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THE BINARY NUMBERING SYSTEM prepared by Burak Galip ASLAN ([email protected]) September, 2006 Information representation Decimal to binary conversion table Base 2 positional numbering system Simple examples 111001 ? 57 23 10111 ? Same amount of information in more number of digits in binary form Arithmetic overflow integer storing capacity 1 1 1 1 1 1 1 1 27 0-255 20 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 215 0 2 0-65535 67890? Arithmetic overflow! Should be handled! Sign-magnitude notation sign bit 0 positive 1 1 1 1 1 1 1 1 1 negative 27 26 20 1 0 1 0 1 0 1 0 170? - 86? ?ball? Interpretation Negation 0 0 0 0 1 1 0 0 + 12 Take one’s complement 1 1 1 1 0 0 1 1 Add 1 1 1 1 1 0 1 0 0 - 12 This is two’s complement technique for negation Binary addition 12 + (- 5) 0 0 0 0 1 1 0 0 + 12 0 0 0 0 0 1 0 1 +5 1 0 1 0 1 0 1 0 1 1 0 1 1 0 +- 57 Binary addition 5 + (- 12) 0 0 0 0 0 1 0 1 +5 0 0 0 0 1 1 0 0 + 12 0 0 1 0 1 0 1 1 1 1 1 --12 ? 7 0 0 0 0 0 1 1 1 Binary multiplication and division (- 3) X 2 6/3 http://courses.cs.vt.edu/~cs1104/BuildingBlocks/divide.010.html Decimal to binary conversion 47 15 7 3 1 0 47 23 11 5 2 1 20 21 22 23 24 25 26 1 0 1 1 1 1 1 0 1 1 1 1 46 22 10 4 0 http://www.math.grinnell.edu/~rebelsky/Courses/CS152/97F/Readings/student-binary.html Storing text / code mapping 0 0 0 0 0 0 0 1 A 0 0 0 0 0 0 1 0 B 0 0 0 0 0 0 1 1 C 8-bit code representation 0 0 0 0 0 1 0 0 D AD 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 A D ASCII coding standard American Standard Code for Information Interchange ASCII art Reliability of binary representation Why not decimal, octal or ternary computers? Nature of electrical systems: bistability! Reliability v.s. conversion time Binary storage devices Two stable energy states (for 0 and 1) States separated by large energy barrier Sense the state without destroying information Switch states by applying energy How about a light switch? bulky and slow, but OK! Binary storage devices Magnetic core memories (1955 – 1975) 128 X 128 bits = 2 KB memory each ring stores 1 bit each ring is ~ 1 mm 16 cm X 16 cm speed up to 1 MHz non-volatile storage Binary storage devices Transistor memories transistor extremely small extremely fast extremely cheap Integrated Circuit Binary storage devices Attacking the complexity problem IC: Integrated Circuits (1950’s) SSI: Small-Scale Integration (1960’s) (10’s) MSI: Medium-Scale Integration (1960’s) (100’s) LSI: Large-Scale Integration (1970’s) (1000’s) VLSI: Very-Large-Scale Integration (1980’s) (millions of transistors) ULSI: Ultra-Large-Scale Integration (> 1M) Binary storage devices Extreme intregration techniques WSI: Wafer-Scale Integration (1980’s) SOC: System-On-Chip Design (today) FPGA: Field Programmable Gate Arrays (today) (ten thousands of LSI circuits) Summary • What is binary representation? • Why is it suitable to be used in computers? • How is it realized? (abstract level) References • An Invitation to Computer Science, 1st Edition (1995) (Schneider & Gersting) (Section 4.2) • Wikipedia & Google September, 2006