Editing Binary number (section)

==Counting in binary==
Counting in binary is similar to counting in any other number system. Beginning with a single digit, counting proceeds through each symbol, in increasing order. Before examining binary counting, it is useful to briefly discuss the more familiar [[decimal]] counting system as a frame of reference.

===Decimal counting===
[[Decimal]] counting uses the ten symbols ''0'' through ''9''. Counting begins with the incremental substitution of the least significant digit (rightmost digit) which is often called the ''first digit''. When the available symbols for this position are exhausted, the least significant digit is reset to ''0'', and the next digit of higher significance (one position to the left) is incremented (''overflow''), and incremental substitution of the low-order digit resumes. This method of reset and overflow is repeated for each digit of significance. Counting progresses as follows:

:000, 001, 002, ... 007, 008, 009, (rightmost digit is reset to zero, and the digit to its left is incremented)
:0'''1'''0, 011, 012, ...
:&nbsp;&nbsp;&nbsp;...
:090, 091, 092, ... 097, 098, 099, (rightmost two digits are reset to zeroes, and next digit is incremented)
:'''1'''00, 101, 102, ...

===Binary counting===
[[File:Binary counter.gif|thumb|This counter shows how to count in binary from numbers zero through thirty-one.]]
[[File:Binary_guess_number_trick_SMIL.svg|thumb|upright|link={{filepath:binary_guess_number_trick_SMIL.svg}}|A party trick to guess a number from which cards it is printed on uses the bits of the binary representation of the number. In the SVG file, click a card to toggle it]]
Binary counting follows the exact same procedure, and again the incremental substitution begins with the least significant binary digit, or ''bit'' (the rightmost one, also called the ''first bit''), except that only the two symbols ''0'' and ''1'' are available. Thus, after a bit reaches 1 in binary, an increment resets it to 0 but also causes an increment of the next bit to the left:

:0000,
:000'''1''', (rightmost bit starts over, and the next bit is incremented)
:00'''1'''0, 0011, (rightmost two bits start over, and the next bit is incremented)
:0'''1'''00, 0101, 0110, 0111, (rightmost three bits start over, and the next bit is incremented)
:'''1'''000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 ...

In the binary system, each bit represents an increasing power of 2, with the rightmost bit representing 2<sup>0</sup>, the next representing 2<sup>1</sup>, then 2<sup>2</sup>, and so on. The value of a binary number is the sum of the powers of 2 represented by each "1" bit. For example, the binary number 100101 is converted to decimal form as follows:

:100101<sub>2</sub> = [ ( '''1''' ) × 2<sup>5</sup> ] + [ ( '''0''' ) × 2<sup>4</sup> ] + [ ( '''0''' ) × 2<sup>3</sup> ] + [ ( '''1''' ) × 2<sup>2</sup> ] + [ ( '''0''' ) × 2<sup>1</sup> ] + [ ( '''1''' ) × 2<sup>0</sup> ]

:100101<sub>2</sub> = [ '''1''' × 32 ] + [ '''0''' × 16 ] + [ '''0''' × 8 ] + [ '''1''' × 4 ] + [ '''0''' × 2 ] + [ '''1''' × 1 ]

:'''100101<sub>2</sub> = 37<sub>10</sub>'''