Base 2 Numeral System

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Base 2 System of Numeration

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


To convert any number "[math]n[/math]" (decimal system) to binary system we write [math]n[/math] as sum of powers of [math]2[/math] as


where [math]d\,\,[/math] can take values [math]0\,\,[/math] or [math]1\,\,[/math].

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

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

[math]\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}[/math]

[math]\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}[/math]

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