[NOTES/QCQI-01001] Binary Number Representation

1. Representation in binary numbers

Binary representation uses a sequence of 0 and 1 to represent a given positive integer. We use the notation that roman alphabets will take values 0,1. Thus $x\in\{0,1)$, similarly $a_1,a_2,\cdots,b_1,\cdots$ etc., will take values in $\{0,1\}$. These numbers will be called bits, or classical bits, or cubits.\\ A sequence of bits such as $$ a_n\cdots\cdots a_1a_0\equiv a $$ will be called multi bit and represents the number $$ N=a_n 2^n+a_{n-1}2^{n-1}+\cdots a_2~ 2+a_0 $$ For example, a 3 bit number 111 is \begin{eqnarray*} 111\mapsto 1\times2^2+1\times2^1+1 &=& 7\\ 11001 \mapsto 1\times2^5+1\times2^3+1 &=& 32+8+1 =41 \end{eqnarray*} A sequence of $n$ classical bits represents numbers in the range 0 to $2^n-1$.\\ For example, using 6 bits, we can represent any integer in the range 0 to $2^6-1=63$.\\ To represent a number $p$, at least $\log_2p$ bits are needed.\\ For example, to represent 1000 as a binary number at least $\log_21000=9.97\rightarrow 10$ bits are required.


2.Boolean algebras 

Several binary operations can be performed on two classical bits.\\ These operations form elements of an algebra known as Boolean algebra  aaaaaaaaa\=aaaaaaaaaaaaaaaaaaaaaa\=aaaaaaaaaaa\kill \>$x+1 =1$ \> $A\cdot\bar{A}=0$\\ \>$x+0 =x$ \> $A+B=B+A$\\ \>$x\cdot1=x$\> $\overline{A+B}=\bar{A}\cdot\bar{B}$\\ \>$x\cdot0=0$\> $\overline{A\cdot B}=\bar{A}+\bar{B}$\\ \>$x+x=x$ \>$x\cdot x=x$\\ \>$x+\bar{x}=1$\> $x\cdot y= x \wedge y$\\ \>NOT $x=\bar{x}$ \> $x+y=x\vee y$\\ \>NOT (NOT $x$)=$x$ \>NOT $\bar{x}$=$x$ \end{tabbing} One should think of bits 1 and 0 as true and false respectively. % $A$ bit is.

3. Basic operations

• {AND:}~ $x$ AND $y$ also written as $x\wedge y$ satisfies $$ x\wedge y = \begin{cases} 1 & \text{if}~ x=y=1\\ 0 & \text{otherwise}\end{cases} $$
• {OR:}~ $x$ OR $y$, also written as $x\vee y$ satisfies $$ x\vee y = \begin{cases}0 & \text{if}~x=y=0\\ 1&\text{otherwise}\end{cases} $$
• {NOT:} NOT $x$, denoted by $\neg x$, is defined by $$\neg x = \begin{cases} 0 & \text{if}~x=1\\ 1 &\text{if}~x=0\end{cases} $$

Question 1  It is correct to say that we can represent the above binary operations on numbers 0 and 1, as arithmetic operations modulo 2. So the question is,``Is it consistent to use the following?''\\ $x\wedge y = x\cdot y$ (multiplication mod 2);\\ $x\vee y = x+y$, (addition modulo 2); and $\neg x = 1-x$.

4.Properties of binary operations 

Distributive property 

$x\wedge(y\vee z) = (x\wedge y)\vee(x\wedge z)$  $x\vee(y\cap z) = (x\vee y)\wedge (x\vee z)$  \it Associative property  $x\cup (y\cup z) = (x\cup y)\cup z$  $x\cap(y\cap z) = (x\cap y)\cap z$ 

Double negation 

$\neg(\neg x) = x$  \it Commutative property}  $x\vee y = y \vee x$  $x\wedge y = y \wedge x$    \QFY{What is being represented in terms of binary numbers? Integers? Rational numbers? Irrational numbers? Any real number?}  Reference: Wikipedia