we can express just two numbers 0 and 1. a) Observe table 3. How many numbers can be represented with 2 bits? with 3 bits? with 4 bits? b) You should get a pattern in question a). From it, can you deduce the number of numbers that can be represented with 5 bits? with 8 bits? with n bits? 2. Observe table 3. Explain how you can distinguish between even and odd numbers if you are given just the binary representation. 3. If you observe the Binary Number column of table 3 you can imagine that this column is split into 4 columns: the column of the first digit (corresponds to the eights), the columns of the second digit (column of the fours), etc.
Load more

Recommended publications
  • TI 83/84: Scientific Notation on Your Calculator
    TI 83/84: Scientific Notation on your calculator this is above the comma, next to the square root! choose the proper MODE : Normal or Sci entering numbers: 2.39 x 106 on calculator: 2.39 2nd EE 6 ENTER reading numbers: 2.39E6 on the calculator means for us. • When you're in Normal mode, the calculator will write regular numbers unless they get too big or too small, when it will switch to scientific notation. • In Sci mode, the calculator displays every answer as scientific notation. • In both modes, you can type in numbers in scientific notation or as regular numbers. Humans should never, ever, ever write scientific notation using the calculator’s E notation! Try these problems. Answer in scientific notation, and round decimals to two places. 2.39 x 1016 (5) 2.39 x 109+ 4.7 x 10 10 (4) 4.7 x 10−3 3.01 103 1.07 10 0 2.39 10 5 4.94 10 10 5.09 1018 − 3.76 10−− 1 2.39 10 5 7.93 10 8 Remember to change your MODE back to Normal when you're done. Using the STORE key: Let's say that you want to store a number so that you can use it later, or that you want to store answers for several different variables, and then use them together in one problem. Here's how: Enter the number, then press STO (above the ON key), then press ALPHA and the letter you want. (The letters are in alphabetical order above the other keys.) Then press ENTER.
    [Show full text]
  • Primitive Number Types
    Primitive number types Values of each of the primitive number types Java has these 4 primitive integral types: byte: A value occupies 1 byte (8 bits). The range of values is -2^7..2^7-1, or -128..127 short: A value occupies 2 bytes (16 bits). The range of values is -2^15..2^15-1 int: A value occupies 4 bytes (32 bits). The range of values is -2^31..2^31-1 long: A value occupies 8 bytes (64 bits). The range of values is -2^63..2^63-1 and two “floating-point” types, whose values are approximations to the real numbers: float: A value occupies 4 bytes (32 bits). double: A value occupies 8 bytes (64 bits). Values of the integral types are maintained in two’s complement notation (see the dictionary entry for two’s complement notation). A discussion of floating-point values is outside the scope of this website, except to say that some bits are used for the mantissa and some for the exponent and that infinity and NaN (not a number) are both floating-point values. We don’t discuss this further. Generally, one uses mainly types int and double. But if you are declaring a large array and you know that the values fit in a byte, you can save ¾ of the space using a byte array instead of an int array, e.g. byte[] b= new byte[1000]; Operations on the primitive integer types Types byte and short have no operations. Instead, operations on their values int and long operations are treated as if the values were of type int.
    [Show full text]
  • C++ Data Types
    Software Design & Programming I Starting Out with C++ (From Control Structures through Objects) 7th Edition Written by: Tony Gaddis Pearson - Addison Wesley ISBN: 13-978-0-132-57625-3 Chapter 2 (Part II) Introduction to C++ The char Data Type (Sample Program) Character and String Constants The char Data Type Program 2-12 assigns character constants to the variable letter. Anytime a program works with a character, it internally works with the code used to represent that character, so this program is still assigning the values 65 and 66 to letter. Character constants can only hold a single character. To store a series of characters in a constant we need a string constant. In the following example, 'H' is a character constant and "Hello" is a string constant. Notice that a character constant is enclosed in single quotation marks whereas a string constant is enclosed in double quotation marks. cout << ‘H’ << endl; cout << “Hello” << endl; The char Data Type Strings, which allow a series of characters to be stored in consecutive memory locations, can be virtually any length. This means that there must be some way for the program to know how long the string is. In C++ this is done by appending an extra byte to the end of string constants. In this last byte, the number 0 is stored. It is called the null terminator or null character and marks the end of the string. Don’t confuse the null terminator with the character '0'. If you look at Appendix A you will see that the character '0' has ASCII code 48, whereas the null terminator has ASCII code 0.
