Week 5

Week 5 - EECE201 Project Integrated Program Module 2, Fall...

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11/23/2009 EECE201 PIP 1 EECE201 Project Integrated Program Module 2, Fall 2009 Week 5 Edmond Cretu Department of Electrical and Computer Engineering edmondc@ece.ubc.ca Ch. 5 – Number representations, binary arithmetic
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11/23/2009 EECE201 PIP 2 Doing Arithmetic Goal : how to represent and implement the usual arithmetic manipulation of numbers using binary symbols and binary logic • It is necessary: – To match the representation of numbers to the symbols {0,1} – Map mathematical operators (+,-,*) to logic combinations of bits, seen as binary signals “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” John Louis von Neumann
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11/23/2009 EECE201 PIP 3 Positional Number Representation Unsigned integers - Numbers that are positive only (> = 0) – Numbers that can also be negative are called signed Positional number representations – Positional decimal numeral system – developed in India (9 th century) and adopted by Persian and Arabic mathematicians (The Hindu–Arabic numeral system ) – Characteristics: use of the same symbol (glyph) for the different orders of magnitude (for example, the "one's place", "ten's place", "hundred's place") => greatly simplified arithmetic Representation : a linear string of digits/symbols, where the weight of each digit is coded into its position in the string – Set of digits (e.g., 0 – 9 and specific multiplicative value corresponding to position of digits (e.g., powers of 10) – Extension to the representation of real numbers
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11/23/2009 EECE201 PIP 4 Most common -- Base 10 (or radix-10 ) – because set of 10 possible digits – Representation: Number D = d n-1 d n-2 …d 1 d 0 Value of number is V(D) = d n-1 X 10 n-1 +d n-2 X 10 n-2 + …+d 1 X 10 1 +d 0 X 10 0 General radix r : Base 2 (or radix-2) – most common when dealing with digital (binary) systems. Each binary digit is called a bit . –B = b n-1 b n-2 …b 1 b 0 where b can take on value of 0 or 1 only V(B) = b n-1 X 2 n-1 + b n-2 X 2 n-2 + …+b 1 X 2 1 + b 0 X 2 0 E.g., (1101) 2 = 8 + 4 + 1 = (13) 10 avoid confusion by indicating the radix Right-most bit least significant bit (LSB) Left-most bit most significant bit (MSB) Base-2 positional representation
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11/23/2009 EECE201 PIP 5 Different Number Systems -- Binary, Octal, and Hexadecimal Bases When dealing with binary systems, common to consider base 2, 8 and 16. Binary is binary!! – series of 0s and 1s Binary 3 bits at a time octal (0 – 7) e.g., 100 111 2 = 47 8 Binary 4 bits at a time hex (0 – F) e.g., 1100 1111 2 = AF 16
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11/23/2009 EECE201 PIP 6 Addition of Unsigned Numbers Sum s 0 1 1 0 Carry c 0 0 0 1 0 0 + 0 1 + 1 0 0 0 1 0 + 1 0 1 1 + 0 1 x y + s c Sum Carry (a) The four possible cases x y 0 0 1 1 0 1 0 1 (b) Truth table x y s c HA x y s c (c) Circuit (d) Graphical symbol Binary addition is performed same way as decimal addition that values of individual digits can only be 0 or 1.
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Week 5 - EECE201 Project Integrated Program Module 2, Fall...

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