Unary, Binary, and Beyond

Unary, Binary, and Beyond

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

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

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

Last time we talked about unary notation.

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

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

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

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

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

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

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.

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+

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

* • 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 * = +[ {}

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

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

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

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

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

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

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

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 

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

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)

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

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]

 = { 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

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

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.

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)]

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

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

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

Let’s define base 10 representation of PU numbers.

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

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))

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.

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

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

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

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

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

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

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

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

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

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

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

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)

Unary, Binary, and Beyond