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Chapter 4: The Building Blocks: Binary Numbers, Boolean Logic, and Gates

Learn about Boolean logic and gates, the binary numbering system, building computer circuits, and control circuits in this chapter.

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Chapter 4: The Building Blocks: Binary Numbers, Boolean Logic, and Gates

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  1. Chapter 4: The Building Blocks: Binary Numbers, Boolean Logic, and Gates Invitation to Computer Science, C++ Version, Third Edition

  2. Objectives In this chapter, you will learn about: • Boolean logic and gates • The binary numbering system • Building computer circuits • Control circuits Invitation to Computer Science, C++ Version, Third Edition

  3. Introduction • Chapter 4 focuses on hardware design (also called logic design) • How to represent and store information inside a computer • How to use the principles of symbolic logic to design gates • How to use gates to construct circuits that perform operations such as adding and comparing numbers, and fetching instructions Invitation to Computer Science, C++ Version, Third Edition

  4. The Reliability of Binary Representation Sec 4.2.3 • Electronic devices are most reliable in a bistable environment • Bistable environment • Distinguishes only two electronic states • Current flowing or not • Direction of flow • Easiest way to make computers reliable was to use a two-state design. Hence they are designed to store and process binary information. Invitation to Computer Science, C++ Version, Third Edition

  5. Binary Storage Devices Sec 4.2.4 • Magnetic core • Historic device for computer memory • Tiny magnetized rings: flow of current sets the direction of magnetic field • Binary values 0 and 1 are represented using the direction of the magnetic field Invitation to Computer Science, C++ Version, Third Edition

  6. Sec 4.2.4 Figure 4.9 Using Magnetic Cores to Represent Binary Values Invitation to Computer Science, C++ Version, Third Edition

  7. Binary Storage Devices (continued) Sec 4.2.4 • Transistors • Solid-state switches: either permits or blocks current flow • A control input causes state change • Constructed from semiconductors Invitation to Computer Science, C++ Version, Third Edition

  8. Sec 4.2.4 Figure 4.11 Simplified Model of a Transistor Invitation to Computer Science, C++ Version, Third Edition

  9. Sec 4.2.4 Invitation to Computer Science, C++ Version, Third Edition

  10. Sec 4.2.4 Invitation to Computer Science, C++ Version, Third Edition

  11. Boolean Logic and Gates: Boolean Logic Sec 4.3.1 • Boolean logic is a branch of mathematics which describes operations on objects with two possible values, True or False • True/False values map easily onto a two-state electronic environment. • Boolean logic operations may be performed on electronic signals using devices built out of transistors and other electronic devices. Invitation to Computer Science, C++ Version, Third Edition

  12. Boolean Logic (continued) Sec 4.3.1 • Let a and b represent objects with two possible values, true and false • Boolean logic operations • a AND b • True only when a is true and b is true • a OR b • True when either a is true or b is true, or both are true • NOT a • True when a is false, and vice versa Invitation to Computer Science, C++ Version, Third Edition

  13. Boolean Logic (continued) Sec 4.3.1 • Boolean expressions • Constructed by combining together Boolean operations • Example: (a AND b) OR ((NOT b) AND (NOT a)) • Truth tables capture the output/value of a Boolean expression • A column for each input plus the output • A row for each combination of input values Invitation to Computer Science, C++ Version, Third Edition

  14. Boolean Logic (continued) Sec 4.3.1 • Example:(a AND b) OR ((NOT b) and (NOT a))_ _alternate expression (a ∙ b) + ( b ∙ a ) Invitation to Computer Science, C++ Version, Third Edition

  15. Gates Sec 4.3.2 • Gates • Hardware devices built from transistors to mimic Boolean logical operations on digital electrical signals • AND gate • Two input lines, one output line • Outputs a 1 when both inputs are 1 Invitation to Computer Science, C++ Version, Third Edition

