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Unary, Binary, and Beyond

Unary, Binary, and Beyond. X n – 1. 1 + X 1 + X 2 + X 3 + … + X n-2 + X n-1 =. X - 1. We are going to need this fundamental sum: The Geometric Series. A Frequently Arising Calculation. ( X -1 ) ( 1 + X 1 + X 2 + X 3 + … + X n-2 + X n-1 )

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Unary, Binary, and Beyond

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  1. Unary, Binary, and Beyond

  2. Xn – 1 1 + X1 + X2 + X 3 + … + Xn-2 + Xn-1 = X- 1 We are going to need this fundamental sum: The Geometric Series

  3. A Frequently Arising Calculation • (X-1) ( 1 + X1 + X2 + X 3 + … + Xn-2 + Xn-1 ) • = X1 + X2 + X 3 + … + Xn-1 + Xn • - 1 - X1 - X2 - X 3 - … - Xn-2 – Xn-1 • = - 1 + Xn • = Xn - 1

  4. The Geometric Series • (X-1) ( 1 + X1 + X2 + X 3 + … + Xn-1 ) = Xn - 1 Xn – 1 1 + X1 + X2 + X 3 + … + Xn-2 + Xn-1 = X- 1 when X1

  5. Last time we talked about unary notation.

  6. nth Triangular Number • n = 1 + 2 + 3 + . . . + n-1 + n • = n(n+1)/2

  7. nth Square Number • n = 1 + 3 + … + 2n-1 • = Sum of first n odd numbers

  8. nth Square Number • n = n + n-1 • = n2

  9. (n)2 = + + . . . + n (n)2 =(n-1)2 + n

  10. We will define sequences of symbols that give us a unary representation of the Natural number.

  11. First we define the general language we use to talk about strings.

  12. Strings Of Symbols. • We take the idea of symbol and sequence of symbols as primitive. • Let  be any fixed finite set of symbols.  is called an alphabet, or a set of symbols. • Examples:  = {0,1,2,3,4} • = {a,b,c,d, …, z} • = all typewriter symbols.

  13. Strings over the alphabet . • A string is a sequence of symbols from . Let s and t be strings. Let st denote the concatenation of s and t, i.e., the string obtained by the string s followed by the string t. • Define + by the following inductive rules: • x2) x2+ • s,t 2+) st 2+

  14. Intuitively, + is the set of all finite strings that we can make using (at least one) letters from .

  15. * • Define  be the empty string. I.e., • XY= XY for all strings X and Y. • is also called the string of length 0. Define 0 = {  } • Define * = +[ {}

  16. Intuitively, * is the set of all finite strings that we can make using letters from , including the empty string.

  17. The Natural Numbers •  = { 0, 1, 2, 3, . . .} Notice that we include 0 as a Natural number.

  18. “God Made The Natural Numbers. Everything Else Is The Work Of Man.” Kronecker •  = { 0, 1, 2, 3, . . .}

  19. Last Time: Unary Notation • 1 • 2 • 3 • 4

  20. To handle the notation for zero, we introduce a small variation on unary.

  21. Peano Unary (PU) • 0 0 • 1 S0 • 2 SS0 • 3 SSS0 • 4 SSSS0 • 5 SSSSS0 • 6 SSSSSS0 Giuseppe Peano [1889]

  22. Each number is a sequence of symbols in {S, 0}+ • 0 0 • 1 S0 • 2 SS0 • 3 SSS0 • 4 SSSS0 • 5 SSSSS0 • 6 SSSSSS0

  23. Peano Unary Representation of Natural Number •  = { 0, S0, SS0, SSS0, . . .} 0 is a natural number called zero. Set notation: 0 2  If X is a natural number, then SX is a natural number called successor of X. Set notation: X 2  ) SX 2 

  24. Inductive Definition of + •  = { 0, S0, SS0, SSS0, . . .} Inductive definition of addition (+): X, Y 2  ) X “+” 0 = X X “+” SY = S(X”+”Y)

  25. S0 + S0 = SS0 (i.e., “1+1=2”) • Proof: • S0 + S0 = S(S0 + 0) = S(S0) = SS0 X, Y 2  ) X “+” 0 = X X “+” SY = S(X”+”Y)

  26. Inductive Definition of * •  = { 0, S0, SS0, SSS0, . . .} Inductive definition of times (*): X, Y 2  ) X “*” 0 = 0 X “*” SY = (X”*”Y) + X