    [Show full text]
  • Floating Point Numbers
    Floating Point Numbers CS031 September 12, 2011 Motivation We’ve seen how unsigned and signed integers are represented by a computer. We’d like to represent decimal numbers like 3.7510 as well. By learning how these numbers are represented in hardware, we can understand and avoid pitfalls of using them in our code. Fixed-Point Binary Representation Fractional numbers are represented in binary much like integers are, but negative exponents and a decimal point are used. 3.7510 = 2+1+0.5+0.25 1 0 -1 -2 = 2 +2 +2 +2 = 11.112 Not all numbers have a finite representation: 0.110 = 0.0625+0.03125+0.0078125+… -4 -5 -8 -9 = 2 +2 +2 +2 +… = 0.00011001100110011… 2 Computer Representation Goals Fixed-point representation assumes no space limitations, so it’s infeasible here. Assume we have 32 bits per number. We want to represent as many decimal numbers as we can, don’t want to sacrifice range. Adding two numbers should be as similar to adding signed integers as possible. Comparing two numbers should be straightforward and intuitive in this representation. The Challenges How many distinct numbers can we represent with 32 bits? Answer: 232 We must decide which numbers to represent. Suppose we want to represent both 3.7510 = 11.112 and 7.510 = 111.12. How will the computer distinguish between them in our representation? Excursus – Scientific Notation 913.8 = 91.38 x 101 = 9.138 x 102 = 0.9138 x 103 We call the final 3 representations scientific notation. It’s standard to use the format 9.138 x 102 (exactly one non-zero digit before the decimal point) for a unique representation.
    [Show full text]
  • Floating Point Numbers and Arithmetic
    Overview Floating Point Numbers & • Floating Point Numbers Arithmetic • Motivation: Decimal Scientific Notation – Binary Scientific Notation • Floating Point Representation inside computer (binary) – Greater range, precision • Decimal to Floating Point conversion, and vice versa • Big Idea: Type is not associated with data • MIPS floating point instructions, registers CS 160 Ward 1 CS 160 Ward 2 Review of Numbers Other Numbers •What about other numbers? • Computers are made to deal with –Very large numbers? (seconds/century) numbers 9 3,155,760,00010 (3.1557610 x 10 ) • What can we represent in N bits? –Very small numbers? (atomic diameter) -8 – Unsigned integers: 0.0000000110 (1.010 x 10 ) 0to2N -1 –Rationals (repeating pattern) 2/3 (0.666666666. .) – Signed Integers (Two’s Complement) (N-1) (N-1) –Irrationals -2 to 2 -1 21/2 (1.414213562373. .) –Transcendentals e (2.718...), π (3.141...) •All represented in scientific notation CS 160 Ward 3 CS 160 Ward 4 Scientific Notation Review Scientific Notation for Binary Numbers mantissa exponent Mantissa exponent 23 -1 6.02 x 10 1.0two x 2 decimal point radix (base) “binary point” radix (base) •Computer arithmetic that supports it called •Normalized form: no leadings 0s floating point, because it represents (exactly one digit to left of decimal point) numbers where binary point is not fixed, as it is for integers •Alternatives to representing 1/1,000,000,000 –Declare such variable in C as float –Normalized: 1.0 x 10-9 –Not normalized: 0.1 x 10-8, 10.0 x 10-10 CS 160 Ward 5 CS 160 Ward 6 Floating
    [Show full text]
  • Princeton University COS 217: Introduction to Programming Systems C Primitive Data Types
    Princeton University COS 217: Introduction to Programming Systems C Primitive Data Types Type: int Description: A (positive or negative) integer. Size: System dependent. On CourseLab with gcc217: 4 bytes. Example Variable Declarations: int iFirst; signed int iSecond; Example Literals (assuming size is 4 bytes): C Literal Binary Representation Note 123 00000000 00000000 00000000 01111011 decimal form -123 11111111 11111111 11111111 10000101 negative form 0173 00000000 00000000 00000000 01111011 octal form 0x7B 00000000 00000000 00000000 01111011 hexadecimal form 2147483647 01111111 11111111 11111111 11111111 largest -2147483648 10000000 00000000 00000000 00000000 smallest Type: unsigned int Description: A non-negative integer. Size: System dependent. sizeof(unsigned int) == sizeof(int). On CourseLab with gcc217: 4 bytes. Example Variable Declaration: unsigned int uiFirst; unsigned uiSecond; Example Literals (assuming size is 4 bytes): C Literal Binary Representation Note 123U 00000000 00000000 00000000 01111011 decimal form 0173U 00000000 00000000 00000000 01111011 octal form 0x7BU 00000000 00000000 00000000 01111011 hexadecimal form 4294967295U 11111111 11111111 11111111 11111111 largest 0U 00000000 00000000 00000000 00000000 smallest Type: long Description: A (positive or negative) integer. Size: System dependent. sizeof(long) >= sizeof(int). On CourseLab with gcc217: 8 bytes. Example Variable Declarations: long lFirst; long int iSecond; signed long lThird; signed long int lFourth; Page 1 of 5 Example Literals (assuming size is 8 bytes):
