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ECE 645: Lecture 4 Number Representation Part 1 Fixed-Point Representations. Endianness. Required Reading B. Parhami, Computer Arithmetic: Algorithms and Hardware Design Chapter 1, Numbers and Arithmetic, Sections 1.1-1.6 Chapter 2, Representing Signed Numbers Endianness, from Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Endianness Historical Representations: Ancient Egyptians Numerals (1) – ~4000 BC – “Sum of Symbols” = (34)10 Historical Representations: Ancient Egyptians Numerals (2) What number does this stone carving from Karnak represent? Historical Representations: Ancient Egyptians Numerals (3) Historical Representations: Ancient Egyptians Numerals (4) Symbol for a fraction (part): Special symbols for most commonly used fractions: Historical Representations: Ancient Babylonian Numerals (1) – First known positional system – 3100 BC – Radix 60 (sexagesimal) – Two symbols: = 1 = 10 − Integers and fractions were represented identically - a radix point was not written but rather made clear by context Babylonian Numerals: Example 1 x 602 20 x 601 56 x 600 = (4,856)10 Historical Representations: Ancient Babylonian Numerals (2) Digits from 1 to 59 Positional Code with Zero • Zero Represented by Space – Partial solution – What about trailing zeros? • Babylonians Introduced New Symbol – or – 4th to 1st Century BC • Zero Allows Representation of Fractions – Fractions started with zero Mixed System • Roman Numerals – Sum of all symbols – I=1 V=5 X=10 L=50 C=100 D=500 M=1000 – Difficult to do arithmetic e.g., MCDXLVII - IX ? Hindu-Arabic Numeral System Brahmi numerals, India, 400 BC-400 AD Evolution of numerals in early Europe Positional Code Decimal System – Documented in the 9th century – Position of coefficient determines its value – Coefficient in position is multiplied by radix (10) raised to the power determined by its position, e.g., 3 2 1 0 4 *10 + 8*10 + 5*10 + 6*10 = (4,856)10 Migration of Positional Notation • ~750 AD – Zero spread from India to Arabic countries • ~1250 AD – Zero spread to Europe • Importance of Zero – Ease of arithmetic which leads to improved commerce Binary Number System • Binary – Positional number system – Two symbols, B = { 0, 1 } – Easily implemented using switches – Easy to implement in electronic circuitry – Algebra invented by George Boole (1815-1864) allows easy manipulation of symbols 3 2 1 0 (0101)2 = 0*2 +1*2 + 0*2 +1*2 = (5)10 Modern Arithmetic and Number Systems • Modern number systems used in digital arithmetic can be broadly classified as: • Fixed-point number representation systems • Integers • Rational numbers of the form x = a/2f, a is an integer, f is a positive integer • Floating-point number representation systems • x * bE, where x is a rational number, b the integer base, and E the exponent • Note that all digital numbers are eventually coded in bits {0,1} on a computer 16 Encoding Numbers in 4-Bits −16 −14 −12 −10 −8 −6 −4 −2 0 2 4 6 8 10 12 14 16 Number format Unsigned integers Signed-magnitude ± fixed 3 + 1 fixed-point, point xxx.x Signed fraction, ±.xxx ± 2’s-compl. fraction, x.xxx 2 + 2 floating-point, s × 2 e e s floating e in [−2, 1], s in [0, 3] point 2 + 2 logarithmic (log = xx.xx) log x 17 Classification of number systems (1) Number system Positional Non-positional Fixed-radix Mixed-radix Conventional Unconventional Binary Signed-digit Decimal Hexadecimal Non-redundant Redundant Classification of number systems (2) Positional k−1 X x w w - weight of the digit x = ∑ i ⋅ i i i i=−l Fixed-radix k−1 X x ri r - radix of the number system = ∑ i ⋅ i=−l Conventional fixed-radix k−1 r integer, r > 0 X = x ⋅ ri ∑ i xi ∈ {0, 1, …, r-1} i=−l Classification of number systems (3) Unconventional fixed-radix k−1 i X x r xi ∈ {-α, …, β } = ∑ i ⋅ i=−l Signed-digit α>0 ⇒ negative digits Non-redundant number of digits = α + β + 1 ≤ r Redundant number of digits = α + β + 1 > r Example of a mixed-radix positional system? What is it? What is it? What is it? Samrat Yantra in Jantar Mantar - Jaipur, India - The largest sundial in the world - 90 feet (27 m) tall - Conceived by Maharaja Sawai Raja Jai Singh II - Built of local stone and marble 1728-1734 - The style's horizontal angle is equal the sundial's geographical latitude 27° north - The time is read by observing where the shadow is sharpest at the time - Accuracy of 2 seconds - Reconstructed in 1901 Jantar Mantar - Jaipur, India Original Units of Time - 1 day = 60 ghadis (1 ghadi = 24 minutes) - 1 ghadi = 60 pals (1 pal = 24 seconds) - 1 pal = 4 breaths (1 breath = 6 seconds) Original weights: 60*60*4 60*4 4 breaths day ghadi pal Current weights: 24*60*60 60*60 60 seconds day hour minute Fixed-point representation Integral and fractional part X = xk-1 xk-2 … x1 x0 . x-1 x-2 … x-l Integral part Fractional part Radix point • NOT stored in the register • understood to be in a fixed position Fixed-Radix Conventional (Unsigned) Representations Fixed-Radix Conventional Number Systems Fixed Point Number system Positional Non-positional Fixed-radix Mixed-radix Conventional Unconventional (signed) (unsigned) Binary Signed-digit Decimal Hexadecimal Non-redundant Redundant 29 Range of numbers Number system Xmin Xmax Decimal k -l X = (xk-1 xk-2 … x1 x0.x-1 … x-l)10 0 10 - 10 Binary k -l X = (xk-1 xk-2 … x1 x0.x-1 … x-l)2 0 2 - 2 Conventional fixed-radix k -l X = (xk-1 xk-2 … x1 x0.x-1 … x-l)r 0 r - r Notation: unit in the least significant position -l ulp = r unit in the last position Number of digits Number of digits in the integer Number system part necessary to cover the range 0..Xmax Decimal k = ⎣log10 X max ⎦+1 = = ⎡log10(X max +1)⎤ Binary k = ⎣log2 X max ⎦+1 = = ⎡log2 (X max +1)⎤ Conventional k = ⎣logr X max ⎦+1 = fixed-radix = ⎡logr (X max +1)⎤ Radix Conversion Whole part Fractional part u = w . v = ( xk–1xk–2 . x1x0 . x–1x–2 . x–l )r Old = ( XK–1XK–2 . X1X0 . X–1X–2 . X–L )R New Example: (31)eight = (25)ten Two methods: Option 1) Radix conversion, using arithmetic in the old radix r Convenient when converting from r = 10 or familiar radix Option 2) Radix conversion, using arithmetic in the new radix R Convenient when converting to R = 10 or familiar radix From: Parhami, Computer Arithmetic: Algorithms and Hardware Design 32 Option 1: Arithmetic in old radix r Converting whole part w: (105)ten = (?)five Repeatedly divide by five Quotient Remainder 105 0 21 1 4 4 0 Therefore, (105)ten = (410)five Converting fractional part v: (105.486)ten = (410.?)five Repeatedly multiply by five Whole Part Fraction .486 2 .430 2 .150 0 .750 3 .750 3 .750 Therefore, (105.486)ten ≅ (410.22033)five From: Parhami, Computer Arithmetic: Algorithms and Hardware Design 33 Radix Conversion of the Integral Part R - destination radix k−1 X = (x x … x x ) = x ⋅ R i = I k-1 k-2 1 0 R ∑ i i=0 = ((...((xk-1 R + xk-2) R + xk-3 ) R + … + x2 ) R + x1 ) R + x0 Quotient Remainder (...((xk-1 R + xk-2) R + xk-3 ) R + … + x2 ) R + x1 x0 ...((xk-1 R + xk-2) R + xk-3 ) R + … + x2 x1 ………. ………. xk-1 R + xk-2 xk-3 xk-1 xk-2 0 xk-1 Radix Conversion of the Fractional Part R - destination radix −1 i X = (. x x … x x ) = x i ⋅ R = F -1 -2 -l+1 -l R ∑ i=−l -1 -1 -1 -1 -1 = R (x-1 + R (x-2 + R (…. + R ( x-l+1 + R x-l )….))) Integer part Fractional part -1 -1 -1 -1 x-1 R (x-2 + R (….. + R ( x-l+1 + R x-l)….)) x -1 -1 -1 -2 R (….. + R ( x-l+1 + R x-l)….) ………. ………. -1 -1 x-l+2 R ( x-l+1 + R x-l) x -1 -l+1 R x-l x-l 0 Option 2: Arithmetic in new radix R Converting fractional part v: (410.22033)five = (105.?)ten 5 (0.22033)five × 5 = (22033)five = (1518)ten 5 1518 / 5 = 1518 / 3125 = 0.48576 Therefore, (410.22033)five = (105.48576)ten Converting (22033)five = (?)ten ! ((((2 × 5) + 2) × 5 + 0) × 5 + 3) × 5 + 3 ! ! |-----| : : : : ! ! 