Computer Arithmetic-II. Signed binary numbers
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1 Computer ArithmeticII Signed binary numbers
2 Binary number representation. Representing binary signed integers: a) Signmagnitude b) One s complement signed magnitude c) Two's complement signed magnitude 2. Representing binary integer/fractions: a) Fixedpoint numbers b) Floatingpoint numbers 2
3 Review Unsigned (positive) numbers
4 Unsigned binary The range is: up to 2 n (min max) Example: n = 3: [2 3 ] = 7 4
5 3bit unsigned binary number Decimal Unsigned
6 8bit unsigned binary numbers Binary Unsigned = 255 6
7 Largest unsigned 32bit integer For a 32bit computer What is the largest unsigned integer that can hold? 32bits 2 32 = 4, 294, 967, Giga 7
8 Binary unsigned numbers (54.5) = (.) (.) 2 = * * * * * 2 + * 2 + * 2 = =
9 . Representing binary signed integers a) Signedmagnitude (review) b) s complement signedmagnitude c) 2 s complement signedmagnitude
10 Unsignedmagnitude (review) A nbit binary number has 2 n distinct values. Decimal Unsigned
11 a) Signedmagnitude << leftmost bit = Negative Number << leftmost bit = Positive Number Decimal signed 3 NEGATIVE +3 POSITIVE
12 a) Signedmagnitude A nbit binary number has 2 n distinct values. Assign about Half to positive integers (Zero MSB) and about Half to negative (One MSB) with two values for the Zero () Decimal Unsigned Decimal signed NEGATIVE POSITIVE 2
13 3bit signedmagnitude numbers A 3bit binary number has a total of 2 3 = 8 numbers. Using signed number representation, we have negative and positive numbers: Divide 8 by 2 4. We have: 4positive (+ +3) 4negative ( 3) numbers. The only «problem» is that we have 2 different zeros (, +). The range is: +3 3 We have 2 ways to represent zero Decimal Unsigned Signed
14 Range: Signedmagnitude numbers The range is: +3 3 In general 2 n to (2 n ) Decimal Unsigned Signed
15 Signedmagnitude Not used in modern computing because there are two representations for the zero. IBM79/ 959 5
16 In general: Range of nbit numbers bits Binary Unsigned Signmagnitude Min Max Min Max n 2 n (2 n ) 2 n 6
17 b) 3bit s complement signedmagnitude The range is: +3 3 In general 2 n to (2 n ) We have 2 ways to represent zero Decimal Unsigned Signed s Comp
18 s complement signedmagnitude Not used in modern computing because there are two representations for the zero CDC6A/96 UNIVAC /22 /962 8
19 c) 3bit 2 s complement signed magnitude The range is: Positive ( + +3) Negative ( 4) we have way to represent zero Decimal Unsigned Signed s Comp. 2 s Comp
20 2 s complement signedmagnitude Today s processors represent signed integers using two s complement Why? Because a 2 s complement signedmagnitude representation has a single representation for zero () 2
21 Range for nbit 2 s complement signed Range: 2 n 2 n For our 3bit [2 3 = 8 divide by 2 4 = 2 2 ] For our example the range is (4 +3): = =
22 22 Modulo6 system (4bits) b 3 b 2 b b Sign and magnitude ' s complement 2' s complement B Value represented
23 In general: Range of nbit numbers bits Binary Unsigned Sign/Magnitude 2 s Complement Min Max Min Max Min Max n 2 n (2 n ) 2 n 2 n 2 n 23
24 Range of 32bit 2 s comp. signed numbers 2 = 2 = 2 = = = (2 3 ) 2 = (2 3 2) 2 3 = 4,294,967,295 4 Giga 24
25 32 bit OS 4 GB RAM 25
26 32bit and 64bit 32bit: 2 32 = 4,294,967,296 4,294,967,296 / (,24 x,24) = 4,96 MB = 4GB (gigabytes) 64bit: 2 64 = 8,446,744,73,79,55,66 8,446,744,73,79,55,66 / (,24 x,24) = 6EB (exabytes) Note that giga >> tera >> peta >> exa 26
27 32bit and 64bit 64bit computers can realistically access 4 GB and 28 GB of RAM. 27
28 64bit CPU Example
29 Apple: A Bionic CPU (iphone8, X) 29
30 Apple: A Bionic CPU (iphone8, X) The A Bionic features an Appledesigned 64bit Two highperformance cores: called Monsoon Four energyefficient cores: called Mistral The A is manufactured by TSMC using a nm FinFET process and it has 4.3 billion transistors FinFET (Fin Field Effect Transistor) 3
31 RealNumbers
32 RealNumbers 32
33 Why? Because we need to Expand the number range and include smaller numbers than Use integers, positive and negative numbers in decimal notation. 33
34 2. FixedPoint Numbers
35 FixedPoint Numbers Fixed point number representation: Every word has the same number of digits and the binary point is always fixed at the same position
36 FixedPoint Arithmetic is used FixedPoint arithmetic is used in applications where speed is more important than precision: Digital Signal/Image Processing Control Systems Mobile smartdevices Games Fixedpoint calculations require less memory and less processor time Fixedpoint hardware are much less complicated than those of floatingpoint hardware 36.
37 FloatingPoint Numbers (next lecture)
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Slide Set 1 for ENEL 339 Fall 2014 Lecture Section 02 Steve Norman, PhD, PEng Electrical & Computer Engineering Schulich School of Engineering University of Calgary Fall Term, 2014 ENEL 353 F14 Section