  16. Gates (continued) Sec 4.3.2 • OR gate • Two input lines, one output line • Outputs a 1 when either input is 1 • NOT gate • One input line, one output line • Outputs a 1 when input is 0 and vice versa Invitation to Computer Science, C++ Version, Third Edition

  17. Sec 4.3.1 Figure 4.15 The Three Basic Gates and Their Symbols Invitation to Computer Science, C++ Version, Third Edition

  18. Sec 4.3.2 NAND gate AND gate Invitation to Computer Science, C++ Version, Third Edition

  19. Sec 4.3.2 NOR gate OR gate Invitation to Computer Science, C++ Version, Third Edition

  20. Gates (continued) Sec 4.3.2 • Abstraction in hardware design • Transform hardware devices to Boolean logical descriptions • Design more complex devices in terms of logic, not electronics • Conversion from logic to hardware design may be automated Invitation to Computer Science, C++ Version, Third Edition

  21. Sec 4.4.1 Figure 4.19 Diagram of a Typical Computer Circuit Invitation to Computer Science, C++ Version, Third Edition

  22. Building Computer Circuits: Introduction Sec 4.4.1 • A circuit is a collection of logic gates: • Transforms a set of binary inputs into a set of binary outputs • Values of the outputs depend only on the current values of the inputs • Combinational circuits have no cycles in them (no outputs feed back into their own inputs) Invitation to Computer Science, C++ Version, Third Edition

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  25. Building Computer Circuits: Intro Cont’d. Sec 4.4.1 We want a 1 only when both inputs are 0. Is there a gate which will do this? An AND gate gives a 1 only when both inputs are 1. An OR gate gives a 1 whenever at least one input is 1 Could we use an AND gate or an OR gate somehow? Invitation to Computer Science, C++ Version, Third Edition

  26. A Circuit Construction Algorithm Sec 4.4.2 • Sum-of-products algorithm is one way to design circuits: • Truth table to Boolean expression to gate layout Invitation to Computer Science, C++ Version, Third Edition

  27. Sec 4.4.2 Figure 4.21 The Sum-of-Products Circuit Construction Algorithm Invitation to Computer Science, C++ Version, Third Edition

  28. A Circuit Construction Algorithm (continued) Sec 4.4.2 • Sum-of-products algorithm • Truth table captures every input/output possible for circuit • Repeat process for each output line • Build a Boolean expression using AND and NOT for each 1 of the output line • Combine together all the expressions with ORs • Build circuit from whole Boolean expression Invitation to Computer Science, C++ Version, Third Edition

  29. The Binary Numbering System • A computer’s internal storage techniques are different from the way people represent information in daily lives • Information inside a digital computer is stored as a collection of binary data Invitation to Computer Science, C++ Version, Third Edition

  30. Binary Representation of Numeric and Textual Information • Binary numbering system • Base-2 • Built from ones and zeros • Each position is a power of 2 1101 = 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20 • Decimal numbering system • Base-10 • Each position is a power of 10 3052 = 3 x 103 + 0 x 102 + 5 x 101 + 2 x 100 Invitation to Computer Science, C++ Version, Third Edition

  31. Figure 4.2 Binary-to-Decimal Conversion Table Invitation to Computer Science, C++ Version, Third Edition

  32. Taxonomy of Integers Invitation to Computer Science, C++ Version, Third Edition

  33. Binary Representation of Numeric and Textual Information (continued) • Representing integers • Decimal integers are converted to binary integers • Given k bits, the largest unsigned integer is 2k - 1 • Given 4 bits, the largest is 24-1 = 15 • Signed integers must also represent the sign (positive or negative) Invitation to Computer Science, C++ Version, Third Edition