    [Show full text]
  • 36 Scientific Notation
    36 Scientific Notation Scientific notation is used to express extremely large or extremely small numbers efficiently. The main idea is to write the number you want as a number between 1 and 10 multiplied by an integer power of 10. Fill in the blanks in the table below. Write the decimal form of the number given in scientific notation. Then give the calculator’s notation for the scientific notation. This is the only time you should write this calculator notation. Calculator notation is not acceptable in your written work; use a human language! The last column is for a word name for the number. In a few rows, you will write the scientific notation yourself. Scientific Decimal Notation Calculator Word name for the notation notation number One billion one billion dollars dollars 1.0 ×10 9 dollars One year 3.16 ×10 7 seconds Distance from earth to moon 3.8 ×10 8 meters 6.24 ×10 23 Mass of Mars 624 sextillion kg kilograms 1.0 ×10 -9 Nanosecond seconds 8 Speed of tv 3.0 ×10 signals meters/second -27 -27 Atomic mass 1.66 ×10 Just say, “1.66 ×10 unit kilograms kg”! × 12 Light year 6 10 miles 15 Light year 9.5 ×10 meters One billion light years meters 16 Parsec 3.1 ×10 meters 6 Megaparsec 10 parsecs Megabyte = 220 = 1,048,576 20 2 bytes bytes bytes Micron One millionth of a meter The number of About one hundred stars in the billion Milky Way -14 Diameter of a 7.5 ×10 carbon-12 meters atom Rosalie A.
    [Show full text]
  • Floating Point CSE410, Winter 2017
    L07: Floating Point CSE410, Winter 2017 Floating Point CSE 410 Winter 2017 Instructor: Teaching Assistants: Justin Hsia Kathryn Chan, Kevin Bi, Ryan Wong, Waylon Huang, Xinyu Sui L07: Floating Point CSE410, Winter 2017 Administrivia Lab 1 due next Thursday (1/26) Homework 2 released today, due 1/31 Lab 0 scores available on Canvas 2 L07: Floating Point CSE410, Winter 2017 Unsigned Multiplication in C u ••• Operands: * bits v ••• True Product: u · v ••• ••• bits Discard bits: UMultw(u , v) ••• bits Standard Multiplication Function . Ignores high order bits Implements Modular Arithmetic w . UMultw(u , v)= u ∙ v mod 2 3 L07: Floating Point CSE410, Winter 2017 Multiplication with shift and add Operation u<<k gives u*2k . Both signed and unsigned u ••• Operands: bits k * 2k 0 ••• 0 1 0 ••• 0 0 True Product: bits u · 2k ••• 0 ••• 0 0 k Discard bits: bits UMultw(u , 2 ) ••• 0 ••• 0 0 k TMultw(u , 2 ) Examples: . u<<3 == u * 8 . u<<5 - u<<3 == u * 24 . Most machines shift and add faster than multiply • Compiler generates this code automatically 4 L07: Floating Point CSE410, Winter 2017 Number Representation Revisited What can we represent in one word? . Signed and Unsigned Integers . Characters (ASCII) . Addresses How do we encode the following: . Real numbers (e.g. 3.14159) . Very large numbers (e.g. 6.02×1023) Floating . Very small numbers (e.g. 6.626×10‐34) Point . Special numbers (e.g. ∞, NaN) 5 L07: Floating Point CSE410, Winter 2017 Floating Point Topics Fractional binary numbers IEEE floating‐point standard Floating‐point operations and rounding Floating‐point in C There are many more details that we won’t cover .
    [Show full text]
  • Fortran Reference Guide
    FORTRAN REFERENCE GUIDE Version 2017 TABLE OF CONTENTS Preface............................................................................................................xiv Audience Description.........................................................................................xiv Compatibility and Conformance to Standards........................................................... xiv Organization.................................................................................................... xv Hardware and Software Constraints.......................................................................xvi Conventions.................................................................................................... xvi Related Publications.........................................................................................xvii Chapter 1. Language Overview............................................................................... 1 1.1. Elements of a Fortran Program Unit.................................................................. 1 1.1.1. Fortran Statements................................................................................. 1 1.1.2. Free and Fixed Source............................................................................. 2 1.1.3. Statement Ordering................................................................................. 2 1.2. The Fortran Character Set.............................................................................. 3 1.3. Free Form Formatting..................................................................................
