# Base 2 Numeral System

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Definition
 Base 2 System of Numeration Also known as binary system of numeration in which only two symbols namely $0$ and $1$ are used to express all numbers. The system of numeration most commonly used is is decimal system or base ten in which ten symbols $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ are used.The number these symbols represent are well known.

## Examples

To convert any number "$n$" (decimal system) to binary system we write $n$ as sum of powers of $2$ as

$n=d\cdot2^0+d\cdot2^1+d\cdot2^2+d\cdot2^3+\ldots\;\!$

where $d\,\,$ can take values $0\,\,$ or $1\,\,$.

Decimal Binary Explanation
$0\,$ $0\,$ $0\cdot2^0$
$1\,$ $1\,$ $1\cdot2^0$
$2\,$ $10\,$ $1\cdot2^{1}+0\cdot2^0$
$3\,$ $11\,$ $1\cdot2^{1}+1\cdot2^0$
$4\,$ $100\,$ $1\cdot2^{2}+0\cdot2^{1}+0\cdot2^0$
$5\,$ $101\,$ $1\cdot2^{2}+0\cdot2^{1}+1\cdot2^0$

Start writing the coefficients of powers of 2 from left to right in ascending order.

\begin{align} 7_{10} & = 1+2+4 \\ & = 2^0+2^{1}+2^{2} \\ & = 1\cdot2^0+1\cdot2^{1}+1\cdot2^{2} \\ & = 111_{2} \end{align}

\begin{align} 26_{10} & = 0+2+0+8+16 \\ & = 0+2^1+0+2^{3}+2^{4} \\ & = 0\cdot2^0+1\cdot2^{1}+0\cdot2^{2}+1\cdot2^{3}+1\cdot2^{4} \\ & = 11010_{2} \end{align}