10 : : : : ! ! |-----------| : : : ! ’ ! 12 : : : !Horner s ! |---------------------| : : ! rule or ! 60 : : ! ! |-------------------------------| : !formula ! 303 : ! ! |-----------------------------------------| ! !! 1518 From: Parhami, Computer Arithmetic: Algorithms and Hardware Design 36 Option 2 cont'd: Horner's rule for fractions Horner’s rule is also applicable: Proceed from right to left and use division instead of multiplication Converting fractional part v: (0.22033)five = (?)ten ! ! ! (((((3 / 5) + 3) / 5 + 0) / 5 + 2) / 5 + 2) / 5! Horner’s |-----| : : : : ! 0.6 : : : : ! rule or |-----------| : : : ! formula 3.6 : : : ! |---------------------| : : ! 0.72 : : ! |-------------------------------| : ! 2.144 : ! |-----------------------------------------| ! 2.4288 ! |-----------------------------------------------|! 0.48576! From: Parhami, Computer Arithmetic: Algorithms and Hardware Design 37 Radix Conversion Shortcut for r=bg, R=bG r=bg → b → R=bG 4=22 → 2 → 8=23 (2301.302)4 = (10 110 001. 110 010)2 = (261.62)8 • Trick here is to first convert to a number in radix b, then to R • Cluster in groups of 3 (because 23 = 8) moving away from binary point 38 Signed Number Representations Representations of signed numbers Signed-magnitude Biased Complement Radix-complement Diminished-radix complement (Digit complement) r=2 r=2 Two’s complement One’s complement Signed- Biased Two’s One’s magnitude complement complement 7 0111 1111 0111 0111 6 0110 1110 0110 0110 5 0101 1101 0101 0101 4 0100 1100 0100 0100 3 0011 1011 0011 0011 2 0010 1010 0010 0010 1 0001 1001 0001 0001 0 0000 1000 0000 0000 -0 1000 1111 -1 1001 0111 1111 1110 -2 1010 0110 1110 1101 -3 1011 0101 1101 1100 -4 1100 0100 1100 1011 -5 1101 0011 1011 1010 -6 1110 0010 1010 1001 -7 1111 0001 1001 1000 -8 0000 1000 Signed-magnitude representation of signed numbers k-1 k-2 0 magnitude sign Advantages: • conceptual simplicity • symmetric range: -(2k-1-1) .
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  • Number Systems

    Number Systems

    1 Number Systems The study of number systems is important from the viewpoint of understanding how data are represented before they can be processed by any digital system including a digital computer. It is one of the most basic topics in digital electronics. In this chapter we will discuss different number systems commonly used to represent data. We will begin the discussion with the decimal number system. Although it is not important from the viewpoint of digital electronics, a brief outline of this will be given to explain some of the underlying concepts used in other number systems. This will then be followed by the more commonly used number systems such as the binary, octal and hexadecimal number systems. 1.1 Analogue Versus Digital There are two basic ways of representing the numerical values of the various physical quantities with which we constantly deal in our day-to-day lives. One of the ways, referred to as analogue,isto express the numerical value of the quantity as a continuous range of values between the two expected extreme values. For example, the temperature of an oven settable anywhere from 0 to 100 °C may be measured to be 65 °C or 64.96 °C or 64.958 °C or even 64.9579 °C and so on, depending upon the accuracy of the measuring instrument. Similarly, voltage across a certain component in an electronic circuit may be measured as 6.5 V or 6.49 V or 6.487 V or 6.4869 V. The underlying concept in this mode of representation is that variation in the numerical value of the quantity is continuous and could have any of the infinite theoretically possible values between the two extremes.