    [Show full text]
  • The Building Blocks: Data Types, Literals, and Variables
    Quigley.book Page 31 Thursday, May 22, 2003 3:19 PM chapter 3 The Building Blocks: Data Types, Literals, and Variables 3.1 Data Types A program can do many things, including calculations, sorting names, preparing phone lists, displaying images, validating forms, ad infinitum. But in order to do anything, the program works with the data that is given to it. Data types specify what kind of data, such as numbers and characters, can be stored and manipulated within a program. Java- Script supports a number of fundamental data types. These types can be broken down into two categories, primitive data types and composite data types. 3.1.1 Primitive Data Types Primitive data types are the simplest building blocks of a program. They are types that can be assigned a single literal value such as the number 5.7, or a string of characters such as "hello". JavaScript supports three core or basic data types: • numeric • string • Boolean In addition to the three core data types, there are two other special types that consist of a single value: • null • undefined Numeric Literals. JavaScript supports both integers and floating-point numbers. Integers are whole numbers and do not contain a decimal point; e.g., 123 and –6. Inte- gers can be expressed in decimal (base 10), octal (base 8), and hexadecimal (base 16), and are either positive or negative values. See Example 3.1. 31 Quigley.book Page 32 Thursday, May 22, 2003 3:19 PM 32 Chapter 3 • The Building Blocks: Data Types, Literals, and Variables Floating-point numbers are fractional numbers such as 123.56 or –2.5.
    [Show full text]
  • Scientific Notation TEACHER NOTES
    Scientific Notation TEACHER NOTES Activity Overview This activity gives students an opportunity to see where large and small numbers are used and how scientific notation offers a convenient method of writing such numbers. Topic: Numbers and Operations Understand numbers, ways of representing numbers, relationships among numbers, and number systems Develop an understanding of large numbers and recognize and This activity utilizes MathPrintTM appropriately use exponential, scientific, and calculator notation functionality and includes screen captures taken from the TI-84 Teacher Preparation and Notes Plus C Silver Edition. It is also To download the data list and student worksheet, go to appropriate for use with the TI-83 education.ti.com/exchange/sn Plus, TI-84 Plus, and TI-84 Plus Silver Edition but slight variances may be found within the directions. Compatible Devices: TI-84 Plus Family TI-84 Plus C Silver Edition Associated Materials: Scientific_Notation_Student.pdf Scientific_Notation_Student.doc Tech Tips: Access free tutorials at http://education.ti.com/calculators /pd/US/Online-Learning/Tutorials Any required calculator files can be distributed to students via handheld-to-handheld transfer. ©2013 Texas Instruments Incorporated 1 education.ti.com Scientific Notation TEACHER NOTES Part 1 – Writing Scientific Notation in Expanded Form Questions 1 -4 Have students write their answers and check them on the calculator. Help students understand the form for scientific notation. Students can enter scientific notation numbers two different ways, using 10 or E. Note that the key is labeled EE but the display is E. 1. _ 3 . 7 7 * 1 0 ^ 8 2. _ 3 . 7 7 `£ 8 To check –3.77108, enter in one of the two ways shown above and see if it matches their answer.
    [Show full text]
  • Floating-Point Arithmetic
    Floating-Point Arithmetic if ((A + A) - A == A) { SelfDestruct() } Reading: Study Chapter 3. Comp 411 – Spring 2013 03/04/2013 L11 – Floating Point 1 Why Floating Point? Aren’t Integers enough? • Many applications require numbers with a VERY large range. (e.g. nanoseconds to centuries) • Most scientific applications need real numbers (e.g. π) But so far we have only used integers. • We *COULD* use integers with a fixed “binary” point • We *COULD* implement rational fractions using two integers (e.g. ½, 1023/102934) • Floating point is a better answer for most applications. Comp 411 – Spring 2013 03/04/2013 L11 – Floating Point 2 Recall Scientific Notation • Let’s start our discussion of floating point by recalling scientific notation from high school Significant Digits • Numbers represented in parts: Exponent 42 = 4.200 x 101 1024 = 1.024 x 103 -0.0625 = -6.250 x 10-2 • Arithmetic is done in pieces 3 Before adding, we must match the 1024 1.024 x 10 exponents, effectively “denormalizing” the smaller magnitude number - 42 - 0.042 x 103 982 0.982 x 103 2 9.820 x 10 We then “normalize” the final result so there is one digit to the left of the decimal point and adjust the exponent accordingly. Comp 411 – Spring 2013 03/04/2013 L11 – Floating Point 3 Multiplication in Scientific Notation • Is straightforward: – Multiply together the significant parts – Add the exponents In multiplication, how far is the most you will – Normalize if required ever normalize? • Examples: 1024 1.024 x 103 In addition? x 0.0625 6.250 x 10-2 64 6.400 x 101 42 4.200
    [Show full text]