Binary

Numbering Systems

The system we usually use is decimal system, i.e. base 10, where we have 10 digits to describe any number (0,1,2,3,4,5,6,7,8,9). When keeping track of amounts larger than 9, multiple digits are necessary, where each digit represents a power of 10. For example, the number $1067$ can be expanded to make this explicit:

$1067=1\cdot 10^{3}+0\cdot 10^{2}+6\cdot 10^{1}+7\cdot 10^{0}$

Now consider the binary system, i.e. base 2, where we only use 2 digits to describe any value/number (0 and 1). When keeping track of values larger than 1, multiple digits are necessary, where each figit represents a power of 2. For example, the number $10110$ can be expanded to make this explicit:

$10110=1\cdot 2^{4}+0\cdot 2^{3}+1\cdot 2^{2}+1\cdot 2^{1}+0\cdot 2^{0}$

Converting a Decimal Number to a Binary Number

Put example here of dividing by 2, over and over, and keeping track of remainders, etc...

Binary Addition

add stuff here

$0$

	${\begin{aligned}(\left\vert \psi \right\rangle )^{*}&=\left({\begin{array}{c}\alpha \\\beta \end{array}}\right)^{*}\\&=\left({\begin{array}{c}\alpha ^{*}\\\beta ^{*}\end{array}}\right)\\&=\alpha ^{*}\left\vert 0\right\rangle +\beta ^{*}\left\vert 1\right\rangle .\end{aligned}}$	(C.1)