  34. Binary to Decimal Conversion Invitation to Computer Science, C++ Version, Third Edition

  35. Decimal to Binary Conversion Invitation to Computer Science, C++ Version, Third Edition

  36. Two’s Complement Representation of Signed Integers • Used by almost all computers today • All places hold the value they would in binary except for the leftmost place which is negative • 8 bit integer: -128 64 32 16 8 4 2 1 • Range: [-128,127] • If last bit (leftmost) • is 0, then positive  looks just as before • is 1, then negative add values for first 7 digits and then subtract 128 • 1000 1101 = 1+4+8-128 = -115 Invitation to Computer Science, C++ Version, Third Edition

  37. Two’s Complement Representation of Signed Integers • Converting from decimal to 2’s complement • For positive numbers: find the binary representation • For negative numbers: • Find the binary representation for its positive equivalent • Flip all bits • Add a 1 • 43 • 43 = 0010 1011 • -43 • 43 = 0010 1011  1101 0100  1101 0101 • 1101 0101 = -128+64+16+4+1 = -43 • We can use the usual laws of binary addition Invitation to Computer Science, C++ Version, Third Edition

  38. Two’s Complement Representation • 125 - 19 = 106 • 0111 1101 1110 1101 ------------- 0110 1010 = 106 ! • What happened to the one that we carried at the end? • Got lost but still we got the right answer! • We carried into and carried out of the leftmost column Invitation to Computer Science, C++ Version, Third Edition

  39. Binary Representation of Numeric and Textual Information (continued) • Representing real numbers • Real numbers may be put into binary scientific notation: a x 2b • Example: 101.11 x 20 • Number then normalized so that first significant digit is immediately to the right of the binary point • Example: .10111 x 23 • Mantissa and exponent then stored Invitation to Computer Science, C++ Version, Third Edition

  40. Binary Representation of Numeric and Textual Information (continued) Invitation to Computer Science, C++ Version, Third Edition

  41. Converting Decimal Fractions To Binary Invitation to Computer Science, C++ Version, Third Edition

  42. Converting Decimal Fractions To Binary • Note that the binary representation of 0.410 is a repeating base two fraction, .01102 • This just says that 0.4 cannot be exactly represented as a sum of inverse powers of two. • .875 = • .400 = Invitation to Computer Science, C++ Version, Third Edition

  43. Binary Representation of Textual Information • Many transformations exist and all are arbitrary • Two popular • EBCDIC (Extended Binary Coded Decimal Interchange Code) by IBM • ASCII (American Standard Code for Information Interchange) by American National Standards Institute (ANSI) • Most widely used • Every letter/symbol is represented by 7 bits • How many letters/symbols do we have in total? • A-Z (26) , a-z (26) , 0-9 (10), symbols (33), control characters (33) • If using 1 byte/character  we have one extra bit • Extended ASCII-8 (more mathematical and graphical symbols or phonetic marks) --- 256 Invitation to Computer Science, C++ Version, Third Edition

  44. Binary Representation of Numeric and Textual Information (continued) • Characters are mapped onto binary numbers • ASCII code set • 8 bits per character; 256 character codes • UNICODE code set • 16 bits per character; 65,536 character codes • Text strings are sequences of characters in some encoding Invitation to Computer Science, C++ Version, Third Edition

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  46. Binary Representation of Images • Representing image data • Images are sampled by reading color and intensity values at even intervals across the image • Each sampled point is a pixel • Image quality depends on number of bits at each pixel Invitation to Computer Science, C++ Version, Third Edition

  47. Binary Representation of Images • Divide the screen into a grid of cells each referred to as a pixel 512*256 image grid has 512 columns and 256 rows • Pixel values and sizes depend on the type of the image • For black & white images, we can use 1 bit for every pixel such that 1  black and 0  white • For grayscale images, we use 1 byte where 255  black and 0  white and anything in between is gray (higher/lower values are closer to black/white) • For color images, we need three values per pixel (depends on the color scheme used): • Red/Green/Blue • We need 1 byte per value  3 bytes per pixel • For an 512x256 image • 512x256x3 = 384K bytes Invitation to Computer Science, C++ Version, Third Edition

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