  27. Inductive Definition of ^ •  = { 0, S0, SS0, SSS0, . . .} Inductive definition of raised to the (^): X, Y 2  ) X “^” 0 = 1 [or X0 = 1 ] X “^” SY = (X”^”Y) * X [or XSY = XY * X]

  28.  = { 0, S0, SS0, SSS0, . . .} • Defining < for : • 8 x,y 2  • “x > y” is TRUE  “y < x” is FALSE • “x > y” is TRUE  “y > x” is FALSE • “x+1 > 0” is TRUE • “x+1 > y+1” is TRUE ) “x > y” is TRUE

  29.  = { 0, 1, 2, 3, . . .} • Defining partial minus for : • 8 x,y 2  • x-0 = x • x>y ) • (x+1) – (y+1) = x-y

  30. a = [a DIV b]*b + [a MOD b] • Defining DIV and MOD for : • 8 a,b 2  • a<b ) • a DIV b = 0 • a¸b>0 ) • a DIV b = 1 + (a-b) DIV b • a MOD b = a – [b*(a DIV b)] The maximum number of times b goes into a without going over. The remainder when a is divided by b.

  31. 45 = [45 DIV 10]*10 + [ 45 MOD 10]= 4*10 + 5 • Defining DIV and MOD for : • 8 a,b 2  • a<b ) • a DIV b = 0 • a¸b>0 ) • a DIV b = 1 + (a-b) DIV b • a MOD b = a – [b*(a DIV b)]

  32. We have defined the Peano representation of the Naturals, with a notion of +, *, ^, <, -, DIV, and MOD.

  33. PU takes size n+1 to represent the number n. Higher bases are much more compact.

  34. 1000000 in Peano Unary takes one million one symbols to write. 1000000 only takes 7 symbols to write in base 10.

  35. Let’s define base 10 representation of PU numbers.

  36. Let = {0, 1,2,3,4,5,6,7,8,9} be our symbol alphabet. Any string in + will be called a decimal number.

  37. Let X be a decimal number. Let’s define the length of X inductively: length() = 0 X= aY, a2, Y2* ) length(X) = S(length(Y))

  38. Let X be decimal. Let n (unary) = length(X). For each unary i<n, we want to be able to talk about the ith symbol of X.

  39. The ith symbol of X. Defining rule: X = PaS where P,S2* and a2. i=length(S).)a is the ith symbol of X.

  40. For any string X, we can define its length n2 PU. For all i2 PU s.t. i<n, we can define the ith symbol ai. X = an-1 an-2 …a0

  41. We define a base conversion function, Base10 to 1 (X), to convert any decimal X to its unary representation.

  42. Initial Cases, length(X)=1 • Base10 to 1 (0) = 0 • Base10 to 1 (1) = S0 • Base10 to 1 (2) = SS0 • Base10 to 1 (3) = SSS0 • …… • Base10 to 1 (9) = SSSSSSSSS0

  43. Suppose n=length(X)>S0 • For all i<n, let ai be the ith symbol of X. Hence, X = an-1 an-2 …a0 . • Define Base10 to 1(X) = • i<n Base10 to 1(ai)*10i • where +, *, and ^ are defined over PU

  44. Example X= 238 • Base10 to 1 (238) = • 2*100 + 3*10 + 8

  45. Base10 to 1 (an-1 an-2 … a0) = an-1*10n-1 + an-2 * 10n-2 +…+ a0 100

  46. Now we want to go back the other way. We want to convert PU to decimal.

  47. No Leading Zero: Let NLZ be the set of decimal numbers with no leading 0 (leftmost symbol  0), or the decimal number 0.

  48. We define Base1 to 10 from PU to NLZ It will turn out to be the inverse function to Base10 to 1.

  49. One digit cases. • Base1 to 10(0) = “0” • Base1 to 10(S0) = “1” • Base1 to 10(SSo) =“2” • …. • Base1 to 10(SSSSSSSSS0) = “9”

  50. Suppose X > 9 • Let n be the smallest unary number such that 10n > X, 10n-1· X. • Let d = X DIV 10n-1 [Notice that 1· d · 9] • Let Y = X MOD 10n-1 • Define Base1 to 10(X) 2 NLZ to be • Base1 to 10 (d) Base1 to 10(Y)

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