
Binary number
In mathematics and digital electronics, a binary number is a number expressed in the binary numeral system, or base2 numeral system, which represents numeric values using two different symbols: typically 0 (zero) and 1 (one). More specifically, the usual base2 system is a positional notation with a radix of 2. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used internally by almost all modern computers and computerbased devices such as mobile phones. Each digit is referred to as a bit.
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 primarily involves incremental manipulation of the "loworder" digit, or the rightmost digit, often called the "first digit". When the available symbols for the loworder digit are exhausted, the nexthigherorder digit (located one position to the left) is incremented, and counting in the loworder digit starts over at 0. In decimal, counting proceeds like so:
 000, 001, 002, ... 007, 008, 009, (rightmost digit starts over, and next digit is incremented)
 010, 011, 012, ...
 ...
 090, 091, 092, ... 097, 098, 099, (rightmost two digits start over, and next digit is incremented)
 100, 101, 102, ...
After a digit reaches 9, an increment resets it to 0 but also causes an increment of the next digit to the left.
Binary counting
In binary, counting follows similar procedure, except that only the two symbols 0 and 1 are used. Thus, after a digit reaches 1 in binary, an increment resets it to 0 but also causes an increment of the next digit to the left:
 0000,
 0001, (rightmost digit starts over, and next digit is incremented)
 0010, 0011, (rightmost two digits start over, and next digit is incremented)
 0100, 0101, 0110, 0111, (rightmost three digits start over, and the next digit is incremented)
 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111 ...
Since binary is a base2 system, each digit represents an increasing power of 2, with the rightmost digit representing 20, the next representing 21, then 22, and so on. To determine the decimal representation of a binary number simply take the sum of the products of the binary digits and the powers of 2 which they represent. For example, the binary number 100101 is converted to decimal form as follows:
 1001012 = [ ( 1 ) × 25 ] + [ ( 0 ) × 24 ] + [ ( 0 ) × 23 ] + [ ( 1 ) × 22 ] + [ ( 0 ) × 21 ] + [ ( 1 ) × 20 ]
 1001012 = [ 1 × 32 ] + [ 0 × 16 ] + [ 0 × 8 ] + [ 1 × 4 ] + [ 0 × 2 ] + [ 1 × 1 ]
 1001012 = 3710
Addition
The simplest arithmetic operation in binary is addition. Adding two singledigit binary numbers is relatively simple, using a form of carrying:
 0 + 0 → 0
 0 + 1 → 1
 1 + 0 → 1
 1 + 1 → 0, carry 1 (since 1 + 1 = 2 = 0 + (1 × 21) )
Adding two "1" digits produces a digit "0", while 1 will have to be added to the next column. This is similar to what happens in decimal when certain singledigit numbers are added together; if the result equals or exceeds the value of the radix (10), the digit to the left is incremented:
 5 + 5 → 0, carry 1 (since 5 + 5 = 10 = 0 + (1 × 101) )
 7 + 9 → 6, carry 1 (since 7 + 9 = 16 = 6 + (1 × 101) )
This is known as carrying. When the result of an addition exceeds the value of a digit, the procedure is to "carry" the excess amount divided by the radix (that is, 10/10) to the left, adding it to the next positional value. This is correct since the next position has a weight that is higher by a factor equal to the radix. Carrying works the same way in binary:
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